Chapter 9Resource Masters
Geometry
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3
ANSWERS FOR WORKBOOKS The answers for Chapter 9 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-860186-X GeometryChapter 9 Resource Masters
1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03
© Glencoe/McGraw-Hill iii Glencoe Geometry
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix
Lesson 9-1Study Guide and Intervention . . . . . . . . 479–480Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 481Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Reading to Learn Mathematics . . . . . . . . . . 483Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 484
Lesson 9-2Study Guide and Intervention . . . . . . . . 485–486Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 487Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 488Reading to Learn Mathematics . . . . . . . . . . 489Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 490
Lesson 9-3Study Guide and Intervention . . . . . . . . 491–492Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 493Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 494Reading to Learn Mathematics . . . . . . . . . . 495Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 496
Lesson 9-4Study Guide and Intervention . . . . . . . . 497–498Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 499Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 500Reading to Learn Mathematics . . . . . . . . . . 501Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 502
Lesson 9-5Study Guide and Intervention . . . . . . . . 503–504Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 505Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 506Reading to Learn Mathematics . . . . . . . . . . 507Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 508
Lesson 9-6Study Guide and Intervention . . . . . . . . 509–510Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 511Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 512Reading to Learn Mathematics . . . . . . . . . . 513Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 514
Lesson 9-7Study Guide and Intervention . . . . . . . . 515–516Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 517Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 518Reading to Learn Mathematics . . . . . . . . . . 519Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 520
Chapter 9 AssessmentChapter 9 Test, Form 1 . . . . . . . . . . . . 521–522Chapter 9 Test, Form 2A . . . . . . . . . . . 523–524Chapter 9 Test, Form 2B . . . . . . . . . . . 525–526Chapter 9 Test, Form 2C . . . . . . . . . . . 527–528Chapter 9 Test, Form 2D . . . . . . . . . . . 529–530Chapter 9 Test, Form 3 . . . . . . . . . . . . 531–532Chapter 9 Open-Ended Assessment . . . . . . 533Chapter 9 Vocabulary Test/Review . . . . . . . 534Chapter 9 Quizzes 1 & 2 . . . . . . . . . . . . . . . 535Chapter 9 Quizzes 3 & 4 . . . . . . . . . . . . . . . 536Chapter 9 Mid-Chapter Test . . . . . . . . . . . . 537Chapter 9 Cumulative Review . . . . . . . . . . . 538Chapter 9 Standardized Test Practice . 539–540
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32
© Glencoe/McGraw-Hill iv Glencoe Geometry
Teacher’s Guide to Using theChapter 9 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 9 Resource Masters includes the core materials neededfor Chapter 9. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 9-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.
Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.
WHEN TO USE Give these pages tostudents before beginning Lesson 9-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.
Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
© Glencoe/McGraw-Hill v Glencoe Geometry
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
Assessment OptionsThe assessment masters in the Chapter 9Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 518–519. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
79
© Glencoe/McGraw-Hill vii Glencoe Geometry
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 9.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
component form
dilation
isometry
line of reflection
line of symmetry
point of symmetry
reflection
regular tessellation
resultant
rotation
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Geometry
Vocabulary Term Found on Page Definition/Description/Example
rotational symmetry
scalar
semi-regular tessellation
similarity transformation
standard position
tessellation
transformation
translation
uniform tessellations
vector
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
99
Learning to Read MathematicsProof Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
99
© Glencoe/McGraw-Hill ix Glencoe Geometry
Proo
f Bu
ilderThis is a list of key theorems and postulates you will learn in Chapter 9. As you
study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 9.1
Theorem 9.2
Postulate 9.1
Study Guide and InterventionReflections
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
© Glencoe/McGraw-Hill 479 Glencoe Geometry
Less
on
9-1
Draw Reflections The transformation called a reflection is a flip of a figure in a point, a line, or a plane. The new figure is the image and the original figure is thepreimage. The preimage and image are congruent, so a reflection is a congruencetransformation or isometry.
Construct the imageof quadrilateral ABCD under areflection in line m .
Draw a perpendicular from each vertexof the quadrilateral to m. Find verticesA�, B�, C�, and D� that are the samedistance from m on the other side of m.The image is A�B�C�D�.
m
C AD
B
A�
B�
C� D�
Quadrilateral DEFG hasvertices D(�2, 3), E(4, 4), F(3, �2), and G(�3, �1). Find the image under reflectionin the x-axis.
To find an image for areflection in the x-axis,use the same x-coordinateand multiply the y-coordinate by �1. Insymbols, (a, b) → (a, �b).The new coordinates areD�(�2, �3), E�(4, �4),F�(3, 2), and G�(�3, 1).The image is D�E�F �G�.
x
y
O
D�E�
F�G�
DE
FG
Example 1Example 1 Example 2Example 2
In Example 2, the notation (a, b) → (a, �b) represents a reflection in the x-axis. Here arethree other common reflections in the coordinate plane.• in the y-axis: (a, b) → (�a, b)• in the line y � x: (a, b) → (b, a)• in the origin: (a, b) → (�a, �b)
Draw the image of each figure under a reflection in line m .
1. 2. 3.
Graph each figure and its image under the given reflection.
4. �DEF with D(�2, �1), E(�1, 3), 5. ABCD with A(1, 4), B(3, 2), C(2, �2),F(3, �1) in the x-axis D(�3, 1) in the y-axis
x
y
OD�
C�
B�
A�
D
C
B
A
x
y
O
F�D�
E�
FD
E
m
Q
R S
TUm
LM
N
OP
m
HJ
K
ExercisesExercises
© Glencoe/McGraw-Hill 480 Glencoe Geometry
Lines and Points of Symmetry If a figure has a line of symmetry, then it can befolded along that line so that the two halves match. If a figure has a point of symmetry, itis the midpoint of all segments between the preimage and image points.
Determine how many lines of symmetry a regular hexagon has. Then determine whether a regular hexagon has point symmetry.There are six lines of symmetry, three that are diagonals through opposite vertices and three that are perpendicular bisectors of opposite sides. The hexagon has point symmetrybecause any line through P identifies two points on the hexagon that can be considered images of each other.
Determine how many lines of symmetry each figure has. Then determine whetherthe figure has point symmetry.
1. 2. 3.
4. 5. 6.
7. 8. 9.
A B
C
DE
FP
Study Guide and Intervention (continued)
Reflections
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
ExampleExample
ExercisesExercises
Skills PracticeReflections
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
© Glencoe/McGraw-Hill 481 Glencoe Geometry
Less
on
9-1
Draw the image of each figure under a reflection in line �.
1. 2.
COORDINATE GEOMETRY Graph each figure and its image under the givenreflection.
3. �ABC with vertices A(�3, 2), B(0, 1), 4. trapezoid DEFG with vertices D(0, �3),and C(�2, �3) in the origin E(1, 3), F(3, 3), and G(4, �3) in the y-axis
5. parallelogram RSTU with vertices 6. square KLMN with vertices K(�1, 0),R(�2, 3), S(2, 4), T(2, �3) and L(�2, 3), M(1, 4), and N(2, 1) in U(�2, �4) in the line y � x the x-axis
Determine how many lines of symmetry each figure has. Then determine whetherthe figure has point symmetry.
7. 8. 9.
M�
K�K
x
y
O
L�
N�
LM
Nx
y
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R�
S�T�
U�
RS
TU
x
y
O
D
E F
GD�
E�F�
G�
x
y
O
A�B�
C�A
B
C
�
�
© Glencoe/McGraw-Hill 482 Glencoe Geometry
Draw the image of each figure under a reflection in line �.
1. 2.
COORDINATE GEOMETRY Graph each figure and its image under the givenreflection.
3. quadrilateral ABCD with vertices 4. �FGH with vertices F(�3, �1), G(0, 4),A(�3, 3), B(1, 4), C(4, 0), and and H(3, �1) in the line y � xD(�3, �3) in the origin
5. rectangle QRST with vertices Q(�3, 2), 6. trapezoid HIJK with vertices H(�2, 5),R(�1, 4), S(2, 1), and T(0, �1) I(2, 5), J(�4, �1), and K(�4, 3) in the x-axis in the y-axis
ROAD SIGNS Determine how many lines of symmetry each sign has. Thendetermine whether the sign has point symmetry.
7. 8. 9.
H I
J
K
H�I�
J�
K�
x
y
OQ�
R�
S�
T�Q
R
S
Tx
y
O
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G�
H�
F
G
Hx
y
O
A�B�
C�
D�AB
C
D
x
y
O
��
Practice Reflections
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
Reading to Learn MathematicsReflections
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
© Glencoe/McGraw-Hill 483 Glencoe Geometry
Less
on
9-1
Pre-Activity Where are reflections found in nature?
Read the introduction to Lesson 9-1 at the top of page 463 in your textbook.
Suppose you draw a line segment connecting a point at the peak of a mountainto its image in the lake. Where will the midpoint of this segment fall?
Reading the Lesson1. Draw the reflected image for each reflection described below.
a. reflection of trapezoid ABCD in the line n b. reflection of �RST in point PLabel the image of ABCD as A�B�C�D�. Label the image of RST as R�S�T�.
c. reflection of pentagon ABCDE in the originLabel the image of ABCDE as A�B�C�D�E�.
2. Determine the image of the given point under the indicated reflection.a. (4, 6); reflection in the y-axisb. (�3, 5); reflection in the x-axisc. (�8, �2); reflection in the line y � xd. (9, �3); reflection in the origin
3. Determine the number of lines of symmetry for each figure described below. Thendetermine whether the figure has point symmetry and indicate this by writing yes or no.a. a square b. an isosceles triangle (not equilateral) c. a regular hexagon d. an isosceles trapezoid e. a rectangle (not a square) f. the letter E
Helping You Remember4. A good way to remember a new geometric term is to relate the word or its parts to
geometric terms you already know. Look up the origins of the two parts of the wordisometry in your dictionary. Explain the meaning of each part and give a term youalready know that shares the origin of that part.
A�B�
C� D�E�
AB
CDE
x
y
O
R�S�
T�
RS P
Tn
A�B�
C�
D�A
BCD
© Glencoe/McGraw-Hill 484 Glencoe Geometry
Reflections in the Coordinate Plane
Study the diagram at the right. It shows how the triangle ABC is mapped onto triangle XYZ by thetransformation (x, y) → (�x � 6, y). Notice that �XYZis the reflection image with respect to the vertical line with equation x � 3.
1. Prove that the vertical line with equation x � 3 is theperpendicular bisector of the segment with endpoints (x, y) and (�x � 6, y). (Hint: Use the midpoint formula.)
2. Every transformation of the form (x, y) → (�x � 2h, y) is a reflection with respect to the vertical line with equation x � h. What kind of transformation is (x, y) → ( x, �y � 2 k)?
Draw the transformation image for each figure and the given transformation. Is it a reflection transformation? If so, with respect to what line?
3. (x, y) → (�x � 4, y) 4. (x, y) → (x, �y � 8)
x
y
Ox
y
O
x � 3
A
B
C
X
Y
Z
(x, y) → (�x � 6, y)
x
y
O
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
Study Guide and InterventionTranslations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
© Glencoe/McGraw-Hill 485 Glencoe Geometry
Less
on
9-2
Translations Using Coordinates A transformation called a translation slides afigure in a given direction. In the coordinate plane, a translation moves every preimagepoint P(x, y) to an image point P(x � a, y � b) for fixed values a and b. In words, atranslation shifts a figure a units horizontally and b units vertically; in symbols,(x, y) → (x � a, y � b).
Rectangle RECT has vertices R(�2, �1),E(�2, 2), C(3, 2), and T(3, �1). Graph RECT and its image for the translation (x, y) → (x � 2, y � 1).The translation moves every point of the preimage right 2 units and down 1 unit.(x, y) → (x � 2, y � 1)R(�2, �1) → R�(�2 � 2, �1 � 1) or R�(0, �2)E(�2, 2) → E�(�2 � 2, 2 � 1) or E�(0, 1)C(3, 2) → C�(3 � 2, 2 � 1) or C�(5, 1)T(3, �1) → T�(3 � 2, �1 � 1) or T�(5, �2)
Graph each figure and its image under the given translation.
1. P�Q� with endpoints P(�1, 3) and Q(2, 2) under the translation left 2 units and up 1 unit
2. �PQR with vertices P(�2, �4), Q(�1, 2), and R(2, 1) under the translation right 2 units and down 2 units
3. square SQUR with vertices S(0, 2), Q(3, 1), U(2, �2), and R(�1, �1) under the translation right 3 units and up 1 unit
x
y
R�
S�Q�
U�R
SQ
U
x
y
P�
Q�
R�
P
QR
x
yP�Q�
PQ
x
y
O
E�
R�
C�
T�
E
R
C
T
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 486 Glencoe Geometry
Translations by Repeated Reflections Another way to find the image of atranslation is to reflect the figure twice in parallel lines. This kind of translation is called acomposite of reflections.
In the figure, m || n . Find the translation image of �ABC.�A�B�C� is the image of �ABC reflected in line m .�A�B�C� is the image of �A�B�C� reflected in line n .The final image, �A�B�C�, is a translation of �ABC.
In each figure, m || n . Find the translation image of each figure by reflecting it inline m and then in line n .
1. 2.
3. 4.
5. 6. m
U
A
D
Q Q �
U �
A�
D �
U�
A�
D�
n
m
n
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T �
S �R �
m
n
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AQ
U
D
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D �
U �T�
m n
P�P P �
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T T �
A A�
m n
A�
AB
C
A�
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C �
m n
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C�
A
B
C
A�
B�
C �
m n
A�
B�
C�
A
B
C
A�
B�
C �
Study Guide and Intervention (continued)
Translations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
ExercisesExercises
ExampleExample
Skills PracticeTranslations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
© Glencoe/McGraw-Hill 487 Glencoe Geometry
Less
on
9-2
In each figure, a || b. Determine whether figure 3 is a translation image of figure 1.Write yes or no. Explain your answer.
1. 2.
3. 4.
COORDINATE GEOMETRY Graph each figure and its image under the giventranslation.
5. �JKL with vertices J(�4, �4), 6. quadrilateral LMNP with vertices L(4, 2),K(�2, �1), and L(2, �4) under the M(4, �1), N(0, �1), and P(1, 4) under the translation (x, y) → (x � 2, y � 5) translation (x, y) → (x � 4, y � 3)
M�
L
MN
P
L�
N�
P�
x
y
OJ�
K�
L�
J
K
L
x
y
O
a b
12
3
a b1
23
a
b
1
2
3a b
1 2 3
© Glencoe/McGraw-Hill 488 Glencoe Geometry
In each figure, c || d . Determine whether figure 3 is a translation image of figure 1.Write yes or no. Explain your answer.
1. 2.
COORDINATE GEOMETRY Graph each figure and its image under the giventranslation.
3. quadrilateral TUWX with vertices 4. pentagon DEFGH with vertices D(�1, �2),T(�1, 1), U(4, 2), W(1, 5), and X(�1, 3) E(2, �1), F(5, �2), G(4, �4), H(1, �4) under the translation under the translation (x, y) → (x � 2, y � 4) (x, y) → (x � 1, y � 5)
ANIMATION Find the translation that moves the figure on the coordinate plane.
5. figure 1 → figure 2
6. figure 2 → figure 3
7. figure 3 → figure 4
x
y
O
1
3
4
2
D�E�
F�
G�H�
DE
F
GH
x
y
O
W
T�U�
W�
X�
T U
X
x
y
O
c
d
1
2
3
c d
12
3
Practice Translations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
Reading to Learn MathematicsTranslations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
© Glencoe/McGraw-Hill 489 Glencoe Geometry
Less
on
9-2
Pre-Activity How are translations used in a marching band show?
Read the introduction to Lesson 9-2 at the top of page 470 in your textbook.
How do band directors get the marching band to maintain the shape of thefigure they originally formed?
Reading the Lesson1. Underline the correct word or phrase to form a true statement.
a. All reflections and translations are (opposites/isometries/equivalent).b. The preimage and image of a figure under a reflection in a line have
(the same orientation/opposite orientations).c. The preimage and image of a figure under a translation have
(the same orientation/opposite orientations).d. The result of successive reflections over two parallel lines is a
(reflection/rotation/translation).e. Collinearity (is/is not) preserved by translations.f. The translation (x, y) → (x � a, y � b) shifts every point a units
(horizontally/vertically) and y units (horizontally/vertically).
2. Find the image of each preimage under the indicated translation.a. (x, y); 5 units right and 3 units upb. (x, y); 2 units left and 4 units downc. (x, y); 1 unit left and 6 units upd. (x, y); 7 units righte. (4, �3); 3 units upf. (�5, 6); 3 units right and 2 units downg. (�7, 5); 7 units right and 5 units downh. (�9, �2); 12 units right and 6 units down
3. �RST has vertices R(�3, 3), S(0, �2), and T(2, 1). Graph �RSTand its image �R�S�T� under the translation (x, y) → (x � 3, y � 2).List the coordinates of the vertices of the image.
Helping You Remember4. A good way to remember a new mathematical term is to relate it to an everyday meaning
of the same word. How is the meaning of translation in geometry related to the idea oftranslation from one language to another?
R�
S�
T�
R
S
T
x
y
O
© Glencoe/McGraw-Hill 490 Glencoe Geometry
Translations in The Coordinate Plane
You can use algebraic descriptions of reflections to show thatthe composite of two reflections with respect to parallel lines isa translation (that is, a slide).
1. Suppose a and b are two different real numbers. Let S and T be thefollowing reflections.
S: (x, y) → (�x � 2 a, y)T: (x, y) → (�x � 2 b, y)
S is reflection with respect to the line with equation x � a, and T isreflection with respect to the line with equation x � b.
a. Find an algebraic description (similar to those above for S andT) to describe the composite transformation “S followed by T.”
b. Find an algebraic description for the composite transformation“T followed by S.”
2. Think about the results you obtained in Exercise 1. What do theytell you about how the distance between two parallel lines isrelated to the distance between a preimage and image point for acomposite of reflections with respect to these lines?
3. Illustrate your answers to Exercises 1 and 2 with sketches. Use aseparate sheet if necessary.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
Study Guide and InterventionRotations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
© Glencoe/McGraw-Hill 491 Glencoe Geometry
Less
on
9-3
Draw Rotations A transformation called a rotation turns a figure through a specifiedangle about a fixed point called the center of rotation. To find the image of a rotation, oneway is to use a protractor. Another way is to reflect a figure twice, in two intersecting lines.
�ABC has vertices A(2, 1), B(3, 4), and C(5, 1). Draw the image of �ABC under a rotation of 90°counterclockwise about the origin.• First draw �ABC. Then draw a segment from O, the origin,
to point A.• Use a protractor to measure 90° counterclockwise with O�A�
as one side.• Draw OR���.• Use a compass to copy O�A� onto OR���. Name the segment O�A���.• Repeat with segments from the origin to points B and C.
Find the image of �ABC under reflection in lines m and n .First reflect �ABC in line m. Label the image �A�B�C�.Reflect �A�B�C� in line n. Label the image �A�B�C�.�A�B�C� is a rotation of �ABC. The center of rotation is theintersection of lines m and n. The angle of rotation is twice themeasure of the acute angle formed by m and n.
Draw the rotation image of each figure 90° in the given direction about the centerpoint and label the coordinates.
1. P�Q� with endpoints P(�1, �2) 2. �PQR with vertices P(�2, �3), Q(2, �1),and Q(1, 3) counterclockwise and R(3, 2) clockwise about the point T(1, 1)about the origin
Find the rotation image of each figure by reflecting it in line m and then in line n.
3. 4. mn
A B
C
m nQ
P
x
yP�
Q�
R�
P
Q
RT
x
y
P�
Q�
P
Q
m
n
A�
B�C�
A�
B�C�
B
C
A
x
y
O
A�B�
C�
A
B
C
R
ExercisesExercises
Example 1Example 1
Example 2Example 2
© Glencoe/McGraw-Hill 492 Glencoe Geometry
Rotational Symmetry When the figure at the right is rotated about point P by 120° or 240°, the image looks like the preimage. The figure has rotational symmetry, which means it can be rotated less than 360° about a point and the preimage and image appear to be the same.
The figure has rotational symmetry of order 3 because there are 3 rotations less than 360° (0°, 120°, 240°) that produce an image that is the same as the original. The magnitude of the rotational symmetry for a figure is 360 degrees divided by the order. For the figure above,the rotational symmetry has magnitude 120 degrees.
Identify the order and magnitude of the rotational symmetry of the design at the right.
The design has rotational symmetry about the center point for rotations of 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°.
There are eight rotations less than 360 degrees, so the order of its rotational symmetry is 8. The quotient 360 � 8 is 45, so the magnitude of its rotational symmetry is 45 degrees.
Identify the order and magnitude of the rotational symmetry of each figure.
1. a square 2. a regular 40-gon
3. 4.
5. 6.
P
Study Guide and Intervention (continued)
Rotations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
ExercisesExercises
ExampleExample
Skills PracticeRotations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
© Glencoe/McGraw-Hill 493 Glencoe Geometry
Less
on
9-3
Rotate each figure about point R under the given angle of rotation and the givendirection. Label the vertices of the rotation image.
1. 90° counterclockwise 2. 90° clockwise
COORDINATE GEOMETRY Draw the rotation image of each figure 90° in the givendirection about the origin and label the coordinates.
3. �STW with vertices S(2, �1), T(5, 1), 4. �DEF with vertices D(�4, 3), E(1, 2),and W(3, 3) counterclockwise and F(�3, �3) clockwise
Use a composition of reflections to find the rotation image with respect to lines kand m. Then find the angle of rotation for each image.
5. 6. k
mL�
M�
N�
O�
LM
NO
k
mA�B�
C�
A
BC
E�
D�F�
F
ED
x
y
OS�
T�
W�
S
T
W
x
y
O
G H
K J R
G�
H�
J�
K�Q
P S
R
Q�
S�
P�
© Glencoe/McGraw-Hill 494 Glencoe Geometry
Rotate each figure about point R under the given angle of rotation and the givendirection. Label the vertices of the rotation image.
1. 80° counterclockwise 2. 100° clockwise
COORDINATE GEOMETRY Draw the rotation image of each figure 90° in the givendirection about the center point and label the coordinates.
3. �RST with vertices R(�3, 3), S(2, 4), 4. �HJK with vertices H(3, 1), J(3, �3),and T(1, 2) clockwise about the and K(�3, �4) counterclockwise about point P(1, 0) the point P(�1, �1)
Use a composition of reflections to find the rotation image with respect to lines pand s. Then find the angle of rotation for each image.
5. 6.
7. STEAMBOATS A paddle wheel on a steamboat is driven by a steam engine and movesfrom one paddle to the next to propel the boat through the water. If a paddle wheelconsists of 18 evenly spaced paddles, identify the order and magnitude of its rotationalsymmetry.
p s
P�
R�
S�
T�
P
R
S
T
ps
E�
F�
G�
E
F
G
K�
J�H�
J
H
K
x
y
OP(–1, –1)
S�
R�
T�T
S
R
x
y
O P(1, 0)
QU
T S
P
R
Q�
S�
T� U�
P�
N
M P
R
M�
N�
P�
Practice Rotations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
Reading to Learn MathematicsRotations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
© Glencoe/McGraw-Hill 495 Glencoe Geometry
Less
on
9-3
Pre-Activity How do some amusement rides illustrate rotations?
Read the introduction to Lesson 9-3 at the top of page 476 in your textbook.
What are two ways that each car rotates?
Reading the Lesson
1. List all of the following types of transformations that satisfy each description: reflection,translation, rotation.a. The transformation is an isometry.b. The transformation preserves the orientation of a figure.c. The transformation is the composite of successive reflections over two intersecting
lines.d. The transformation is the composite of successive reflections over two parallel
lines.e. A specific transformation is defined by a fixed point and a specified angle.f. A specific transformation is defined by a fixed point, a fixed line, or a fixed plane.
g. A specific transformation is defined by (x, y) → (x � a, x � b), for fixed values of a and b.
h. The transformation is also called a slide.i. The transformation is also called a flip.j. The transformation is also called a turn.
2. Determine the order and magnitude of the rotational symmetry for each figure.
a. b.
c. d.
Helping You Remember
3. What is an easy way to remember the order and magnitude of the rotational symmetryof a regular polygon?
© Glencoe/McGraw-Hill 496 Glencoe Geometry
Finding the Center of RotationSuppose you are told that � X�Y�Z� is the rotation image of � XYZ, but you are not toldwhere the center of rotation is nor the measure of the angle of rotation. Can you find them? Yes,you can. Connect two pairs of correspondingvertices with segments. In the figure, the segments YY� and ZZ� are used. Draw theperpendicular bisectors, � and m, of thesesegments. The point C where � and m intersect is the center of rotation.
1. How can you find the measure of the angle of rotation in the figure above?
Locate the center of rotation for the rotation that maps WXYZ onto W�X�Y�Z�.Then find the measure of the angle of rotation.
2. 3.
Z
Z�
W
W�
X�
X
Y
Y�
center ofrotation
angle ofrotation:about 110
Z�
Z
W
W�X�
X
Y�
Y
center ofrotation
angle ofrotation:about 100
Z�
Z
C
X�X
Y�
Y
m�
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
Study Guide and InterventionTessellations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
© Glencoe/McGraw-Hill 497 Glencoe Geometry
Less
on
9-4
Regular Tessellations A pattern that covers a plane with repeating copies of one ormore figures so that there are no overlapping or empty spaces is a tessellation. A regulartessellation uses only one type of regular polygon. In a tessellation, the sum of the measuresof the angles of the polygons surrounding a vertex is 360. If a regular polygon has an interiorangle that is a factor of 360, then the polygon will tessellate.
regular tessellation tessellation Copies of a regular hexagon Copies of a regular pentagon can form a tessellation. cannot form a tessellation.
Determine whether a regular 16-gon tessellates the plane. Explain.If m�1 is the measure of one interior angle of a regular polygon, then a formula for m�1
is m�1 � �180(n
n� 2)�. Use the formula with n � 16.
m�1 � �180(n
n� 2)�
� �180(1
166
� 2)�
� 157.5
The value 157.5 is not a factor of 360, so the 16-gon will not tessellate.
Determine whether each polygon tessellates the plane. If so, draw a sample figure.
1. scalene right triangle 2. isosceles trapezoid
Determine whether each regular polygon tessellates the plane. Explain.
3. square 4. 20-gon
5. septagon 6. 15-gon
7. octagon 8. pentagon
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 498 Glencoe Geometry
Tessellations with Specific Attributes A tessellation pattern can contain any typeof polygon. If the arrangement of shapes and angles at each vertex in the tessellation is thesame, the tessellation is uniform. A semi-regular tessellation is a uniform tessellationthat contains two or more regular polygons.
Determine whether a kite will tessellate the plane. If so, describe the tessellation as uniform, regular, semi-regular, or not uniform.A kite will tessellate the plane. At each vertex the sum of the measures is a � b � b � c, which is 360. The tessellation is uniform.
Determine whether a semi-regular tessellation can be created from each set offigures. If so, sketch the tessellation. Assume that each figure has a side length of1 unit.
1. rhombus, equilateral triangle, 2. square and equilateral triangleand octagon
Determine whether each polygon tessellates the plane. If so, describe thetessellation as uniform, not uniform, regular, or semi-regular.
3. rectangle 4. hexagon and square
c� a�
b�
b�
c�
c�
a�
a�b�
b�
b�
c�a�
b�
b�
c�a�
b�
b�
c�a�
b�
b�
b�
c� a�
b�
b�
Study Guide and Intervention (continued)
Tessellations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
ExercisesExercises
ExampleExample
Skills PracticeTessellations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
© Glencoe/McGraw-Hill 499 Glencoe Geometry
Less
on
9-4
Determine whether each regular polygon tessellates the plane. Explain.
1. 15-gon 2. 18-gon
3. square 4. 20-gon
Determine whether a semi-regular tessellation can be created from each set offigures. Assume each figure has a side length of 1 unit.
5. regular pentagons and equilateral triangles
6. regular dodecagons and equilateral triangles
7. regular octagons and equilateral triangles
Determine whether each polygon tessellates the plane. If so, describe thetessellation as uniform, not uniform, regular, or semi-regular.
8. rhombus 9. isosceles trapezoid and square
Determine whether each pattern is a tessellation. If so, describe it as uniform, notuniform, regular, or semi-regular.
10. 11.
© Glencoe/McGraw-Hill 500 Glencoe Geometry
Determine whether each regular polygon tessellates the plane. Explain.
1. 22-gon 2. 40-gon
Determine whether a semi-regular tessellation can be created from each set offigures. Assume each figure has a side length of 1 unit.
3. regular pentagons and regular decagons
4. regular dodecagons, regular hexagons, and squares
Determine whether each polygon tessellates the plane. If so, describe thetessellation as uniform, not uniform, regular, or semi-regular.
5. kite 6. octagon and decagon
Determine whether each pattern is a tessellation. If so, describe it as uniform, notuniform, regular, or semi-regular.
7. 8.
FLOOR TILES For Exercises 9 and 10, use the following information.Mr. Martinez chose the pattern of tile shown to retile his kitchen floor.
9. Determine whether the pattern is a tessellation. Explain
10. Is the pattern uniform, regular, or semi-regular?
Practice Tessellations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
Reading to Learn MathematicsTesselations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
© Glencoe/McGraw-Hill 501 Glencoe Geometry
Less
on
9-4
Pre-Activity How are tessellations used in art?
Read the introduction to Lesson 9-4 at the top of page 483 in your textbook.
• In the pattern shown in the picture in your textbook, how many smallequilateral triangles make up one regular hexagon?
• In this pattern, how many fish make up one equilateral triangle?
Reading the Lesson
1. Underline the correct word, phrase, or number to form a true statement.
a. A tessellation is a pattern that covers a plane with the same figure or set of figures sothat there are no (congruent angles/overlapping or empty spaces/right angles).
b. A tessellation that uses only one type of regular polygon is called a(uniform/regular/semi-regular) tessellation.
c. The sum of the measures of the angles at any vertex in any tessellation is(90/180/360).
d. A tessellation that contains the same arrangement of shapes and angles at everyvertex is called a (uniform/regular/semi-regular) tessellation.
e. In a regular tessellation made up of hexagons, there are (3/4/6) hexagons meeting ateach vertex, and the measure of each of the angles at any vertex is (60/90/120).
f. A uniform tessellation formed using two or more regular polygons is called a(rotational/regular/semi-regular) tessellation.
g. In a regular tessellation made up of triangles, there are (3/4/6) triangles meeting ateach vertex, and the measure of each of the angles at any vertex is (30/60/120).
h. If a regular tessellation is made up of quadrilaterals, all of the quadrilaterals must becongruent (rectangles/parallelograms/squares/trapezoids).
2. Write all of the following words that describe each tessellation: uniform, non-uniform,regular, semi-regular.
a. b.
Helping You Remember
3. Often the everyday meanings of a word can help you to remember its mathematicalmeaning. Look up uniform in your dictionary. How can its everyday meanings help youto remember the meaning of a uniform tessellation?
© Glencoe/McGraw-Hill 502 Glencoe Geometry
Polygonal NumbersCertain numbers related to regular polygons are called polygonal numbers. The chartshows several triangular, square, and pentagonal numbers. The rank of a polygon numberis the number of dots on each “side” of the outer polygon. For example, the pentagonalnumber 22 has a rank of 4.
Polygonal numbers can be described with formulas. For example, a triangular number T
of rank r can be described by T � �r(r
2� 1)�.
Answer each question.
1. Draw a diagram to find the 2. Draw a diagram to find the triangular number of rank 5. pentagonal number of rank 5.
3. Write a formula for a square number 4. Write a formula for a pentagonalS of rank r. number P of rank r.
5. What is the rank of the pentagonal 6. List the hexagonal numbers for ranksnumber 70? 1 to 5. (Hint: Draw a diagram.)
Rank 1
Triangle
1 3 6 10
1 4 9 16
1 5 12 22
Square
Pentagon
Rank 2 Rank 3 Rank 4
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
Study Guide and InterventionDilations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
© Glencoe/McGraw-Hill 503 Glencoe Geometry
Less
on
9-5
Classify Dilations A dilation is a transformation in which the image may be adifferent size than the preimage. A dilation requires a center point and a scale factor, r.
Let r represent the scale factor of a dilation.
If | r | � 1, then the dilation is an enlargement.
If | r | � 1, then the dilation is a congruence transformation.
If 0 | r | 1, then the dilation is a reduction.
Draw the dilation image of �ABC with center O and r � 2.Draw OA���, OB���, and OC���. Label points A�, B�, and C�so that OA� � 2(OA), OB� � 2(OB), and OC� � 2(OC). �A�B�C� is a dilation of �ABC.
Draw the dilation image of each figure with center C and the given scale factor.Describe each transformation as an enlargement, congruence, or reduction.
1. r � 2 2. r � �12�
3. r � 1 4. r � 3
5. r � �23� 6. r � 1
K J
HF
C
G
M
C
P
R
V
W
M�
P�R�
V�
W�
CS
T S�
T�
C
W
Y
XZ
C
S T
R S� T�
R�C
B
A
A�
B�
O
B
A
CO
B
A
C
A�
B�
C�
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 504 Glencoe Geometry
Identify the Scale Factor If you know corresponding measurements for a preimageand its dilation image, you can find the scale factor.
Determine the scale factor for the dilation of X�Y� to A�B�. Determine whether the dilation is an enlargement,reduction, or congruence transformation.
scale factor � �priemimag
aege
lelnegntghth�
� �84
uu
nn
iittss�
� 2The scale factor is greater than 1, so the dilation is an enlargement.
Determine the scale factor for each dilation with center C. Determine whether thedilation is an enlargement, reduction, or congruence transformation.
1. CGHJ is a dilation image of CDEF. 2. �CKL is a dilation image of �CKL.
3. STUVWX is a dilation image 4. �CPQ is a dilation image of �CYZ.of MNOPQR.
5. �EFG is a dilation image of �ABC. 6. �HJK is a dilation image of �HJK.
C
J
K
H
CB
DG
F
EA
C
Z
Y
P
QC
M N
O
PQ
RUX
S T
VW
C
K L
G
C
D
H
F
E
J
A
B
X
Y
C
Study Guide and Intervention (continued)
Dilations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
ExampleExample
ExercisesExercises
Skills PracticeDilations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
© Glencoe/McGraw-Hill 505 Glencoe Geometry
Less
on
9-5
Draw the dilation image of each figure with center C and the given scale factor.
1. r � 2 2. r � �14�
Find the measure of the dilation image M���N��� or of the preimage M�N� using thegiven scale factor.
3. MN � 3, r � 3 4. M�N� � 7, r � 21
COORDINATE GEOMETRY Find the image of each polygon, given the vertices, aftera dilation centered at the origin with a scale factor of 2. Then graph a dilation centered at the origin with a scale factor of �
12�.
5. J(2, 4), K(4, 4), P(3, 2) 6. D(�2, 0), G(0, 2), F(2, �2)
Determine the scale factor for each dilation with center C. Determine whether thedilation is an enlargement, reduction, or congruence transformation. The dashedfigure is the dilation image.
7. 8.
CC
G�
F�
D�
G�
F �
D �x
y
O
J� K�
P�J � K �
P �x
y
O
CC
© Glencoe/McGraw-Hill 506 Glencoe Geometry
Draw the dilation image of each figure with center C and the given scale factor.
1. r � �32� 2. r � �
23�
Find the measure of the dilation image A���T��� or of the preimage A�T� using the givenscale factor.
3. AT � 15, r � �35� 4. AT � 30, r � ��
16� 5. A�T� � 12, r � �
43�
COORDINATE GEOMETRY Find the image of each polygon, given the vertices, aftera dilation centered at the origin with a scale factor of 2. Then graph a dilation centered at the origin with a scale factor of �
12�.
6. A(1, 1), C(2, 3), D(4, 2), E(3, 1) 7. Q(�1, �1), R(0, 2), S(2, 1)
Determine the scale factor for each dilation with center C. Determine whether thedilation is an enlargement, reduction, or congruence transformation. The dottedfigure is the dilation image.
8. 9.
10. PHOTOGRAPHY Estebe enlarged a 4-inch by 6-inch photograph by a factor of �52�. What
are the new dimensions of the photograph?
CC
S�
Q�
R�
S �
Q �
R �
x
y
O
C�
D�
E�A�
C �D �
E �A�x
y
O
C
C
Practice Dilations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
Reading to Learn MathematicsDilations
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
© Glencoe/McGraw-Hill 507 Glencoe Geometry
Less
on
9-5
Pre-Activity How do you use dilations when you use a computer?
Read the introduction to Lesson 9-5 at the top of page 490 in your textbook.
In addition to the example given in your textbook, give two everydayexamples of scaling an object, one that makes the object larger and anotherthat makes it smaller.
Reading the Lesson1. Each of the values of r given below represents the scale factor for a dilation. In each
case, determine whether the dilation is an enlargement, a reduction, or a congruencetransformation.a. r � 3 b. r � 0.5c. r � �0.75 d. r � �1
e. r � �23� f. r � ��
32�
g. r � �1.01 h. r � 0.999
2. Determine whether each sentence is always, sometimes, or never true. If the sentence isnot always true, explain why.a. A dilation requires a center point and a scale factor.
b. A dilation changes the size of a figure.
c. A dilation changes the shape of a figure.
d. The image of a figure under a dilation lies on the opposite side of the center from thepreimage.
e. A similarity transformation is a congruence transformation.
f. The center of a dilation is its own image.
g. A dilation is an isometry.
h. The scale factor for a dilation is a positive number.
i. Dilations produce similar figures.
Helping You Remember3. A good way to remember something is to explain it to someone else. Suppose that your
classmate Lydia is having trouble understanding the relationship between similaritytransformations and congruence transformations. How can you explain this to her?
© Glencoe/McGraw-Hill 508 Glencoe Geometry
Similar CirclesYou may be surprised to learn that two noncongruent circles that lie inthe same plane and have no common interior points can be mappedone onto the other by more than one dilation.
1. Here is diagram that suggests one way to map a smaller circle onto a larger one using a dilation. The circles are given. The linessuggest how to find the center for the dilation. Describe how the center is found.Use segments in the diagram to name the scale factor.
2. Here is another pair of noncongruent circles with no commoninterior point. From Exercise 1, you know you can locate a point off to the left of the smaller circle that is the center for a dilationmapping �C onto �C�. Find another center for another dilationthat maps �C onto �C�. Mark and label segments to name thescale factor.
X
X�
C�CO
A �
M�
M
AO
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
Study Guide and InterventionVectors
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
© Glencoe/McGraw-Hill 509 Glencoe Geometry
Less
on
9-6
Magnitude and Direction A vector is a directed segment representing a quantity that has both magnitude, or length,and direction. For example, the speed and direction of an airplane can be represented by a vector. In symbols, a vector is written as AB�, where A is the initial point and B is the endpoint,or as v�.
A vector in standard position has its initial point at (0, 0) and can be represented by the ordered pair for point B. The vector at the right can be expressed as v� � �5, 3�.
You can use the Distance Formula to find the magnitude | AB� | of a vector. You can describe the direction of a vector bymeasuring the angle that the vector forms with the positivex-axis or with any other horizontal line.
Find the magnitude and direction of AB� for A(5, 2) and B(8, 7).Find the magnitude.
|AB | � �(x2 ��x1)2 �� ( y2 �� y1)2�� �(8 � 5�)2 � (�7 � 2�)2�� �34� or about 5.8 units
To find the direction, use the tangent ratio.
tan A � �53� The tangent ratio is opposite over adjacent.
m�A� 59.0 Use a calculator.
The magnitude of the vector is about 5.8 units and its direction is 59°.
Find the magnitude and direction of AB� for the given coordinates. Round to thenearest tenth.
1. A(3, 1), B(�2, 3) 2. A(0, 0), B(�2, 1)
3. A(0, 1), B(3, 5) 4. A(�2, 2), B(3, 1)
5. A(3, 4), B(0, 0) 6. A(4, 2), B(0, 3)
x
y
O
B(8,7)
A(5, 2)
x
y
O
B(5, 3)
A(0, 0)
V�
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 510 Glencoe Geometry
Translations with Vectors Recall that the transformation (a, b) → (a � 2, b � 3)represents a translation right 2 units and down 3 units. The vector �2, �3� is another way todescribe that translation. Also, two vectors can be added: �a, b� � �c, d� � �a � c, b � d�. Thesum of two vectors is called the resultant.
Graph the image of parallelogram RSTUunder the translation by the vectors m� � �3, �1� and n� � ��2, �4�.Find the sum of the vectors.m�� n� � �3, �1� � ��2, �4�
� �3 � 2, �1 � 4�� �1, �5�
Translate each vertex of parallelogram RSTU right 1 unit and down 5 units.
Graph the image of each figure under a translation by the given vector(s).
1. �ABC with vertices A(�1, 2), B(0, 0), 2. ABCD with vertices A(�4, 1), B(�2, 3),and C(2, 3); m�� �2, �3� C(1, 1), and D(�1, �1); n� � �3 �3�
3. ABCD with vertices A(�3, 3), B(1, 3), C(1, 1), and D(�3, 1); the sum of p� � ��2, 1� and q� � �5, �4�
Given m� � �1, �2� and n� � ��3, �4�, represent each of the following as a singlevector.
4. m�� n� 5. n� � m�
x
y
D�
A�B�
C�
D
A B
C
x
y
D�
A�
B�
C�D
A
B
C
x
y
A�
B�
C�
A
B
C
x
y
R�S�
T�
U�
R
S
TU
Study Guide and Intervention (continued)
Vectors
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
ExampleExample
ExercisesExercises
Skills PracticeVectors
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
© Glencoe/McGraw-Hill 511 Glencoe Geometry
Less
on
9-6
Write the component form of each vector.
1. 2.
Find the magnitude and direction of RS� for the given coordinates. Round to thenearest tenth.
3. R(2, �3), S(4, 9) 4. R(0, 2), S(3, 12)
5. R(5, 4), S(�3, 1) 6. R(1, 5), S(�4, �6)
Graph the image of each figure under a translation by the given vector(s).
7. �ABC with vertices A(�4, 3), B(�1, 4), 8. trapezoid with vertices T(�4, �2),C(�1, 1); t� � �4, �3� R(�1, �2), S(�2, �3), Y(�3, �3); a� � �3, 1�
and b� � �2, 4�
Find the magnitude and direction of each resultant for the given vectors.
9. y� � �7, 0�, z� � �0, 6� 10. b� � �3, 2�, c� � ��2, 3�
S�Y�
T� R�
SY
T Rx
y
O
C�
B�A�C
B
A
x
y
O
x
y
O
B(–2, 2)
D(3, –3)
x
y
OC(1, –1)
E(4, 3)
© Glencoe/McGraw-Hill 512 Glencoe Geometry
Write the component form of each vector.
1. 2.
Find the magnitude and direction of FG� for the given coordinates. Round to thenearest tenth.
3. F(�8, �5), G(�2, 7) 4. F(�4, 1), G(5, �6)
Graph the image of each figure under a translation by the given vector(s).
5. �QRT with vertices Q(�1, 1), R(1, 4), 6. trapezoid with vertices J(�4, �1),T(5, 1); s� � ��2, �5� K(0, �1), L(�1, �3), M(�2, �3); c� � �5, 4�
and d� � ��2, 1�
Find the magnitude and direction of each resultant for the given vectors.
7. a� � ��6, 4�, b� � �4, 6� 8. e� � ��4, �5�, f� � ��1, 3�
AVIATION For Exercises 9 and 10, use the following information.A jet begins a flight along a path due north at 300 miles per hour. A wind is blowing due westat 30 miles per hour.
9. Find the resultant velocity of the plane.
10. Find the resultant direction of the plane.
L�M�
J� K�
LM
JK
x
y
O
T�
R�
Q�
T
R
Qx
y
O
x
y
O
K(–2, 4)
L(3, –4)
x
y
O
A(–2, –2)
B(4, 2)
Practice Vectors
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
Reading to Learn MathematicsVectors
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
© Glencoe/McGraw-Hill 513 Glencoe Geometry
Less
on
9-6
Pre-Activity How do vectors help a pilot plan a flight?
Read the introduction to Lesson 9-6 at the top of page 498 in your textbook.
Why do pilots often head their planes in a slightly different direction fromtheir destination?
Reading the Lesson1. Supply the missing words or phrases to complete the following sentences.
a. A is a directed segment representing a quantity that has bothmagnitude and direction.
b. The length of a vector is called its .
c. Two vectors are parallel if and only if they have the same or direction.
d. A vector is in if it is drawn with initial point at the origin.
e. Two vectors are equal if and only if they have the same and the
same .
f. The sum of two vectors is called the .
g. A vector is written in if it is expressed as an ordered pair.
h. The process of multiplying a vector by a constant is called .
2. Write each vector described below in component form.
a. a vector in standard position with endpoint (a, b)
b. a vector with initial point (a, b) and endpoint (c, d)
c. a vector in standard position with endpoint (�3, 5)
d. a vector with initial point (2, �3) and endpoint (6, �8)
e. a� � b� if a� � ��3, 5� and b� � �6, �4�
f. 5u� if u� � �8, �6�
g. ��13�v� if v� � ��15, 24�
h. 0.5u� � 1.5v� if u� � �10, �10� and v� � ��8, 6�
Helping You Remember3. A good way to remember a new mathematical term is to relate it to a term you already
know. You learned about scale factors when you studied similarity and dilations. How is theidea of a scalar related to scale factors?
© Glencoe/McGraw-Hill 514 Glencoe Geometry
Reading MathematicsMany quantities in nature can be thought of as vectors. The science ofphysics involves many vector quantities. In reading about applicationsof mathematics, ask yourself whether the quantities involve onlymagnitude or both magnitude and direction. The first kind of quantityis called scalar. The second kind is a vector.
Classify each of the following. Write scalar or vector.
1. the mass of a book
2. a car traveling north at 55 mph
3. a balloon rising 24 feet per minute
4. the size of a shoe
5. a room temperature of 22 degrees Celsius
6. a west wind of 15 mph
7. the batting average of a baseball player
8. a car traveling 60 mph
9. a rock falling at 10 mph
10. your age
11. the force of Earth’s gravity acting on a moving satellite
12. the area of a record rotating on a turntable
13. the length of a vector in the coordinate plane
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
Study Guide and InterventionTransformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
9-79-7
© Glencoe/McGraw-Hill 515 Glencoe Geometry
Less
on
9-7
Translations and Dilations A vector can be represented by the ordered pair �x, y� or
by the column matrix . When the ordered pairs for all the vertices of a polygon are
placed together, the resulting matrix is called the vertex matrix for the polygon.
For �ABC with A(�2, 2), B(2, 1), and C(�1, �1), the vertex
matrix for the triangle is .
For �ABC above, use a matrix to find the coordinates of thevertices of the image of �ABC under the translation (x, y) → (x � 3, y � 1).To translate the figure 3 units to the right, add 3 to each x-coordinate. To translate thefigure 1 unit down, add �1 to each y-coordinate.
Vertex Matrix Translation Vertex Matrixof �ABC Matrix of �A�B�C�
� �
The coordinates are A�(1, 1), B�(5, 0), and C�(2, �2).
For �ABC above, use a matrix to find the coordinates of the verticesof the image of �ABC for a dilation centered at the origin with scale factor 3.
Scale Vertex Matrix Vertex MatrixFactor of �ABC of �A�B�C�
3 �
The coordinates are A�(�6, 6), B�(6, 3), and C�(�3, �3).
Use a matrix to find the coordinates of the vertices of the image of each figureunder the given translations or dilations.
1. �ABC with A(3, 1), B(�2, 4), C(�2, �1); (x, y) → (x � 1, y � 2)
2. parallelogram RSTU with R(�4, �2), S(�3, 1), T(3, 4), U(2, 1); (x, y) → (x � 4, y � 3)
3. rectangle PQRS with P(4, 0), Q(3, �3), R(�3, �1), S(�2, 2); (x, y) → (x � 2, y � 1)
4. �ABC with A(�2, �1), B(�2, �3), C(2, �1); dilation centered at the origin with scalefactor 2
5. parallelogram RSTU with R(4, �2), S(�4, �1), T(�2, 3), U(6, 2); dilation centered at theorigin with scale factor 1.5
�6 6 �3 6 3 �3
�2 2 �1 2 1 �1
1 5 21 0 �2
3 3 3�1 �1 �1
�2 2 �1 2 1 �1
�2 2 �1 2 1 �1
x
y
B(2, 1)
C(–1, –1)
A(–2, 2)
x y
ExercisesExercises
Example 1Example 1
Example 2Example 2
© Glencoe/McGraw-Hill 516 Glencoe Geometry
Reflections and Rotations When you reflect an image, one way to find thecoordinates of the reflected vertices is to multiply the vertex matrix of the object by areflection matrix. To perform more than one reflection, multiply by one reflection matrixto find the first image. Then multiply by the second matrix to find the final image. Thematrices for reflections in the axes, the origin, and the line y � x are shown below.
For a reflection in the: x-axis y-axis origin line y � x
Multiply the vertex matrix by:
�ABC has vertices A(�2, 3), B(1, 4), and C(3, 0). Use a matrix to find the coordinates of the vertices of the image of �ABC after a reflection in the x-axis.To reflect in the x-axis, multiply the vertex matrix of �ABC by thereflection matrix for the x-axis.
Reflection Matrix Vertex Matrix Vertex Matrix for x-axis of �ABC of �A’B’C’
�
Use a matrix to find the coordinates of the vertices of the image of each figureunder the given reflection.
1. �ABC with A(�3, 2), B(�1, 3), C(1, 0); reflection in the x-axis
2. �XYZ with X(2, �1), Y(4, �3), Z(�2, 1); reflection in the y-axis
3. �ABC with A(3, 4), B(�1, 0), C(�2, 4); reflection in the origin
4. parallelogram RSTU with R(�3, 2), S(3, 2), T(5, �1), U(�1, �1); reflection in the line y � x
5. �ABC with A(2, 3), B(�1, 2), C(1, �1); reflection in the origin, then reflection in the liney � x
6. parallelogram RSTU with R(0, 2), S(4, 2), T(3, �2), U(�1, �2); reflection in the x-axis,then reflection in the y-axis
�2 1 3�3 �4 0
�2 1 3 3 4 0
1 00 �1
x
y
A�B�
AB
C
0 11 0
�1 0 0 �1
�1 0 0 1
1 00 �1
Study Guide and Intervention (continued)
Transformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
9-79-7
ExampleExample
ExercisesExercises
Skills PracticeTransformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
9-79-7
© Glencoe/McGraw-Hill 517 Glencoe Geometry
Less
on
9-7
Use a matrix to find the coordinates of the vertices of the image of each figureunder the given translations.
1. �STU with S(6, 4), T(9, 7), and U(14, 2); (x, y) → (x � 4, y � 3)
2. �GHI with G(�5, 0), H(�3, 6), and I(�2, 1); (x, y) → (x � 2, y � 6)
Use scalar multiplication to find the coordinates of the vertices of each figure fora dilation centered at the origin with the given scale factor.
3. �DEF with D(2, 1), E(5, 4), and F(7, 2); r � 4
4. quadrilateral WXYZ with W(�9, 6), X(�6, 3), Y(3, 12), and Z(�6, 15); r � �13�
Use a matrix to find the coordinates of the vertices of the image of each figureunder the given reflection.
5. �MNO with M(�5, 1), N(�2, 3), and O(2, 0); y-axis
6. quadrilateral ABCD with A(3, 1), B(6, �2), C(5, �5), and D(1, �6); x-axis
Use a matrix to find the coordinates of the vertices of the image of each figureunder the given rotation.
7. �RST with R(�2, �2), S(�3, 3), and T(2, 2); 90°counterclockwise
8. �LMNP with L(3, 4), M(7, 4), N(9, �3), and P(5, �3); 180° counterclockwise
© Glencoe/McGraw-Hill 518 Glencoe Geometry
Use a matrix to find the coordinates of the vertices of the image of each figureunder the given translations.
1. �KLM with K(�7, �3), L(4, 9), and M(9, �6); (x, y) → (x � 7, y � 2)
2. �ABCD with A(�4, 3), B(�2, 8), C(3, 10), and D(1, 5); (x, y) → (x � 3, y � 9)
Use scalar multiplication to find the coordinates of the vertices of each figure fora dilation centered at the origin with the given scale factor.
3. quadrilateral HIJK with H(�2, 3), I(2, 6), J(8, 3), and K(3, �4); r � ��13�
4. pentagon DEFGH with D(�8, �4), E(�8, 2), F(2, 6), G(8, 0), and H(4, �6); r � �54�
Use a matrix to find the coordinates of the vertices of the image of each figureunder the given reflection.
5. �QRS with Q(�5, �4), R(�1, �1), and S(2, �6); x-axis
6. quadrilateral VXYZ with V(�4, �2), X(�3, 4), Y(2, 1), and Z(4, �3); y � x
Use a matrix to find the coordinates of the vertices of the image of each figureunder the given rotation.
7. �EFGH with E(�5 �4), F(�3, �1), G(5, �1), and H(3, �4); 90° counterclockwise
8. quadrilateral PSTU with P(�3, 5), S(2, 6), T(8, 1), and U(�6, �4); 270° counterclockwise
9. FORESTRY A research botanist mapped a section of forested land on a coordinate gridto keep track of endangered plants in the region. The vertices of the map are A(�2, 6),B(9, 8), C(14, 4), and D(1, �1). After a month, the botanist has decided to decrease the
research area to �34� of its original size. If the center for the reduction is O(0, 0),what are
the coordinates of the new research area?
Practice Transformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
9-79-7
Reading to Learn MathematicsTransformations with Matrices
NAME ______________________________________________ DATE ____________ PERIOD _____
9-79-7
© Glencoe/McGraw-Hill 519 Glencoe Geometry
Less
on
9-7
Pre-Activity How can matrices be used to make movies?
Read the introduction to Lesson 9-7 at the top of page 506 in your textbook.
• What kind of transformation should be used to move a polygon?
• What kind of transformation should be used to resize a polygon?
Reading the Lesson1. Write a vertex matrix for each figure.
a. �ABC b. parallelogram ABCD
2. Match each transformation from the first column with the corresponding matrix fromthe second or third column. In each case, the vertex matrix for the preimage of a figureis multiplied on the left by one of the matrices below to obtain the image of the figure.All rotations listed are counterclockwise through the origin. (Some matrices may be usedmore than once or not at all.)
a. reflection over the y-axis i. v.b. 90° rotationc. reflection over the line y � x ii. vi.d. 270° rotatione. reflection over the origin iii. vii.f. 180° rotationg. reflection over the x-axis iv. viii.h. 360° rotation
Helping You Remember3. How can you remember or quickly figure out the matrices for the transformations in
Exercise 2?
0 �11 0
0 �1�1 0
1 00 �1
0 11 0
�1 0 0 1
�1 0 0 �1
0 1�1 0
1 00 1
B C
A D x
y
O
B
C
A
x
y
O
© Glencoe/McGraw-Hill 520 Glencoe Geometry
Vector AdditionVectors are physical quantities with magnitude and direction. Force and velocity are twoexamples. We will investigate adding vector quantities. The sum of two vectors is called aresultant vector or just the resultant.
Two separate forces, one measuring 20 units and the other measuring 40 units, act on an object. If the angle between the forces is 50°, find themagnitude and direction of the resultant force.
First, the vectors must be rearranged by placing the tail of the 20-unit vector at the head of the 40-unit vector. Since these vectors are not perpendicular, the horizontal and verticalcomponents of one of the vectors must be found. Using trigonometry, the horizontal component must be (20 cos 50°) units and the vertical component must be (20 sin 50°) units.Replacing the 20-unit vector with these components, we can now form two vectors perpendicular and use the Pythagorean Theorem to find the resultant.
r2 � (40 � 20 cos 50°)2 � (20 sin 50°)2
r2 � (52.9)2 � (15.3)2
r2 � 3032.5r2 � 55.1
tan O � �402�0
2s0in
c5os0°
50°�
� 0.2898m�O � 16
Therefore, the resultant force is 55.1 units directed 16° from the 40-unit force.
Solve. Round all angle measures to the nearest degree. Round all other measuresto the nearest tenth.
1. A plane flies due west at 250 kilometers per hour while the wind blows south at 70kilometers per hour. Find the plane’s resultant velocity.
2. A plane flies east for 200 km, then 60° south of east for 80 km. Find the plane’s distanceand direction from its starting point.
3. One force of 100 units acts on an object. Another force of 80 units acts on the object at a40° angle from the first force. Find the resultant force on the object.
O
20
40 � 20 cos 50�
20 sin 50�r
20
20 cos 50�
20 sin 50�
50�
O
20 units
40 units50�
O
20 units
40 units50�
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-79-7
ExampleExample
Chapter 9 Test, Form 199
© Glencoe/McGraw-Hill 521 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
1. Given A(3, �7), under which reflection is A�(3, 7)?A. reflection in the x-axis B. reflection in the y-axisC. reflection in the origin D. reflection in y � x
2. Name the image of B�C� under reflection in line m.A. B�C� B. B�A�C. A�C� D. line �
3. How many lines of symmetry does a square have?A. 0 B. 2 C. 4 D. 8
4. Which of the following will result in a translation?A. reflecting in two parallel linesB. reflecting in two intersecting linesC. reflecting in two perpendicular linesD. turning the figure upside down
5. Which transformation moves all points the same distance in the samedirection?A. rotation B. translationC. reflection D. dilation
6. What is the image of X(3, 5) under the translation (x, y) → (x � 4, y � 6)?A. X�(7, �1) B. X�(�1, �1)C. X�(7, 11) D. X�(�1, 11)
7. Find the measure of the angle between two intersecting lines if successivereflections in these lines creates a rotation of 80°.A. 180 B. 160 C. 80 D. 40
8. Find the angle of rotation if the preimage is reflected in perpendicular lines.A. 45° B. 90° C. 180° D. 360°
9. Which figure could tessellate the plane?A. regular pentagon B. regular hexagonC. regular octagon D. regular heptagon
10. Which term describes this tessellation?A. regular B. semi-regularC. not uniform D. uniform
11. What type of dilation occurs with a scale factor of �32�?
A. enlargement B. reductionC. congruence transformation D. inverse transformation
B
CA�
m
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 522 Glencoe Geometry
Chapter 9 Test, Form 1 (continued)99
12.
13.
14.
15.
16.
17.
18.
19.
20.
12. If �A�B�C� is the image of �ABC under a dilation with center at (0, 0). Find the scale factor.
A. 3 B. �23�
C. �13� D. ��
13�
13. Joe’s old graphing calculator had 96 pixels across the screen. His newcalculator has 144 pixels. Find the scale factor by which he increased hisscreen size.
A. �12� B. �
23� C. �
32� D. 48
14. Find the component form of AB� with A(2, 3) and B(�4, 6).A. ��2, 9� B. �2, �9� C. ��6, 3� D. �6, �3�
15. Find the magnitude of AB� with A(3, 4) and B(�1, 7).A. �4, �3� B. 5 C. �13� D. 25
16. Find the direction of AB� with A(3, 4) and B(�1, 7) to the nearest tenth.A. 36.9° B. 53.1° C. 126.9° D. 143.1°
17. Find the image of A(3, 7) under a translation by a� � ��4, 2�.A. A�(�7, �5) B. A�(�1, 9) C. A�(7, 5) D. A�(1, �9)
18. Which translation matrix could be used to translate �ABC 5 units to theright and 7 units up?
A. B. C. D.
19. Find the coordinates of X� with X(6, 5) for a dilation centered at the originwith a scale factor of �2.A. X�(�10, �12) B. X�(10, 12) C. X�(12, 10) D. X�(�12, �10)
20. Which reflection matrix could you use to reflect a figure in the x-axis?
A. B. C. D.
Bonus Describe a rotation that movestriangle 1 to triangle 2.
1 2
0 11 0
�1 0 0 �1
�1 0 0 1
1 00 �1
�5 �5 �5�7 �7 �7
5 5 5�7 �7 �7
�5 �5 �5 7 7 7
5 5 57 7 7
x
y
O
A�
B� C�
A
B C
B:
NAME DATE PERIOD
Chapter 9 Test, Form 2A99
© Glencoe/McGraw-Hill 523 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
1. Given B(�4, �6), under which reflection is B�(4, 6)?A. reflected in the x-axis B. reflected in the y-axisC. reflected in the origin D. reflected in y � x
2. Name the image of E�F� under reflection in line �.A. F�G� B. H�G�C. E�H� D. F�E�
3. How many lines of symmetry does a regular decagon have?A. 0 B. 2 C. 5 D. 10
4. Which property is changed by a translation?A. collinearity B. angle measureC. distance measure D. position
5. What is the image of Y(�4, 7) under the translation (x, y) → (x � 3, y � 5)?A. Y�(�1, 2) B. Y�(�1, 12) C. Y�(�7, 2) D. Y�(�7, 12)
6. What is a transformation called that turns every point of the preimagethrough a specified angle and direction about a fixed point?A. reflection B. rotation C. translation D. dilation
7. Find the angle of rotation for a figure reflected in two lines that intersect toform a 72° angle.A. 36° B. 72° C. 144° D. 288°
8. Find the sum of the measures of the angles of the polygons in a tessellationat a vertex.A. 90 B. 180 C. 360 D. 720
9. Describe this tessellation.A. uniform and regularB. uniform and semi-regularC. not uniform but regularD. not uniform and semi-regular
10. What type of dilation occurs with a scale factor of �14�?
A. enlargement B. reductionC. congruence transformation D. inverse transformation
E
G
F
H
�
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 524 Glencoe Geometry
Chapter 9 Test, Form 2A (continued)99
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
11. Find the scale factor if �D�E�F� is the image of �DEF under a dilation with center C.A. 2 B. 1C. �1 D. �2
12. Sue scans a 4-inch picture into her computer. She stretches the picture’slength to 10 inches. Find the scale factor she used.
A. 6 B. �52� C. 2 D. �
25�
13. Find the component form of CD� with C(5, �7) and D(�3, 9).A. ��2, 2� B. �2, 2� C. �8, �16� D. ��8, 16�
14. Find the magnitude of CD� with C(5, 2) and D(�1, 6).A. ��6, 4� B. 2�13� C. 4�5� D. 10
15. Find the direction of CD� with C(5, 2) and D(�1, 6) to the nearest tenth.A. 33.7° B. 56.3° C. 123.7° D. 146.3°
16. Find the image of P(�2, 4) under a translation by the vector b� � �6, 5�.A. P�(4, 9) B. P�(�4, �9) C. P�(�8, �1) D. P�(8, 1)
17. HIJK is a trapezoid with H(5, 4), I(10, �2), J(�8, �2), and K(�3, 4). Find thecoordinates of the image of H under (x, y) → (x � 10, y � 11).A. (20, �13) B. (15, �7) C. (�5, 15) D. (7, �7)
18. The vertex matrix of a figure is multiplied on the left by . Find theangle of rotation about the origin.A. 90° counterclockwise B. 180° counterclockwiseC. 270° counterclockwise D. 360° counterclockwise
19. Find the matrix that is used to rotate a figure 270° counterclockwise aboutthe origin.
A. B. C. D.
20. Find the matrix you could use to reflect a figure in the origin.
A. B. C. D.
Bonus Find a vector in component form with magnitude 1 in a direction opposite to a� � ��3, �4�.
0 11 0
�1 0 0 �1
1 00 �1
�1 0 0 1
0 �1�1 0
0 1�1 0
�1 0 0 �1
0 �11 0
0 �11 0
x
y
O
E�
D�
F�D
F E
C
B:
NAME DATE PERIOD
Chapter 9 Test, Form 2B99
© Glencoe/McGraw-Hill 525 Glencoe Geometry
Ass
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Write the letter for the correct answer in the blank at the right of eachquestion.
1. Given C(2, 9), under which reflection is C�(9, 2)?A. reflected in the x-axis B. reflected in the y-axisC. reflected in the origin D. reflected in y � x
2. Name the image of K�L� under reflection in point N.A. L�K� B. M�L�C. M�J� D. J�K�
3. How many lines of symmetry does a regular 12-gon have?A. 4 B. 6 C. 12 D. 24
4. Which of the following figures shows a translation?A. B.
C. D.
5. Name the image of C(6, �4) under rotation 90° counterclockwise about theorigin.A. C�(4, 6) B. C�(�4, �6) C. C�(6, 4) D. C�(�6, �4)
6. Name the image of Z(�11, �6) under the translation (x, y) → (x � 1, y � 7).A. Z�(�12, 1) B. Z�(�12, �13) C. Z�(�10, 1) D. Z�(�10, �13)
7. Find the angle of rotation of a figure that is reflected in two lines thatintersect to form a 66° angle.A. 264° B. 132° C. 66° D. 33°
8. Which could be used to create a regular tessellation?A. square B. rectangle C. trapezoid D. all of these
9. Describe this tessellation.A. uniform and regularB. uniform and semi-regularC. not uniform but regularD. not uniform and semi-regular
10. What type of dilation occurs with a scale factor of 1?A. enlargement B. reductionC. congruence transformation D. inverse transformation
J
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K
M L
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© Glencoe/McGraw-Hill 526 Glencoe Geometry
Chapter 9 Test, Form 2B (continued)99
11.
12.
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15.
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17.
18.
19.
20.
11. Find the scale factor if �X�Y�Z� is the image of �XYZ under a dilation with center C.A. 6 B. 3
C. 2 D. �13�
12. Stella carved an equilateral triangle that is 3 centimeters on each side of apumpkin. Then she carved another equilateral triangle that is 5 centimeters oneach side. What scale factor did Stella use to increase the size of the triangle?
A. 2 B. �53� C. �
23� D. �
35�
13. Find the component form of EF� with E(�11, �3) and F(7, �4).A. ��18, 1� B. �18, �1� C. ��4, �7� D. �4, 7�
14. Find the magnitude of GH� with G(�3, 0) and H(4, 5).A. ��7, �5� B. �7, 5� C. �26� D. �74�
15. Find the direction of GH� with G(�3, 0) and H(4, 5) to the nearest tenth.A. 35.5° B. 54.5° C. 125.5° D. 144.5°
16. Find the image of D(3, 4) under a translation by the vector c� � ��7, �2�.A. D�(10, 6) B. D�(�10, �6) C. D�(4, �2) D. D�(�4, 2)
17. Find the image Y(�7, 4) for a dilation centered at the origin with a scalefactor of �3.A. Y�(�21, 12) B. Y�(21, �12) C. Y�(12, �21) D. Y�(�12, 21)
18. LMNO is a trapezoid with L(�1, 4), M(5, 12), N(5, �3), and O(�1, �5). Findthe coordinates for the image of N under the translation (x, y) → (x � 7, y � 9).A. (�2, 6) B. (12, �12) C. (�8, 13) D. (�2, 21)
19. If you multiply a vertex matrix of a figure on the left by , under a
rotation counterclockwise about the origin, of what magnitude will you findthe vertices of the image?A. 90° B. 180° C. 270° D. 360°
20. To reflect a figure in the line y � x you can multiply a vertex matrix of afigure on the left by which of the following matrices?
A. B. C. D.
Bonus Find a vector in component form in the same direction as ��3, �4� with magnitude 15.
0 11 0
�1 0 0 �1
1 00 �1
�1 0 0 1
�1 0 0 �1
x
y
OX�Z�
Y�
Z
C
XY
B:
NAME DATE PERIOD
Chapter 9 Test, Form 2C99
© Glencoe/McGraw-Hill 527 Glencoe Geometry
Ass
essm
ents
1. Write the coordinates of the image of P(�2, 5) reflected in theline y � x.
2. Graph �ABC with vertices A(4, 4), B(3, �2), and C(�1, �1).Then graph the image of �ABC reflected in the y-axis.
3. How many lines of symmetry does this figure have?
4. Determine whether �A�B�C� is a translation image of �ABC. Explainyour answer.
5. Find the image of W�X� with W(7, 1) and X(�4, 5) under thetranslation (x, y) → (x � 4, y � 3).
6. Find the image of A�B� with A(�3, 1) and B(�1, 5) under arotation of 90° clockwise about the origin.
7. Find the coordinates of L� if �LMN with L(3, 1), M(�1, 6), andN(�3, 2) is reflected in the line y � x and then in the x-axis.
8. Determine whether a regular 12-gon tessellates the plane.Explain.
9. Describe this tessellation as uniform,not uniform, regular, or semi-regular.
10. If AB � 10 and A�B� � 5, is the dilation an enlargement,reduction, or congruence transformation?
aA�
B�C�
A
BC
A�
B �C �
b
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A�
B�
C�
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© Glencoe/McGraw-Hill 528 Glencoe Geometry
Chapter 9 Test, Form 2C (continued)99
11. Find the measure of the image of S�T� if ST � 4 under a
dilation with a scale factor of �34�.
12. Draw the image of �MNO under a dilation with center C anda scale factor of 2.
13. If ST � 12 and S�T� � 9, find the scale factor of the dilation.
14. Find the direction of n� � ��7, �3� to the nearest tenth.
15. Find the magnitude of k� � �6, 8�.
16. Write the component form of AB�.
17. If a� and m� have opposite directions, are they parallel?
18. Find the image of the point at (5, 1) under the translation bym�� ��9, 6�.
19. Use a matrix to find the coordinates of the vertices of theimage of �EFG with E(6, 1), F(�1, �3), and G(2, 4), under thetranslation (x, y) → (x � 3, y � 2).
20. Use a matrix to find the coordinates of the vertices of theimage of �ABC with A(0, 2), B(3, 6), and C(5, 0), after areflection in the y-axis.
Bonus An airplane is flying at 400 miles per hour due west. Thewind is blowing from due north at 30 miles per hour. Findthe resultant speed and direction of the plane.
x
y
O
A
B
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M�
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N
OC
Chapter 9 Test, Form 2D99
© Glencoe/McGraw-Hill 529 Glencoe Geometry
Ass
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ents
1. Write the coordinates of the image of Q(�3, �6) reflected inthe origin.
2. Graph �PQR with vertices P(3, 4), Q(5, �1), and R(�3, 0).Then graph the image of �PQR reflected in the x-axis.
3. How many lines of symmetry does this figure have?
4. Determine whether W�X�Y�Z� is a translation image of WXYZ. Explain.
5. Find the image of U�V� with U(�3, 5) and V(0, 8) under thetranslation (x, y) → (x � 2, y � 5).
6. Find the image of C�D� with C(0, 4) and D(3, 4) under a rotationof 90° counterclockwise about the origin.
7. Find the coordinates of Q� if �OPQ with O(4, 2), P(5, 0), andQ(1, �2) is reflected in the x-axis and then in the y-axis.
8. Determine whether a regular 15-gon tessellates the plane.Explain.
9. Describe this tessellation as uniform,not uniform, regular, or semi-regular.
10. If CD � 3 and C �D� � 8, is the dilation an enlargement,reduction, or congruence transformation?
W� X�
Y�Z�
WYZX
X �
Y �Z �
W �
a
b
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10.
R�
P�
Q�
P
QR
NAME DATE PERIOD
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© Glencoe/McGraw-Hill 530 Glencoe Geometry
Chapter 9 Test, Form 2D (continued)99
11. Find the measure of the image of G�H� if GH � 7 under adilation with a scale factor of 5.
12. Draw the image of �CDE under dilation with center G and a scale factor of �
13�.
13. Find the scale factor of the dilation if OP � 15 and O�P� � 20.
14. Find the direction of o� � ��6, �4� to the nearest tenth.
15. Find the magnitude of l�� �5, 12�.
16. Write the component form of EF�.
17. If w� and x� are equal, do they have the same magnitude anddirection?
18. Find the image of the point at (�11, �7) under a translationby r� � ��2, 3�.
19. Use a matrix to find the coordinates of the vertices of theimage of �JKL with J(�5, 4), K(6, 8), and L(�2, �3), underthe translation (x, y) → (x � 6, y � 5).
20. Use a matrix to find the coordinates of the vertices of theimage of �DEF with D(�2, 1), E(�1, 6), and F(3, 2), after areflection in the x-axis.
Bonus An airplane is flying at 300 miles per hour due south. Thewind is blowing from due west at 40 miles per hour. Findthe resultant speed and direction of the plane.
x
y
O
E
F
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C�
D�
E�
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D
E
G
Chapter 9 Test, Form 399
© Glencoe/McGraw-Hill 531 Glencoe Geometry
Ass
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ents
1. Draw the lines of symmetry.
2. Graph �TUV with vertices T(3, 3), U(6, �1), and V(�2, 1).Then graph the image of �TUV reflected in the line y � 2.
3. Name two properties that are preserved by a reflection.
4. Draw the translation image of �ABC in lines � and m.
5. Find the preimage of C���D��� with C�(4, 6) and D�(�1, 2) underthe translation (x, y) → (x � 3, y � 7).
6. Identify the order and magnitude of the rotational symmetryin a regular 20-gon.
7. Find the coordinates of the image and the measure of theangle of rotation if �RST with R(5, 3), S(7, 8), and T(10, 1) isreflected in the line y � x and then in the x-axis.
8. Determine whether a regular decagon can tessellate a plane.Explain.
9. Describe this tessellation as uniform,not uniform, regular, or semi-regular.
10. Find the measure of the dilation image of P�Q� if PQ � 12 under a dilation with a scale factor of ��
34�.
1.
2.
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4.
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9.
10.
�
AB
C
A� B�
C�
m
V�
T�
U�
T
U
V
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 532 Glencoe Geometry
Chapter 9 Test, Form 3 (continued)99
11. Draw the image of this figure under a dilation with center Cand a scale factor of �
13�.
12. An 11-inch by 14-inch picture is being reduced on a printer bya scale factor of 60%. Find the dimensions of the image.
13. If WX � �23� and W�X� � �
45�, find the scale factor of the dilation.
14. Name the series of reflections that would result in the sameimage as a figure rotated 180° counterclockwise about the origin.
15. Find the direction of p� � �6, �3� to the nearest tenth.
16. Find the magnitude of m�� ��7, 2� to the nearest tenth.
17. Write the component form of XY� with X(�11, �3), and Y(�8, 5).
18. Find the image of the point at (4, �7) under the translation byw�� �a, b�.
19. Use matrices to find the coordinates of the image of �JKLwith J(�6, �2), K(2, 10), and L(�2, �2), under a dilation with
a scale factor of ��12�, then a reflection in the line y � x, then
the translation (x, y) → (x � 3, y � 2).
20. Use a matrix to find the coordinates of the vertices of theimage of �GHI with G(�5, �4), H(1, 1), and I(6, �2), after areflection in the origin.
Bonus D(0, 6) is rotated 30° clockwise about the origin. Find thecoordinates of its image.
NAME DATE PERIOD
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Chapter 9 Open-Ended Assessment99
© Glencoe/McGraw-Hill 533 Glencoe Geometry
Ass
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Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.
1. Draw a uniform regular tessellation. Explain why your figure fits thiscategory.
2. Name an object that has at least one line of symmetry. Describe the linesof symmetry in this object.
3. a. Show the matrix used to determine the vertices of a figure’s image aftera rotation of 90° counterclockwise about the origin.
b. Find the coordinates of the image of B(1, 0) rotated 90°counterclockwise about the origin.
c. Find the coordinates of the image of A(0, 1) rotated 90°counterclockwise about the origin.
d. Compare the answers in parts b and c to the answer in part a.
4. Draw a tessellation that is not uniform.
5. Graph �ABC and label the coordinates of its vertices. Find the image of�ABC under a composition of a reflection, rotation, translation, anddilation. Name the transformations you used.
6. Draw CD� with one endpoint at the origin and the other in the firstquadrant. Find the magnitude and direction of CD� to the nearest tenth.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 534 Glencoe Geometry
Chapter 9 Vocabulary Test/Review99
Choose from the terms above to complete each sentence.
1. A congruence transformation is called a(n) .
2. A transformation representing a flip of a figure is called a(n).
3. A(n) is a transformation that turns every point of apreimage through a specified angle and direction about a fixedpoint.
4. A vector is in when the initial point is at the origin.
5. A(n) is a transformation that moves all points of afigure the same distance in the same direction.
6. A(n) is a directed segment representing a quantity.
7. When a figure can be folded so that the two halves matchexactly, the fold is called a(n) .
8. A(n) is a transformation that requires a center pointand a scale factor.
9. Two vectors are if and only if they have the same oropposite directions.
10. Expressing a vector as an ordered pair is the of thevector.
Define each term.
11. center of rotation
12. scalar
13. tessellation
?
?
?
?
?
?
?
?
?
? 1.
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angle of rotationcenter of rotationcolumn matrixcomponent form composition of reflectionscongruence transformationdilationdirect isometrydirectionequal vectors
glide reflectionindirect isometryisometryline of reflectionline of symmetrymagnitudeparallel vectorspoint of symmetryreflection
reflection matrixregular tessellationresultantrotationrotation matrixrotational symmetryscalarscalar multiplicationsemi-regular tessellation
similarity transformationstandard positiontessellationtransformationtranslationtranslation matrixuniformvectorvertex matrix
NAME DATE PERIOD
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Chapter 9 Quiz (Lessons 9–1 and 9–2)
99
© Glencoe/McGraw-Hill 535 Glencoe Geometry
Ass
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NAME DATE PERIOD
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1.
2.
3.
4.
5.
1. Name the coordinates of the image of Q(6, �4) reflected in thex-axis.
2. How many lines of symmetry does an isosceles triangle have?
3. Why is �A�B�C� with vertices A�(�1, �2), B�(0, 0), and C�(�6, 0) not a translation image of �ABC with A(1, 2),B(0, 0), and C(6, 0)?
4. Complete this statement. The image of A(�3, �5) under (x, y) → is A�(6, �1).
5. Draw the image of �PQR with P(0, 4), Q(2, 8), and R(�3, 6)reflected in the line x � 1.
?
Q�
P�
R�
Chapter 9 Quiz (Lessons 9–3 and 9–4)
99
1.
2.
3.
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5.
1. Find the image of A(�2, 3) reflected in the line y � �x andthen in the x-axis.
2. Draw the image of �PQR under a rotation 45° clockwise aboutpoint T.
3. Determine whether a scalene triangle will tessellate the plane.
4. Determine whether a regular octagon and a square, bothhaving sides 2 units long, can be used to draw a semi-regulartessellation.
5. A figure is reflected in two intersecting lines forming arotation of magnitude 84°. Find the measure of the acuteangle formed by the intersecting lines of reflection.
P�Q�
R�R
Q
P
T
NAME DATE PERIOD
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© Glencoe/McGraw-Hill 536 Glencoe Geometry
Chapter 9 Quiz (Lessons 9–5 and 9–6)
99
1.
2.
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4.
5.
1. Name a property that is not preserved by a dilation.
2. �XYZ has vertices X(�1, 3), Y(5, 7), and Z(2, �4). Find thecoordinates of the image of �XYZ after a dilation centered atthe origin with a scale factor of 4.
3. Find the magnitude and direction to the nearest degree of x� � �1, �4�.
4. Graph the image of �DEF with vertices D(�1, 2), E(1, 0), andF(�3, �2) under a translation by w�� �0, 2�.
5. STANDARDIZED TEST PRACTICE Reva wants to enlarge a3-inch by 4-inch picture by a scale factor of 2.5. Find thedimensions of the enlarged image.A. 6 in. by 8 in. B. 7.5 in. by 10 in.C. 8 in. by 10 in. D. 9 in. by 12 in.
D�
F�
E�D
E
F
NAME DATE PERIOD
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Chapter 9 Quiz (Lesson 9–7)
99
1.
2.
3.
4.
5.
1. Use scalar multiplication to dilate Y�Z� with Z(�2, 3) and Y(�1, 4) so that its length is 4 times the length of Y�Z�.
2. Use a matrix to find the coordinates of the vertices of theimage of �MNO with M(�5, 4), N(�3, 3), and O(�4, 7) underthe translation (x, y) → (x � 2, y � 1).
3. Use a matrix to find the coordinates of the vertices of theimage of �ABC with A(�3, �1), B(8, 2), and C(5, 7) after areflection in the y-axis.
4. Find the coordinates of the image of �DEF with D(2, 3),E(4, �1), and F(�2, �3) after a dilation with a scale factor of�12� and then a reflection in the x-axis.
5. Name the matrix by which you would multiply a vertex matrixof a figure on the left to find the figure’s image under arotation 90° counterclockwise about the origin.
NAME DATE PERIOD
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Chapter 9 Mid-Chapter Test (Lessons 9–1 through 9–4)
99
© Glencoe/McGraw-Hill 537 Glencoe Geometry
Ass
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ents
1. How many lines of symmetry does a parallelogram have?A. 0 B. 1 C. 2 D. 4
2. Which translation moves every point of a preimage 4 units left and 6 units up?A. (x, y) → (x � 4, y � 6) B. (x, y) → (x � 4, y � 6)C. (x, y) → (x � 6, y � 4) D. (x, y) → (x � 6, y � 4)
3. Find the magnitude of the rotational symmetry in a square.A. 45° B. 90° C. 180° D. 360°
4. Which pair of figures could be used to create a semi-regular tessellation?A. trapezoid, square B. kite, squareC. equilateral triangle, square D. rectangle, square
5. Which action represents the reflection of a figure?A. slide B. shift C. turn D. flip
6.
7.
8.
9.
10.
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1.
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Part II
6. Write the coordinates of the image of S(�7, 1) reflected in they-axis.
7. DEFG is a square with D(1, 1), E(1, 6), F(6, 6), and G(6, 1) andis first reflected in the line x � 1 and then in the y-axis. Findthe coordinates of D�, E �, F �, and G�.
8. Determine whether �P�Q�R� is a translation image of �PQR. Explain.
9. Find the image of B(4, 7) reflected in the y-axis and then inthe line y � �x.
10. Determine whether a regular 30-gon will tessellate the plane.Explain.
P�
Q�
R�
R
Q
P
Part I Write the letter for the correct answer in the blank at the right of each question.
© Glencoe/McGraw-Hill 538 Glencoe Geometry
Chapter 9 Cumulative Review(Chapters 1–9)
99
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12.
1. If �T and �S form a linear pair and m�T is twelve more thaneleven times m�S, find m�T and m�S. (Lesson 1-5)
2. Determine whether the statement If C�D� � D�E�, then D is themidpoint of C�E� is always, sometimes, or never true. (Lesson 2-5)
3. Find the distance between the parallel lines � and m whoseequations are y � x � 6 and y � x � 2, respectively. (Lesson 3-6)
4. Name the missing coordinates of isosceles �DFG with a base n units long. (Lesson 4-7)
5. Two sides of a triangle measure 45 inches and 68 inches, andthe length of the third side measures x inches. Find the rangefor x. (Lesson 5-4)
6. Find x so that W�V� || Y�Z�. (Lesson 6-4) 7. Find b. (Lesson 7-1)
8. In �STV, s � 40, t � 52, and m�T � 82. Find m�S, m�V,and v to the nearest tenth. (Lesson 7-7)
9. Determine whether ABCD is a rectangle with vertices A(2, 8),B(0, 1), C(1, �8), and D(5, 6). Justify your answer. (Lesson 8-4)
10. Find a, b, and m�HJK if HJKL is a parallelogram. (Lesson 8-5)
11. Draw the image of �PQR under a 90° counterclockwiserotation about M(�1, 2). (Lesson 9-3)
12. Write the component form of AB� with A(�4, 7) and B(9, �2).(Lesson 9-6)
(5a � 1)�
(91 � 4a)�
5b � 4 4b
H J
L K
9 5
a cb
21
6xX ZV
YW28
x
y F(?, ?)
G(n, 0)
b
D
P�
Q�
R�
x
y
O
P
QR
M
NAME DATE PERIOD
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Standardized Test Practice (Chapters 1–9)
© Glencoe/McGraw-Hill 539 Glencoe Geometry
1. Which statement has a false converse? (Lesson 2-3)
A. If a quadrilateral is a rectangle, then the diagonals of thequadrilateral are congruent.
B. If a quadrilateral has diagonals that bisect each other, then thequadrilateral is a parallelogram.
C. If a quadrilateral is a rectangle, then all angles of thequadrilateral are right angles.
D. If a quadrilateral is a parallelogram, then both pairs of oppositesides of the quadrilateral are congruent.
2. If �FGH is an isosceles triangle with �G as the vertex angle andm�G � 52, find m�F and m�H. (Lesson 4-6)
E. m�F � 72, m�H � 56 F. m�F � 64, m�H � 64G. m�F � 52, m�H � 76 H. m�F � 40, m�H � 88
3. What can you conclude from the figure? (Lesson 5-5)
A. m�W � m�D by SSSInequalityB. m�G � m�V by SSS InequalityC. DF � WV by SAS InequalityD. GF � VT by SAS Inequality
4. Find x. (Lesson 6-3)
E. 8.68 F. 20.25G. 31.24 H. 42.31
5. If Q�R� is the hypotenuse of right �PQR,PQ � 18, and QR � 24, find RP. (Lesson 7-2)
A. �63� B. �252� C. 30 D. 60
6. If A�B� is a median of trapezoid WXYZ, find m�1, m�2, and m�3. (Lesson 8-6)
E. m�1 � 72, m�2 � 108, m�3 � 72F. m�1 � 108, m�2 � 72, m�3 � 72G. m�1 � 108, m�2 � 72, m�3 � 108H. m�1 � 108, m�2 � 108, m�3 � 72
7. If A(c, d) is reflected in the y-axis, find the coordinates of A�.(Lesson 9-1)
A. A�(c, �d) B. A�(�c, d) C. A�(�c, �d) D. A�(d, c)
W X
BA
Z Y72�123
12.2
13.5
18.325.4�
25.4�
x
22
2014
149
9
W T
V
G F
D
NAME DATE PERIOD
SCORE 99
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
1.
2.
3.
4.
5.
6.
7. A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
Ass
essm
ents
© Glencoe/McGraw-Hill 540 Glencoe Geometry
Standardized Test Practice (continued)
8. Find x so that the line containing (�7, 8) and(6, x) is perpendicular to RT��� with R(3, 9) andT(4, �4). (Lesson 3-3)
9. Find x. (Lesson 4-2)
10. Quadrilateral QRTV � quadrilateral HJKL,QR � 5, RT � 10, JK � 14, and KL � 8. Findthe scale factor as a fraction of quadrilateralQRTV to quadrilateral HJKL. (Lesson 6-2)
11. Find cos M. (Lesson 7-4)
12. Find the measure of the preimage of dilated J�K� if J�K� � 62 and r � ��
13�. (Lesson 9-5)
12
159
K
L M
97.8�
22.1�x �
NAME DATE PERIOD
99
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
Part 3: Short Response
Instructions: Show your work or explain in words how you found your answer.
8. 9.
10. 11.
12.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
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87654321
87654321
13. Two parallel lines are cut by a transversal, �1 and �2 areconsecutive interior angles, m�1 � 6y � 5, and m�2 � 10y � 17. Find m�1 and m�2. (Lesson 3-2)
14. Parallelogram ABCD has vertices A(c, a), C(b, 0), and D(0, 0).Find the coordinates for vertex B. (Lesson 8-7)
15. Triangle DEF with D(7, �12), E(2, 10), and F(�11, �8) istranslated so that D�(12, �16). Find the coordinates of E�and F �. (Lesson 9-2)
13.
14.
15.
9 6 0 . 1
5 / 7
1 8 6
0 . 8
Standardized Test PracticeStudent Record Sheet (Use with pages 518–519 of the Student Edition.)
99
© Glencoe/McGraw-Hill A1 Glencoe Geometry
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7
2 5
3 6 DCBADCBA
DCBADCBA
DCBADCBADCBA
NAME DATE PERIOD
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 3 Open-EndedPart 3 Open-Ended
Solve the problem and write your answer in the blank.
For Questions 10 and 11, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.
8 9 10
9 (grid in)
10 (grid in)0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
Record your answers for Questions 11–12 on the back of this paper.
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Ref
lect
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
©G
lenc
oe/M
cGra
w-H
ill47
9G
lenc
oe G
eom
etry
Lesson 9-1
Dra
w R
efle
ctio
ns
Th
e tr
ansf
orm
atio
n c
alle
d a
refl
ecti
onis
a f
lip
of a
fig
ure
in
a
poin
t,a
lin
e,or
a p
lan
e.T
he
new
fig
ure
is
the
imag
ean
d th
e or
igin
al f
igu
re i
s th
ep
reim
age.
Th
e pr
eim
age
and
imag
e ar
e co
ngr
uen
t,so
a r
efle
ctio
n i
s a
con
gru
ence
tran
sfor
mat
ion
or i
som
etry
.
Con
stru
ct t
he
imag
eof
qu
adri
late
ral
AB
CD
un
der
are
flec
tion
in
lin
em
.
Dra
w a
per
pend
icul
ar f
rom
eac
h ve
rtex
of t
he q
uadr
ilat
eral
to
m.F
ind
vert
ices
A�,
B�,
C�,
and
D�
that
are
the
sam
edi
stan
ce f
rom
mon
the
oth
er s
ide
of m
.T
he i
mag
e is
A�B
�C�D
�.
m
CA
D
B
A�
B�
C�
D�
Qu
adri
late
ral
DE
FG
has
vert
ices
D(�
2,3)
,E(4
,4),
F(3
,�2)
,an
d
G(�
3,�
1).F
ind
th
e im
age
un
der
ref
lect
ion
in t
he
x-ax
is.
To
fin
d an
im
age
for
are
flec
tion
in
th
e x-
axis
,us
e th
e sa
me
x-co
ordi
nate
and
mu
ltip
ly t
he
y-co
ordi
nat
e by
�1.
Insy
mbo
ls,(
a,b)
→(a
,�b)
.T
he
new
coo
rdin
ates
are
D�(
�2,
�3)
,E�(
4,�
4),
F�(
3,2)
,an
d G
�(�
3,1)
.T
he
imag
e is
D�E
�F�G
�.
x
y
O
D�
E�
F�
G�D
E
FG
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
In E
xam
ple
2,th
e n
otat
ion
(a,
b) →
(a,�
b) r
epre
sen
ts a
ref
lect
ion
in
th
e x-
axis
.Her
e ar
eth
ree
oth
er c
omm
on r
efle
ctio
ns
in t
he
coor
din
ate
plan
e.•
in t
he
y-ax
is:
(a,b
) →
(�a,
b)•
in t
he
lin
e y
�x:
(a,b
) →
(b,a
)•
in t
he
orig
in:
(a,b
) →
(�a,
�b)
Dra
w t
he
imag
e of
eac
h f
igu
re u
nd
er a
ref
lect
ion
in
lin
em
.
1.2.
3.
Gra
ph
eac
h f
igu
re a
nd
its
im
age
un
der
th
e gi
ven
ref
lect
ion
.
4.�
DE
Fw
ith
D(�
2,�
1),E
(�1,
3),
5.A
BC
Dw
ith
A(1
,4),
B(3
,2),
C(2
,�2)
,F
(3,�
1) i
n t
he
x-ax
isD
(�3,
1) i
n t
he
y-ax
is x
y
OD
�
C�
B�
A�
D
C
B
A
x
y
O
F�
D� E
�
FD
E
m
R�
T�
S�
Q
RS
TU
m
P�
N�
M�
LM
N
OP
mH
�
J�K
�
HJ
K
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill48
0G
lenc
oe G
eom
etry
Lin
es a
nd
Po
ints
of
Sym
met
ryIf
a f
igu
re h
as a
lin
e of
sym
met
ry,t
hen
it
can
be
fold
ed a
lon
g th
at l
ine
so t
hat
th
e tw
o h
alve
s m
atch
.If
a fi
gure
has
a p
oin
t of
sym
met
ry,i
tis
th
e m
idpo
int
of a
ll s
egm
ents
bet
wee
n t
he
prei
mag
e an
d im
age
poin
ts.
Det
erm
ine
how
man
y li
nes
of
sym
met
ry
a re
gula
r h
exag
on h
as.T
hen
det
erm
ine
wh
eth
er a
re
gula
r h
exag
on h
as p
oin
t sy
mm
etry
.T
her
e ar
e si
x li
nes
of
sym
met
ry,t
hre
e th
at a
re d
iago
nal
s th
rou
gh o
ppos
ite
vert
ices
an
d th
ree
that
are
per
pen
dicu
lar
bise
ctor
s of
opp
osit
e si
des.
Th
e h
exag
on h
as p
oin
t sy
mm
etry
beca
use
an
y li
ne
thro
ugh
Pid
enti
fies
tw
o po
ints
on
th
e h
exag
on t
hat
can
be
con
side
red
imag
es o
f ea
ch o
ther
.
Det
erm
ine
how
man
y li
nes
of
sym
met
ry e
ach
fig
ure
has
.Th
en d
eter
min
e w
het
her
the
figu
re h
as p
oin
t sy
mm
etry
.
1.2.
3.
4;ye
s3;
no
2;ye
s
4.5.
6.
5;n
o2;
yes
0;n
o
7.8.
9.
1;n
o1;
no
1;n
o
AB
C
DE
FP
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Ref
lect
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 9-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Ref
lect
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
©G
lenc
oe/M
cGra
w-H
ill48
1G
lenc
oe G
eom
etry
Lesson 9-1
Dra
w t
he
imag
e of
eac
h f
igu
re u
nd
er a
ref
lect
ion
in
lin
e �.
1.2.
CO
OR
DIN
ATE
GEO
MET
RYG
rap
h e
ach
fig
ure
an
d i
ts i
mag
e u
nd
er t
he
give
nre
flec
tion
.
3.�
AB
Cw
ith
ver
tice
s A
(�3,
2),B
(0,1
),4.
trap
ezoi
d D
EF
Gw
ith
ver
tice
s D
(0,�
3),
and
C(�
2,�
3) i
n t
he
orig
inE
(1,3
),F
(3,3
),an
d G
(4,�
3) i
n t
he
y-ax
is
5.pa
rall
elog
ram
RS
TU
wit
h v
erti
ces
6.sq
uar
e K
LM
Nw
ith
ver
tice
s K
(�1,
0),
R(�
2,3)
,S(2
,4),
T(2
,�3)
an
d L
(�2,
3),M
(1,4
),an
d N
(2,1
) in
U
(�2,
�4)
in
th
e li
ne
y�
xth
e x-
axis
Det
erm
ine
how
man
y li
nes
of
sym
met
ry e
ach
fig
ure
has
.Th
en d
eter
min
e w
het
her
the
figu
re h
as p
oin
t sy
mm
etry
.
7.8.
9.
1;n
o4;
yes
0;ye
s
M�
K�
Kx
y
O
L�
N�
LM
Nx
y
O
R�S
�T
�
U�
RS T
U
x
y
O
DEF
GD
�
E�
F�
G�
x
y
O
A�
B�
C�
AB
C
�
�
©G
lenc
oe/M
cGra
w-H
ill48
2G
lenc
oe G
eom
etry
Dra
w t
he
imag
e of
eac
h f
igu
re u
nd
er a
ref
lect
ion
in
lin
e �.
1.2.
CO
OR
DIN
ATE
GEO
MET
RYG
rap
h e
ach
fig
ure
an
d i
ts i
mag
e u
nd
er t
he
give
nre
flec
tion
.
3.qu
adri
late
ral
AB
CD
wit
h v
erti
ces
4.�
FG
Hw
ith
ver
tice
s F
(�3,
�1)
,G(0
,4),
A(�
3,3)
,B(1
,4),
C(4
,0),
and
and
H(3
,�1)
in
th
e li
ne
y�
xD
(�3,
�3)
in
th
e or
igin
5.re
ctan
gle
QR
ST
wit
h v
erti
ces
Q(�
3,2)
,6.
trap
ezoi
d H
IJK
wit
h v
erti
ces
H(�
2,5)
,R
(�1,
4),S
(2,1
),an
d T
(0,�
1)
I(2,
5),J
(�4,
�1)
,an
d K
(�4,
3)
in t
he
x-ax
isin
th
e y-
axis
RO
AD
SIG
NS
Det
erm
ine
how
man
y li
nes
of
sym
met
ry e
ach
sig
n h
as.T
hen
det
erm
ine
wh
eth
er t
he
sign
has
poi
nt
sym
met
ry.
7.8.
9.
0;ye
s1;
no
4;ye
s
HI
JK
H�
I�
J�K� x
y
OQ
�
R�
S�
T�
Q
R
S
Tx
y
O
F�
G�
H�
F
G
Hx
y
O
A�
B�
C�
D�
AB
C
D
x
y
O
��Pra
ctic
e (
Ave
rag
e)
Ref
lect
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
Answers (Lesson 9-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csR
efle
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
©G
lenc
oe/M
cGra
w-H
ill48
3G
lenc
oe G
eom
etry
Lesson 9-1
Pre-
Act
ivit
yW
her
e ar
e re
flec
tion
s fo
un
d i
n n
atu
re?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-1
at
the
top
of p
age
463
in y
our
text
book
.
Sup
pose
you
dra
w a
line
seg
men
t co
nnec
ting
a p
oint
at
the
peak
of a
mou
ntai
nto
its
imag
e in
the
lake
.Whe
re w
ill t
he m
idpo
int
of t
his
segm
ent
fall?
on
th
eb
ou
nd
ary
line
bet
wee
n t
he
sho
re a
nd
th
e su
rfac
e o
f th
e la
ke
Rea
din
g t
he
Less
on
1.D
raw
th
e re
flec
ted
imag
e fo
r ea
ch r
efle
ctio
n d
escr
ibed
bel
ow.
a.re
flec
tion
of
trap
ezoi
d A
BC
Din
th
e li
ne
nb
.re
flec
tion
of
�R
ST
in p
oin
t P
Lab
el t
he
imag
e of
AB
CD
as A
�B�C
�D�.
Lab
el t
he
imag
e of
RS
Tas
R�S
�T�.
c.re
flec
tion
of
pen
tago
n A
BC
DE
in t
he
orig
inL
abel
th
e im
age
of A
BC
DE
as A
�B�C
�D�E
�.
2.D
eter
min
e th
e im
age
of t
he
give
n p
oin
t u
nde
r th
e in
dica
ted
refl
ecti
on.
a.(4
,6);
refl
ecti
on i
n t
he
y-ax
is(�
4,6)
b.
(�3,
5);r
efle
ctio
n i
n t
he
x-ax
is(�
3,�
5)c.
(�8,
�2)
;ref
lect
ion
in
th
e li
ne
y�
x(�
2,�
8)d
.(9
,�3)
;ref
lect
ion
in
th
e or
igin
(�9,
3)
3.D
eter
min
e th
e n
um
ber
of l
ines
of
sym
met
ry f
or e
ach
fig
ure
des
crib
ed b
elow
.Th
ende
term
ine
wh
eth
er t
he
figu
re h
as p
oin
t sy
mm
etry
an
d in
dica
te t
his
by
wri
tin
g ye
sor
no.
a.a
squ
are
4;ye
sb
.an
iso
scel
es t
rian
gle
(not
equ
ilat
eral
) 1;
no
c.a
regu
lar
hex
agon
6;
yes
d.a
n i
sosc
eles
tra
pezo
id 1
;n
oe.
a re
ctan
gle
(not
a s
quar
e) 2
;ye
sf.
the
lett
er E
1;
no
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
geo
met
ric
term
is
to r
elat
e th
e w
ord
or i
ts p
arts
to
geom
etri
c te
rms
you
alr
eady
kn
ow.L
ook
up
the
orig
ins
of t
he
two
part
s of
th
e w
ord
isom
etry
in y
our
dict
ion
ary.
Exp
lain
th
e m
ean
ing
of e
ach
par
t an
d gi
ve a
ter
m y
oual
read
y kn
ow t
hat
sh
ares
th
e or
igin
of
that
par
t.S
amp
le a
nsw
er:T
he
firs
t p
art
com
es f
rom
iso
s,w
hic
h m
ean
s eq
ual
,as
in is
osc
eles
.Th
e se
con
d p
art
com
es f
rom
met
ron
,wh
ich
mea
ns
mea
sure
,as
in g
eom
etry
.
A�
B�
C�
D�
E�
AB
CD
E
x
y
O
R�
S�
T�
RS
P
Tn
A�
B�
C�
D�
A
BC D
©G
lenc
oe/M
cGra
w-H
ill48
4G
lenc
oe G
eom
etry
Ref
lect
ion
s in
th
e C
oo
rdin
ate
Pla
ne
Stu
dy
the
dia
gram
at
the
righ
t.It
sh
ows
how
th
e tr
ian
gle
AB
Cis
map
ped
on
to t
rian
gle
XY
Zb
y th
etr
ansf
orm
atio
n (
x,y)
→(�
x�
6,y)
.Not
ice
that
�X
YZ
is t
he
refl
ecti
on i
mag
e w
ith
res
pec
t to
th
e ve
rtic
al
lin
e w
ith
eq
uat
ion
x�
3.
1.P
rove
th
at t
he
vert
ical
lin
e w
ith
equ
atio
n x
�3
is t
he
perp
endi
cula
r bi
sect
or o
f th
e se
gmen
t w
ith
en
dpoi
nts
(x
,y)
and
(�x
�6,
y).(
Hin
t:U
se t
he
mid
poin
t fo
rmu
la.)
Mid
po
int
���x
�(�
2x�
6)�
,�y
� 2y
��o
r (3
,y)
Th
e se
gm
ent
join
ing
(x,
y)
and
(�
x�
6,y
) is
h
ori
zon
tal a
nd
hen
ce is
per
pen
dic
ula
r to
th
e ve
rtic
al li
ne.
2.E
very
tra
nsf
orm
atio
n o
f th
e fo
rm (
x,y)
→(�
x�
2h
,y)
is
a re
flec
tion
wit
h r
espe
ct t
o th
e ve
rtic
al l
ine
wit
h e
quat
ion
x
�h
.Wh
at k
ind
of t
ran
sfor
mat
ion
is
(x,y
) →
(x,�
y�
2k)
?
refl
ecti
on
wit
h r
esp
ect
to t
he
ho
rizo
nta
l lin
e w
ith
eq
uat
ion
y�
k
Dra
w t
he
tran
sfor
mat
ion
im
age
for
each
fig
ure
an
d t
he
give
n
tran
sfor
mat
ion
.Is
it a
ref
lect
ion
tra
nsf
orm
atio
n?
If s
o,w
ith
re
spec
t to
wh
at l
ine?
3.(x
,y)
→(�
x�
4,y)
4.(x
,y)
→(x
,�y
�8)
yes;
x�
�2
yes;
y�
4
x
y
Ox
y
O
x �
3
A
B
C
X
Y
Z
(x, y
) → (�
x �
6, y
)
x
y
O
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-1
9-1
Answers (Lesson 9-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Tran
slat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
©G
lenc
oe/M
cGra
w-H
ill48
5G
lenc
oe G
eom
etry
Lesson 9-2
Tran
slat
ion
s U
sin
g C
oo
rdin
ates
A t
ran
sfor
mat
ion
cal
led
a tr
ansl
atio
nsl
ides
afi
gure
in
a g
iven
dir
ecti
on.I
n t
he
coor
din
ate
plan
e,a
tran
slat
ion
mov
es e
very
pre
imag
epo
int
P(x
,y)
to a
n i
mag
e po
int
P(x
�a,
y�
b) f
or f
ixed
val
ues
aan
d b.
In w
ords
,atr
ansl
atio
n s
hif
ts a
fig
ure
au
nit
s h
oriz
onta
lly
and
bu
nit
s ve
rtic
ally
;in
sym
bols
,(x
,y)
→(x
�a,
y�
b).
Rec
tan
gle
RE
CT
has
ver
tice
s R
(�2,
�1)
,E
(�2,
2),C
(3,2
),an
d T
(3,�
1).G
rap
h R
EC
Tan
d i
ts i
mag
e fo
r th
e tr
ansl
atio
n (
x,y)
→(x
�2,
y�
1).
Th
e tr
ansl
atio
n m
oves
eve
ry p
oin
t of
th
e pr
eim
age
righ
t 2
un
its
and
dow
n 1
un
it.
(x,y
) →
(x�
2,y
�1)
R(�
2,�
1) →
R�(
�2
�2,
�1
�1)
or
R�(
0,�
2)E
(�2,
2) →
E�(
�2
�2,
2 �
1) o
r E
�(0,
1)C
(3,2
) →
C�(
3 �
2,2
�1)
or
C�(
5,1)
T(3
,�1)
→T
�(3
�2,
�1
�1)
or
T�(
5,�
2)
Gra
ph
eac
h f
igu
re a
nd
its
im
age
un
der
th
e gi
ven
tra
nsl
atio
n.
1.P �
Q�w
ith
en
dpoi
nts
P(�
1,3)
an
d Q
(2,2
) u
nde
r th
e tr
ansl
atio
n
left
2 u
nit
s an
d u
p 1
un
it
2.�
PQ
Rw
ith
ver
tice
s P
(�2,
�4)
,Q(�
1,2)
,an
d R
(2,1
) u
nde
r th
e tr
ansl
atio
n r
igh
t 2
un
its
and
dow
n 2
un
its
3.sq
uar
e S
QU
Rw
ith
ver
tice
s S
(0,2
),Q
(3,1
),U
(2,�
2),a
nd
R(�
1,�
1) u
nde
r th
e tr
ansl
atio
n r
igh
t 3
un
its
and
up
1 u
nit
x
y
R�S
�Q
�
U�
R
SQ
U
x
y P�
Q�
R�
P
QR
x
yP
�Q
�P
Q
x
y
OE�
R�
C�
T�
E R
C T
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill48
6G
lenc
oe G
eom
etry
Tran
slat
ion
s b
y R
epea
ted
Ref
lect
ion
sA
not
her
way
to
fin
d th
e im
age
of a
tran
slat
ion
is
to r
efle
ct t
he
figu
re t
wic
e in
par
alle
l li
nes
.Th
is k
ind
of t
ran
slat
ion
is
call
ed a
com
pos
ite
of r
efle
ctio
ns.
In t
he
figu
re, m
|| n.F
ind
th
e tr
ansl
atio
n i
mag
e of
�A
BC
.�
A�B
�C�
is t
he
imag
e of
�A
BC
refl
ecte
d in
lin
e m
.�
A�B
�C�
is t
he
imag
e of
�A�
B�C
�re
flec
ted
in l
ine
n.T
he
fin
al i
mag
e,�
A�B
�C�,
is a
tra
nsl
atio
n o
f �
AB
C.
In e
ach
fig
ure
, m|| n
.Fin
d t
he
tran
slat
ion
im
age
of e
ach
fig
ure
by
refl
ecti
ng
it i
nli
ne
man
d t
hen
in
lin
e n.
1.2.
3.4.
5.6.
m
U
A D
�
U�
A� D
�
U�
A�
D�
n
m n
U
T
S
R U�
T�
S�
R�
m
n
D�
Q�
AQ
U
D
A�
Q�
D�
U�
T�
mn
P�
PP
�
E�
EE
�N
N�
TT
�
AA
�
mn
A�
AB
C
A�
B�
C�
mn
A�
B�
C�
A
B C
A�
B�
C�
mn
A�
B�
C�
A
B C
A�
B�
C�
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Tran
slat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 9-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
Tran
slat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
©G
lenc
oe/M
cGra
w-H
ill48
7G
lenc
oe G
eom
etry
Lesson 9-2
In e
ach
fig
ure
,a|| b
.Det
erm
ine
wh
eth
er f
igu
re 3
is
a tr
ansl
atio
n i
mag
e of
fig
ure
1.
Wri
te y
esor
no.
Exp
lain
you
r an
swer
.
1.2.
Yes;
refl
ect
fig
ure
1 in
lin
e a
toN
o;
it is
ori
ente
d d
iffe
ren
tly
than
g
et f
igu
re 2
an
d f
igu
re 2
in li
ne
bfi
gu
re 1
.to
get
fig
ure
3.
3.4.
Yes;
refl
ect
fig
ure
1 in
lin
e a
to
No
;it
is a
ref
lect
ion
of
a tr
ansl
atio
n
get
fig
ure
2 a
nd
fig
ure
2 in
lin
e b
and
is o
rien
ted
dif
fere
ntl
y th
an
to g
et f
igu
re 3
.fi
gu
re 1
.
CO
OR
DIN
ATE
GEO
MET
RYG
rap
h e
ach
fig
ure
an
d i
ts i
mag
e u
nd
er t
he
give
ntr
ansl
atio
n.
5.�
JK
Lw
ith
ver
tice
s J
(�4,
�4)
,6.
quad
rila
tera
l L
MN
Pw
ith
ver
tice
s L
(4,2
),K
(�2,
�1)
,an
d L
(2,�
4) u
nde
r th
e M
(4,�
1),N
(0,�
1),a
nd
P(1
,4)
un
der
the
tran
slat
ion
(x,
y) →
(x�
2,y
�5)
tran
slat
ion
(x,
y) →
(x�
4,y
�3)
M�
L MN
P
L �
N�P
�
x
y
OJ�
K�
L�
J
K
L
x
y
O
ab
12
3
ab
1
23
a b
1 2 3a
b
12
3
©G
lenc
oe/M
cGra
w-H
ill48
8G
lenc
oe G
eom
etry
In e
ach
fig
ure
,c|| d
.Det
erm
ine
wh
eth
er f
igu
re 3
is
a tr
ansl
atio
n i
mag
e of
fig
ure
1.
Wri
te y
esor
no.
Exp
lain
you
r an
swer
.
1.2.
No
;it
is a
tra
nsl
atio
n o
f a
Yes;
it is
a r
efle
ctio
n o
f a
refl
ecti
on
re
flec
tio
n a
nd
is o
rien
ted
wit
h r
esp
ect
to t
he
par
alle
l lin
es.
dif
fere
ntl
y th
an f
igu
re 1
.
CO
OR
DIN
ATE
GEO
MET
RYG
rap
h e
ach
fig
ure
an
d i
ts i
mag
e u
nd
er t
he
give
ntr
ansl
atio
n.
3.qu
adri
late
ral
TU
WX
wit
h v
erti
ces
4.pe
nta
gon
DE
FG
Hw
ith
ver
tice
s D
(�1,
�2)
,T
(�1,
1),U
(4,2
),W
(1,5
),an
d X
(�1,
3)
E(2
,�1)
,F(5
,�2)
,G(4
,�4)
,H(1
,�4)
u
nde
r th
e tr
ansl
atio
n
un
der
the
tran
slat
ion
(x
,y)
→(x
�2,
y�
4)(x
,y)
→(x
�1,
y�
5)
AN
IMA
TIO
NF
ind
th
e tr
ansl
atio
n t
hat
mov
es t
he
figu
re
on t
he
coor
din
ate
pla
ne.
5.fi
gure
1 →
figu
re 2
(x�
4,y
�2)
6.fi
gure
2 →
figu
re 3
(x�
4,y
�1)
7.fi
gure
3 →
figu
re 4
(x�
3,y
�3)
x
y
O
13
4
2
D�
E�
F�
G�
H�
DE
F
GH
x
y
O
W
T�
U�
W�
X�
TU
X
x
y
O
c d
1 2 3
cd
12
3
Pra
ctic
e (
Ave
rag
e)
Tran
slat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
Answers (Lesson 9-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csTr
ansl
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
©G
lenc
oe/M
cGra
w-H
ill48
9G
lenc
oe G
eom
etry
Lesson 9-2
Pre-
Act
ivit
yH
ow a
re t
ran
slat
ion
s u
sed
in
a m
arch
ing
ban
d s
how
?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-2
at
the
top
of p
age
470
in y
our
text
book
.
How
do
ban
d di
rect
ors
get
the
mar
chin
g ba
nd
to m
ain
tain
th
e sh
ape
of t
he
figu
re t
hey
ori
gin
ally
for
med
?S
amp
le a
nsw
er:T
he
ban
d p
ract
ices
mar
chin
g in
un
iso
n,w
ith
eve
ryo
ne
mak
ing
iden
tica
l mov
es a
tth
e sa
me
tim
e.R
ead
ing
th
e Le
sso
n1.
Un
derl
ine
the
corr
ect
wor
d or
ph
rase
to
form
a t
rue
stat
emen
t.a.
All
ref
lect
ion
s an
d tr
ansl
atio
ns
are
(opp
osit
es/is
omet
ries
/equ
ival
ent)
.b
.T
he
prei
mag
e an
d im
age
of a
fig
ure
un
der
a re
flec
tion
in
a l
ine
hav
e (t
he
sam
e or
ien
tati
on/o
ppos
ite
orie
nta
tion
s).
c.T
he
prei
mag
e an
d im
age
of a
fig
ure
un
der
a tr
ansl
atio
n h
ave
(th
e sa
me
orie
nta
tion
/opp
osit
e or
ien
tati
ons)
.d
.T
he
resu
lt o
f su
cces
sive
ref
lect
ion
s ov
er t
wo
para
llel
lin
es i
s a
(ref
lect
ion
/rot
atio
n/t
ran
slat
ion
).e.
Col
lin
eari
ty (
is/i
s n
ot)
pres
erve
d by
tra
nsl
atio
ns.
f.T
he
tran
slat
ion
(x,
y) →
(x�
a,y
�b)
sh
ifts
eve
ry p
oin
t a
un
its
(hor
izon
tall
y/ve
rtic
ally
) an
d y
un
its
(hor
izon
tall
y/ve
rtic
ally
).
2.F
ind
the
imag
e of
eac
h p
reim
age
un
der
the
indi
cate
d tr
ansl
atio
n.
a.(x
,y);
5 u
nit
s ri
ght
and
3 u
nit
s u
p(x
�5,
y�
3)b
.(x
,y);
2 u
nit
s le
ft a
nd
4 u
nit
s do
wn
(x�
2,y
�4)
c.(x
,y);
1 u
nit
lef
t an
d 6
un
its
up
(x�
1,y
�6)
d.
(x,y
);7
un
its
righ
t(x
�7,
y)e.
(4,�
3);3
un
its
up
(4,0
)f.
(�5,
6);3
un
its
righ
t an
d 2
un
its
dow
n(�
2,4)
g.(�
7,5)
;7 u
nit
s ri
ght
and
5 u
nit
s do
wn
(0,0
)h
.(�
9,�
2);1
2 u
nit
s ri
ght
and
6 u
nit
s do
wn
(3,�
8)
3.�
RS
Th
as v
erti
ces
R(�
3,3)
,S(0
,�2)
,an
d T
(2,1
).G
raph
�R
ST
and
its
imag
e �
R�S
�T�
unde
r th
e tr
ansl
atio
n (x
,y)
→(x
�3,
y�
2).
Lis
t th
e co
ordi
nat
es o
f th
e ve
rtic
es o
f th
e im
age.
R�(
0,1)
,S�(
3,�
4),T
�(5,
�1)
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al t
erm
is
to r
elat
e it
to
an e
very
day
mea
ning
of t
he
sam
e w
ord.
How
is
the
mea
nin
g of
tra
nsl
atio
nin
geo
met
ry r
elat
ed t
o th
e id
ea o
ftr
ansl
atio
nfr
om o
ne
lan
guag
e to
an
oth
er?
Sam
ple
an
swer
:Wh
en y
ou
tra
nsl
ate
fro
m o
ne
lan
gu
age
to a
no
ther
,yo
u c
arry
ove
rth
e m
ean
ing
fro
m o
ne
lan
gu
age
to a
no
ther
.Wh
en y
ou
tra
nsl
ate
a g
eom
etri
c fig
ure
,yo
u c
arry
over
the
figu
re f
rom
on
e p
osi
tion
to
an
oth
er w
itho
ut
chan
gin
g it
s b
asic
R�
S�
T�
R
S
T
x
y
O
©G
lenc
oe/M
cGra
w-H
ill49
0G
lenc
oe G
eom
etry
Tran
slat
ion
s in
Th
e C
oo
rdin
ate
Pla
ne
You
can
use
alg
ebra
ic d
escr
ipti
ons
of r
efle
ctio
ns
to s
how
th
atth
e co
mp
osit
e of
tw
o re
flec
tion
s w
ith
res
pec
t to
par
alle
l li
nes
is
a tr
ansl
atio
n (
that
is,
a sl
ide)
.
1.S
upp
ose
aan
d b
are
two
diff
eren
t re
al n
um
bers
.Let
San
d T
be t
he
foll
owin
g re
flec
tion
s.
S:(
x,y)
→(�
x�
2a,
y)T
:(x,
y) →
(�x
�2
b,y)
Sis
ref
lect
ion
wit
h r
espe
ct t
o th
e li
ne
wit
h e
quat
ion
x�
a,an
d T
isre
flec
tion
wit
h r
espe
ct t
o th
e li
ne
wit
h e
quat
ion
x�
b.
a.F
ind
an a
lgeb
raic
des
crip
tion
(si
mil
ar t
o th
ose
abov
e fo
r S
and
T)
to d
escr
ibe
the
com
posi
te t
ran
sfor
mat
ion
“S
foll
owed
by
T.”
(x,y
) →
(x�
2 (b
�a)
,y)
b.F
ind
an a
lgeb
raic
des
crip
tion
for
th
e co
mpo
site
tra
nsf
orm
atio
n“T
foll
owed
by
S.”
(x,y
) →
(x�
2 (a
�b
),y
)
2.T
hin
k ab
out
the
resu
lts
you
obt
ain
ed i
n E
xerc
ise
1.W
hat
do
they
tell
you
abo
ut
how
th
e di
stan
ce b
etw
een
tw
o pa
rall
el l
ines
is
rela
ted
to t
he
dist
ance
bet
wee
n a
pre
imag
e an
d im
age
poin
t fo
r a
com
posi
te o
f re
flec
tion
s w
ith
res
pect
to
thes
e li
nes
?
Th
e d
ista
nce
bet
wee
n a
pre
imag
e an
d it
s im
age
po
int
istw
ice
the
dis
tan
ce b
etw
een
th
e p
aral
lel l
ines
.
3.Il
lust
rate
you
r an
swer
s to
Exe
rcis
es 1
an
d 2
wit
h s
ketc
hes
.Use
ase
para
te s
hee
t if
nec
essa
ry.
See
stu
den
ts’w
ork
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-2
9-2
Answers (Lesson 9-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Ro
tati
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
©G
lenc
oe/M
cGra
w-H
ill49
1G
lenc
oe G
eom
etry
Lesson 9-3
Dra
w R
ota
tio
ns
A t
ran
sfor
mat
ion
cal
led
a ro
tati
ontu
rns
a fi
gure
th
rou
gh a
spe
cifi
edan
gle
abou
t a
fixe
d po
int
call
ed t
he
cen
ter
of r
otat
ion
.To
fin
d th
e im
age
of a
rot
atio
n,o
ne
way
is
to u
se a
pro
trac
tor.
An
oth
er w
ay i
s to
ref
lect
a f
igu
re t
wic
e,in
tw
o in
ters
ecti
ng
lin
es.
�A
BC
has
ver
tice
s A
(2,1
),B
(3,4
),an
d
C(5
,1).
Dra
w t
he
imag
e of
�A
BC
un
der
a r
otat
ion
of
90°
cou
nte
rclo
ckw
ise
abou
t th
e or
igin
.•
Fir
st d
raw
�A
BC
.Th
en d
raw
a s
egm
ent
from
O,t
he
orig
in,
to p
oin
t A
.•
Use
a p
rotr
acto
r to
mea
sure
90°
cou
nte
rclo
ckw
ise
wit
h O �
A�as
on
e si
de.
•D
raw
OR
� �� .
•U
se a
com
pass
to
copy
O �A�
onto
OR
��� .
Nam
e th
e se
gmen
t O�
A���.
•R
epea
t w
ith
seg
men
ts f
rom
th
e or
igin
to
poin
ts B
and
C.
Fin
d t
he
imag
e of
�A
BC
un
der
ref
lect
ion
in
lin
es m
and
n.
Fir
st r
efle
ct �
AB
Cin
lin
e m
.Lab
el t
he
imag
e �
A�B
�C�.
Ref
lect
�A�
B�C
�in
lin
e n.
Lab
el t
he
imag
e �
A�B
�C�.
�A
�B�C
�is
a r
otat
ion
of
�A
BC
.Th
e ce
nte
r of
rot
atio
n i
s th
ein
ters
ecti
on o
f li
nes
man
d n.
Th
e an
gle
of r
otat
ion
is
twic
e th
em
easu
re o
f th
e ac
ute
an
gle
form
ed b
y m
and
n.
Dra
w t
he
rota
tion
im
age
of e
ach
fig
ure
90°
in t
he
give
n d
irec
tion
ab
out
the
cen
ter
poi
nt
and
lab
el t
he
coor
din
ates
.
1.P�
Q�w
ith
en
dpoi
nts
P(�
1,�
2)
2.�
PQ
Rw
ith
ver
tice
s P
(�2,
�3)
,Q(2
,�1)
,an
d Q
(1,3
) co
un
terc
lock
wis
e an
d R
(3,2
) cl
ockw
ise
abou
t th
e po
int
T(1
,1)
abou
t th
e or
igin
Fin
d t
he
rota
tion
im
age
of e
ach
fig
ure
by
refl
ecti
ng
it i
n l
ine
man
d t
hen
in
lin
e n
.
3.4.
mn
B�
C�
A�
AB
C
B�
C�
A�
mn
Q�
P�
Q
PQ
�
P�
x
yP
�
Q�
R�
P
Q
RT
x
y
P�
Q�
P
Q
m
n
A� B
�C
�
A�
B�
C�
B
C
A
x
y
O
A�
B�
C�
A
B
C
R
Exer
cises
Exer
cises
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
©G
lenc
oe/M
cGra
w-H
ill49
2G
lenc
oe G
eom
etry
Ro
tati
on
al S
ymm
etry
Wh
en t
he
figu
re a
t th
e ri
ght
is r
otat
ed
abou
t po
int
Pby
120
°or
240
°,th
e im
age
look
s li
ke t
he
prei
mag
e.T
he
figu
re h
as r
otat
ion
al s
ymm
etry
,wh
ich
mea
ns
it c
an b
e ro
tate
d le
ss
than
360
°ab
out
a po
int
and
the
prei
mag
e an
d im
age
appe
ar t
o be
th
e sa
me.
Th
e fi
gure
has
rot
atio
nal
sym
met
ry o
f or
der
3 be
cau
se t
her
e ar
e 3
rota
tion
s le
ss t
han
360
°(0
°,12
0°,2
40°)
th
at p
rodu
ce a
n i
mag
e th
at
is t
he
sam
e as
th
e or
igin
al.T
he
mag
nit
ud
eof
th
e ro
tati
onal
sym
met
ry
for
a fi
gure
is
360
degr
ees
divi
ded
by t
he
orde
r.F
or t
he
figu
re a
bove
,th
e ro
tati
onal
sym
met
ry h
as m
agn
itu
de 1
20 d
egre
es.
Iden
tify
th
e or
der
an
d m
agn
itu
de
of t
he
rota
tion
al
sym
met
ry o
f th
e d
esig
n a
t th
e ri
ght.
Th
e de
sign
has
rot
atio
nal
sym
met
ry a
bou
t th
e ce
nte
r po
int
for
rota
tion
s of
0°,
45°,
90°,
135°
,180
°,22
5°,2
70°,
and
315°
.
Th
ere
are
eigh
t ro
tati
ons
less
th
an 3
60 d
egre
es,s
o th
e or
der
of i
ts
rota
tion
al s
ymm
etry
is
8.T
he
quot
ien
t 36
0 �
8 is
45,
so t
he
mag
nit
ude
of
its
rot
atio
nal
sym
met
ry i
s 45
deg
rees
.
Iden
tify
th
e or
der
an
d m
agn
itu
de
of t
he
rota
tion
al s
ymm
etry
of
each
fig
ure
.
1.a
squ
are
2.a
regu
lar
40-g
on
ord
er:
4;m
agn
itu
de:
90°
ord
er:
40;
mag
nit
ud
e:9°
3.4.
ord
er:
2;m
agn
itu
de:
180°
ord
er:
5;m
agn
itu
de:
72°
5.6.
ord
er:
16;
mag
nit
ud
e:22
.5°
ord
er:
3;m
agn
itu
de:
120°
P
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Ro
tati
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 9-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Ro
tati
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
©G
lenc
oe/M
cGra
w-H
ill49
3G
lenc
oe G
eom
etry
Lesson 9-3
Rot
ate
each
fig
ure
ab
out
poi
nt
Ru
nd
er t
he
give
n a
ngl
e of
rot
atio
n a
nd
th
e gi
ven
dir
ecti
on.L
abel
th
e ve
rtic
es o
f th
e ro
tati
on i
mag
e.
1.90
°co
un
terc
lock
wis
e2.
90°
cloc
kwis
e
CO
OR
DIN
ATE
GEO
MET
RYD
raw
th
e ro
tati
on i
mag
e of
eac
h f
igu
re 9
0°in
th
e gi
ven
dir
ecti
on a
bou
t th
e or
igin
an
d l
abel
th
e co
ord
inat
es.
3.�
ST
Ww
ith
ver
tice
s S
(2,�
1),T
(5,1
),4.
�D
EF
wit
h v
erti
ces
D(�
4,3)
,E(1
,2),
and
W(3
,3)
cou
nte
rclo
ckw
ise
and
F(�
3,�
3) c
lock
wis
e
Use
a c
omp
osit
ion
of
refl
ecti
ons
to f
ind
th
e ro
tati
on i
mag
e w
ith
res
pec
t to
lin
es k
and
m.T
hen
fin
d t
he
angl
e of
rot
atio
n f
or e
ach
im
age.
5.6.
120°
clo
ckw
ise
140°
cou
nte
rclo
ckw
ise
k mL�
M�
N� O
�
LM N
Ok m
A�
B�C�A
BC
E�
D�
F�
F
ED
x
y
OS
�
T�
W�
S
T
W
x
y
O
GH
KJ
R
G�
H�
J�K�
Q
PS
R
Q�
S�
P�
©G
lenc
oe/M
cGra
w-H
ill49
4G
lenc
oe G
eom
etry
Rot
ate
each
fig
ure
ab
out
poi
nt
Ru
nd
er t
he
give
n a
ngl
e of
rot
atio
n a
nd
th
e gi
ven
dir
ecti
on.L
abel
th
e ve
rtic
es o
f th
e ro
tati
on i
mag
e.
1.80
°co
un
terc
lock
wis
e2.
100°
cloc
kwis
e
CO
OR
DIN
ATE
GEO
MET
RYD
raw
th
e ro
tati
on i
mag
e of
eac
h f
igu
re 9
0°in
th
e gi
ven
dir
ecti
on a
bou
t th
e ce
nte
r p
oin
t an
d l
abel
th
e co
ord
inat
es.
3.�
RS
Tw
ith
ver
tice
s R
(�3,
3),S
(2,4
),4.
�H
JK
wit
h v
erti
ces
H(3
,1),
J(3
,�3)
,an
d T
(1,2
) cl
ockw
ise
abou
t th
e an
d K
(�3,
�4)
cou
nte
rclo
ckw
ise
abou
t po
int
P(1
,0)
the
poin
t P
(�1,
�1)
Use
a c
omp
osit
ion
of
refl
ecti
ons
to f
ind
th
e ro
tati
on i
mag
e w
ith
res
pec
t to
lin
es p
and
s.T
hen
fin
d t
he
angl
e of
rot
atio
n f
or e
ach
im
age.
5.6.
160°
cou
nte
rclo
ckw
ise
100°
cou
nte
rclo
ckw
ise
7.ST
EAM
BO
ATS
A p
addl
e w
hee
l on
a s
team
boat
is
driv
en b
y a
stea
m e
ngi
ne
and
mov
esfr
om o
ne
padd
le t
o th
e n
ext
to p
rope
l th
e bo
at t
hro
ugh
th
e w
ater
.If
a pa
ddle
wh
eel
con
sist
s of
18
even
ly s
pace
d pa
ddle
s,id
enti
fy t
he
orde
r an
d m
agn
itu
de o
f it
s ro
tati
onal
sym
met
ry.
ord
er 1
8 an
d m
agn
itu
de
20°
ps
P�
R�
S�
T�
P
R
S
T
ps
E�
F�
G�
EF
G
K�
J�H
�
JH
K
x
y
OP
( –1,
–1)
S�
R�
T�
T
S
R
x
y
OP
( 1, 0
)
QU
TS
P
RQ�
S�T
�U
� P�
N
MP
R
M�
N�
P�
Pra
ctic
e (
Ave
rag
e)
Ro
tati
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
Answers (Lesson 9-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csR
ota
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
©G
lenc
oe/M
cGra
w-H
ill49
5G
lenc
oe G
eom
etry
Lesson 9-3
Pre-
Act
ivit
yH
ow d
o so
me
amu
sem
ent
rid
es i
llu
stra
te r
otat
ion
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-3
at
the
top
of p
age
476
in y
our
text
book
.
Wh
at a
re t
wo
way
s th
at e
ach
car
rot
ates
?
Eac
h c
ar s
pin
s ar
ou
nd
its
ow
n c
ente
r an
d e
ach
car
ro
tate
sar
ou
nd
a p
oin
t in
th
e ce
nte
r o
f th
e ci
rcu
lar
trac
k.
Rea
din
g t
he
Less
on
1.L
ist
all
of t
he
foll
owin
g ty
pes
of t
ran
sfor
mat
ion
s th
at s
atis
fy e
ach
des
crip
tion
:ref
lect
ion
,tr
ansl
atio
n,r
otat
ion
.a.
Th
e tr
ansf
orm
atio
n i
s an
iso
met
ry.
refl
ecti
on
,tra
nsl
atio
n,r
ota
tio
nb
.T
he
tran
sfor
mat
ion
pre
serv
es t
he
orie
nta
tion
of
a fi
gure
.tr
ansl
atio
n,r
ota
tio
nc.
Th
e tr
ansf
orm
atio
n i
s th
e co
mpo
site
of
succ
essi
ve r
efle
ctio
ns
over
tw
o in
ters
ecti
ng
lin
es.
rota
tio
nd
.T
he
tran
sfor
mat
ion
is
the
com
posi
te o
f su
cces
sive
ref
lect
ion
s ov
er t
wo
para
llel
li
nes
.tr
ansl
atio
ne.
A s
peci
fic
tran
sfor
mat
ion
is
defi
ned
by
a fi
xed
poin
t an
d a
spec
ifie
d an
gle.
rota
tio
nf.
A s
peci
fic
tran
sfor
mat
ion
is
defi
ned
by
a fi
xed
poin
t,a
fixe
d li
ne,
or a
fix
ed p
lan
e.re
flec
tio
ng.
A s
peci
fic
tran
sfor
mat
ion
is
defi
ned
by
(x,y
) →
(x�
a,x
�b)
,for
fix
ed v
alu
es o
f a
and
b.tr
ansl
atio
nh
.T
he
tran
sfor
mat
ion
is
also
cal
led
a sl
ide.
tran
slat
ion
i.T
he
tran
sfor
mat
ion
is
also
cal
led
a fl
ip.
refl
ecti
on
j.T
he
tran
sfor
mat
ion
is
also
cal
led
a tu
rn.
rota
tio
n
2.D
eter
min
e th
e or
der
and
mag
nit
ude
of
the
rota
tion
al s
ymm
etry
for
eac
h f
igu
re.
a.b
.
ord
er 3
;m
agn
itu
de
120°
ord
er 2
;m
agn
itu
de
180°
c.d
.
ord
er 5
;m
agn
itu
de
72°
ord
er 8
;m
agn
itu
de
45°
Hel
pin
g Y
ou
Rem
emb
er
3.W
hat
is
an e
asy
way
to
rem
embe
r th
e or
der
and
mag
nit
ude
of
the
rota
tion
al s
ymm
etry
of a
reg
ula
r po
lygo
n?
Sam
ple
an
swer
:Th
e o
rder
is t
he
sam
e as
th
e n
um
ber
of
sid
es.T
o f
ind
th
em
agn
itu
de,
div
ide
360
by t
he
nu
mb
er o
f si
des
.
©G
lenc
oe/M
cGra
w-H
ill49
6G
lenc
oe G
eom
etry
Fin
din
g t
he
Cen
ter
of
Ro
tati
on
Su
ppos
e yo
u a
re t
old
that
�X
�Y�Z
�is
th
e ro
tati
on i
mag
e of
�X
YZ
,bu
t yo
u a
re n
ot t
old
wh
ere
the
cen
ter
of r
otat
ion
is
nor
th
e m
easu
re
of t
he
angl
e of
rot
atio
n.C
an y
ou f
ind
them
? Ye
s,yo
u c
an.C
onn
ect
two
pair
s of
cor
resp
ondi
ng
vert
ices
wit
h s
egm
ents
.In
th
e fi
gure
,th
e se
gmen
ts Y
Y�
and
ZZ
�ar
e u
sed.
Dra
w t
he
perp
endi
cula
r bi
sect
ors,
�an
d m
,of
thes
ese
gmen
ts.T
he
poin
t C
wh
ere
�an
d m
inte
rsec
t is
th
e ce
nte
r of
rot
atio
n.
1.H
ow c
an y
ou f
ind
the
mea
sure
of
the
angl
e of
rot
atio
n i
n t
he
figu
re a
bove
?D
raw
�X
CX
�(�
YC
Y�
or
�Z
CZ
�w
ou
ld
also
do
) an
d m
easu
re it
wit
h a
pro
trac
tor.
Loc
ate
the
cen
ter
of r
otat
ion
for
th
e ro
tati
on t
hat
map
s W
XY
Zon
to W
�X�Y
�Z�.
Th
en f
ind
th
e m
easu
re o
f th
e an
gle
of r
otat
ion
.
2.3.
Th
e se
gm
ents
stu
den
ts c
ho
ose
to
bis
ect
may
var
y.S
ee s
tud
ents
’wo
rk.
Z
Z�
W
W�
X�
X
Y
Y�
cen
ter
of
rota
tio
n
ang
le o
fro
tati
on
:ab
ou
t 11
0
Z�
Z
W
W�
X�
X
Y�
Y
cen
ter
of
rota
tio
n
ang
le o
fro
tati
on
:ab
ou
t 10
0
Z�
Z
C
X�
X
Y�
Y
m�
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-3
9-3
Answers (Lesson 9-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Tess
ella
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
©G
lenc
oe/M
cGra
w-H
ill49
7G
lenc
oe G
eom
etry
Lesson 9-4
Reg
ula
r Te
ssel
lati
on
sA
pat
tern
th
at c
over
s a
plan
e w
ith
rep
eati
ng
copi
es o
f on
e or
mor
e fi
gure
s so
th
at t
her
e ar
e n
o ov
erla
ppin
g or
em
pty
spac
es i
s a
tess
ella
tion
.A r
egu
lar
tess
ella
tion
uses
onl
y on
e ty
pe o
f re
gula
r po
lygo
n.In
a t
esse
llat
ion,
the
sum
of
the
mea
sure
sof
the
ang
les
of t
he p
olyg
ons
surr
ound
ing
a ve
rtex
is 3
60.I
f a
regu
lar
poly
gon
has
an
in
teri
oran
gle
that
is
a fa
ctor
of
360,
then
th
e po
lygo
n w
ill
tess
ella
te.
regu
lar
tess
ella
tion
tess
ella
tion
Cop
ies
of a
reg
ular
hex
agon
C
opie
s of
a r
egul
ar p
enta
gon
can
form
a t
esse
llatio
n.ca
nnot
for
m a
tes
sella
tion.
Det
erm
ine
wh
eth
er a
reg
ula
r 16
-gon
tes
sell
ates
th
e p
lan
e.E
xpla
in.
If m
�1
is t
he
mea
sure
of
one
inte
rior
an
gle
of a
reg
ula
r po
lygo
n,t
hen
a f
orm
ula
for
m�
1
is m
�1
��18
0(n n
�2)
�.U
se t
he
form
ula
wit
h n
�16
.
m�
1��18
0(n n
�2)
�
��18
0(1 16 6
�2)
�
�15
7.5
Th
e va
lue
157.
5 is
not
a f
acto
r of
360
,so
the
16-g
on w
ill
not
tes
sell
ate.
Det
erm
ine
wh
eth
er e
ach
pol
ygon
tes
sell
ates
th
e p
lan
e.If
so,
dra
w a
sam
ple
fig
ure
.
1.sc
alen
e ri
ght
tria
ngl
eye
s2.
isos
cele
s tr
apez
oid
yes
Det
erm
ine
wh
eth
er e
ach
reg
ula
r p
olyg
on t
esse
llat
es t
he
pla
ne.
Exp
lain
.
3.sq
uar
e4.
20-g
on
Yes;
the
mea
sure
of
each
inte
rio
r N
o;
the
mea
sure
of
each
inte
rio
r an
gle
is 9
0,an
d 9
0 is
a f
acto
r an
gle
is 1
62,a
nd
162
is n
ot
a fa
cto
r o
f 36
0.o
f 36
0.
5.se
ptag
on6.
15-g
on
No
;th
e m
easu
re o
f ea
ch in
teri
or
No
;th
e m
easu
re o
f ea
ch in
teri
or
ang
le is
128
.6,a
nd
128
.6 is
no
t a
ang
le is
156
,an
d 1
56 is
no
t a
fact
or
fact
or
of
360.
of
360.
7.oc
tago
n8.
pen
tago
n
No
;th
e m
easu
re o
f ea
ch in
teri
or
No
;th
e m
easu
re o
f ea
ch in
teri
or
ang
le is
135
,an
d 1
35 is
no
t a
ang
le is
108
,an
d 1
08 is
no
t a
fact
or
fact
or
of
360.
of
360.
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill49
8G
lenc
oe G
eom
etry
Tess
ella
tio
ns
wit
h S
pec
ific
Att
rib
ute
sA
tes
sell
atio
n p
atte
rn c
an c
onta
in a
ny
type
of p
olyg
on.I
f th
e ar
ran
gem
ent
of s
hap
es a
nd
angl
es a
t ea
ch v
erte
x in
th
e te
ssel
lati
on i
s th
esa
me,
the
tess
ella
tion
is
un
ifor
m.A
sem
i-re
gula
r te
ssel
lati
onis
a u
nif
orm
tes
sell
atio
nth
at c
onta
ins
two
or m
ore
regu
lar
poly
gon
s.
Det
erm
ine
wh
eth
er a
kit
e w
ill
tess
ella
te t
he
pla
ne.
If s
o,d
escr
ibe
the
tess
ella
tion
as
un
ifor
m,r
egu
lar,
sem
i-re
gula
r,or
not
un
ifor
m.
A k
ite
wil
l te
ssel
late
th
e pl
ane.
At
each
ver
tex
the
sum
of
th
e m
easu
res
is a
�b
�b
�c,
wh
ich
is
360.
Th
e te
ssel
lati
on i
s u
nif
orm
.
Det
erm
ine
wh
eth
er a
sem
i-re
gula
r te
ssel
lati
on c
an b
e cr
eate
d f
rom
eac
h s
et o
ffi
gure
s.If
so,
sket
ch t
he
tess
ella
tion
.Ass
um
e th
at e
ach
fig
ure
has
a s
ide
len
gth
of
1 u
nit
.
1.rh
ombu
s,eq
uil
ater
al t
rian
gle,
2.sq
uar
e an
d eq
uil
ater
al t
rian
gle
and
octa
gon
no
yes
Det
erm
ine
wh
eth
er e
ach
pol
ygon
tes
sell
ates
th
e p
lan
e.If
so,
des
crib
e th
ete
ssel
lati
on a
s u
nif
orm
,not
un
ifor
m,r
egu
lar,
or s
emi-
regu
lar.
3.re
ctan
gle
4.h
exag
on a
nd
squ
are
yes,
un
iform
no
c�a�
b� b�
c�c�
a�
a�b�
b� b�
c�a�
b� b�
c�a�
b� b�
c�a�
b� b�
b�
c�a�
b� b�
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Tess
ella
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 9-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
Skil
ls P
ract
ice
Tess
ella
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
©G
lenc
oe/M
cGra
w-H
ill49
9G
lenc
oe G
eom
etry
Lesson 9-4
Det
erm
ine
wh
eth
er e
ach
reg
ula
r p
olyg
on t
esse
llat
es t
he
pla
ne.
Exp
lain
.
1.15
-gon
2.18
-gon
no
,in
teri
or
ang
le �
156
no
,in
teri
or
ang
le �
160
3.sq
uar
e4.
20-g
onye
s,in
teri
or
ang
le �
90n
o,i
nte
rio
r an
gle
�16
2
Det
erm
ine
wh
eth
er a
sem
i-re
gula
r te
ssel
lati
on c
an b
e cr
eate
d f
rom
eac
h s
et o
ffi
gure
s.A
ssu
me
each
fig
ure
has
a s
ide
len
gth
of
1 u
nit
.
5.re
gula
r pe
nta
gon
s an
d eq
uil
ater
al t
rian
gles
no
6.re
gula
r do
deca
gon
s an
d eq
uil
ater
al t
rian
gles
yes
7.re
gula
r oc
tago
ns
and
equ
ilat
eral
tri
angl
esn
o
Det
erm
ine
wh
eth
er e
ach
pol
ygon
tes
sell
ates
th
e p
lan
e.If
so,
des
crib
e th
ete
ssel
lati
on a
s u
nif
orm
,not
un
ifor
m,r
egu
lar,
or s
emi-
regu
lar.
8.rh
ombu
s9.
isos
cele
s tr
apez
oid
and
squ
are
yes;
un
iform
no
Det
erm
ine
wh
eth
er e
ach
pat
tern
is
a te
ssel
lati
on.I
f so
,des
crib
e it
as
un
ifor
m,n
otu
nif
orm
,reg
ula
r,or
sem
i-re
gula
r.
10.
11.
yes;
un
iform
,sem
i-re
gu
lar
yes;
un
iform
©G
lenc
oe/M
cGra
w-H
ill50
0G
lenc
oe G
eom
etry
Det
erm
ine
wh
eth
er e
ach
reg
ula
r p
olyg
on t
esse
llat
es t
he
pla
ne.
Exp
lain
.
1.22
-gon
2.40
-gon
no
,in
teri
or
ang
le ≈
163.
6n
o,i
nte
rio
r an
gle
�17
1
Det
erm
ine
wh
eth
er a
sem
i-re
gula
r te
ssel
lati
on c
an b
e cr
eate
d f
rom
eac
h s
et o
ffi
gure
s.A
ssu
me
each
fig
ure
has
a s
ide
len
gth
of
1 u
nit
.
3.re
gula
r pe
nta
gon
s an
d re
gula
r de
cago
ns
no
(w
ill n
ot
tess
ella
te t
he
pla
ne)
4.re
gula
r do
deca
gon
s,re
gula
r h
exag
ons,
and
squ
ares
yes
Det
erm
ine
wh
eth
er e
ach
pol
ygon
tes
sell
ates
th
e p
lan
e.If
so,
des
crib
e th
ete
ssel
lati
on a
s u
nif
orm
,not
un
ifor
m,r
egu
lar,
or s
emi-
regu
lar.
5.ki
te6.
octa
gon
an
d de
cago
n
yes;
un
iform
no
Det
erm
ine
wh
eth
er e
ach
pat
tern
is
a te
ssel
lati
on.I
f so
,des
crib
e it
as
un
ifor
m,n
otu
nif
orm
,reg
ula
r,or
sem
i-re
gula
r.
7.8.
yes;
un
iform
,sem
i-re
gu
lar
yes,
no
t u
nifo
rm
FLO
OR
TIL
ESF
or E
xerc
ises
9 a
nd
10,
use
th
e fo
llow
ing
info
rmat
ion
.M
r.M
arti
nez
ch
ose
the
patt
ern
of
tile
sh
own
to
reti
le h
is k
itch
en f
loor
.
9.D
eter
min
e w
het
her
th
e pa
tter
n i
s a
tess
ella
tion
.Exp
lain
Yes;
sin
ce t
her
e ar
e 2
squ
ares
an
d 3
tri
ang
les
at e
ach
ve
rtex
,th
e su
m o
f th
e an
gle
s at
th
e ve
rtic
es is
360
°.
10.I
s th
e pa
tter
n u
nif
orm
,reg
ula
r,or
sem
i-re
gula
r?u
nifo
rm a
nd
sem
i-re
gu
lar
Pra
ctic
e (
Ave
rag
e)
Tess
ella
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
Answers (Lesson 9-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csTe
ssel
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
©G
lenc
oe/M
cGra
w-H
ill50
1G
lenc
oe G
eom
etry
Lesson 9-4
Pre-
Act
ivit
yH
ow a
re t
esse
llat
ion
s u
sed
in
art
?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-4
at
the
top
of p
age
483
in y
our
text
book
.
•In
th
e pa
tter
n s
how
n i
n t
he
pict
ure
in
you
r te
xtbo
ok,h
ow m
any
smal
leq
uil
ater
al t
rian
gles
mak
e u
p on
e re
gula
r h
exag
on?
6•
In t
his
pat
tern
,how
man
y fi
sh m
ake
up
one
equ
ilat
eral
tri
angl
e?1�
1 2�
Rea
din
g t
he
Less
on
1.U
nde
rlin
e th
e co
rrec
t w
ord,
phra
se,o
r n
um
ber
to f
orm
a t
rue
stat
emen
t.
a.A
tes
sell
atio
n i
s a
patt
ern
th
at c
over
s a
plan
e w
ith
th
e sa
me
figu
re o
r se
t of
fig
ure
s so
that
th
ere
are
no
(con
gru
ent
angl
es/o
verl
appi
ng
or e
mpt
y sp
aces
/rig
ht
angl
es).
b.
A t
esse
llat
ion
th
at u
ses
only
on
e ty
pe o
f re
gula
r po
lygo
n i
s ca
lled
a(u
nif
orm
/reg
ula
r/se
mi-
regu
lar)
tes
sell
atio
n.
c.T
he
sum
of
the
mea
sure
s of
th
e an
gles
at
any
vert
ex i
n a
ny
tess
ella
tion
is
(90/
180/
360)
.
d.
A t
esse
llat
ion
th
at c
onta
ins
the
sam
e ar
ran
gem
ent
of s
hap
es a
nd
angl
es a
t ev
ery
vert
ex i
s ca
lled
a (
un
ifor
m/r
egu
lar/
sem
i-re
gula
r) t
esse
llat
ion
.
e.In
a r
egu
lar
tess
ella
tion
mad
e u
p of
hex
agon
s,th
ere
are
(3/4
/6)
hex
agon
s m
eeti
ng
atea
ch v
erte
x,an
d th
e m
easu
re o
f ea
ch o
f th
e an
gles
at
any
vert
ex i
s (6
0/90
/120
).
f.A
un
ifor
m t
esse
llat
ion
for
med
usi
ng
two
or m
ore
regu
lar
poly
gon
s is
cal
led
a(r
otat
ion
al/r
egu
lar/
sem
i-re
gula
r) t
esse
llat
ion
.
g.In
a r
egu
lar
tess
ella
tion
mad
e u
p of
tri
angl
es,t
her
e ar
e (3
/4/6
) tr
ian
gles
mee
tin
g at
each
ver
tex,
and
the
mea
sure
of
each
of
the
angl
es a
t an
y ve
rtex
is
(30/
60/1
20).
h.
If a
reg
ula
r te
ssel
lati
on i
s m
ade
up
of q
uad
rila
tera
ls,a
ll o
f th
e qu
adri
late
rals
mu
st b
eco
ngr
uen
t (r
ecta
ngl
es/p
aral
lelo
gram
s/sq
uar
es/t
rape
zoid
s).
2.W
rite
all
of
the
foll
owin
g w
ords
th
at d
escr
ibe
each
tes
sell
atio
n:u
nif
orm
,non
-un
ifor
m,
regu
lar,
sem
i-re
gula
r.
a.n
on
-un
iform
b.
un
iform
,sem
i-re
gu
lar
Hel
pin
g Y
ou
Rem
emb
er
3.O
ften
th
e ev
eryd
ay m
ean
ings
of
a w
ord
can
hel
p yo
u t
o re
mem
ber
its
mat
hem
atic
alm
ean
ing.
Loo
k u
p u
nif
orm
in y
our
dict
ion
ary.
How
can
its
eve
ryda
y m
ean
ings
hel
p yo
uto
rem
embe
r th
e m
ean
ing
of a
un
ifor
mte
ssel
lati
on?
Sam
ple
an
swer
:Th
ree
sim
ilar
mea
nin
gs
of
un
iform
are
unv
aryi
ng
,co
nsi
sten
t,an
d id
enti
cal.
In a
un
iform
tes
sella
tio
n,t
he
arra
ng
emen
t o
f sh
apes
an
d a
ng
les
at e
ach
vert
ex is
th
e sa
me.
Eac
h o
f th
e th
ree
ever
yday
mea
nin
gs
des
crib
es t
his
situ
atio
n.
©G
lenc
oe/M
cGra
w-H
ill50
2G
lenc
oe G
eom
etry
Po
lyg
on
al N
um
ber
sC
erta
in n
um
bers
rel
ated
to
regu
lar
poly
gon
s ar
e ca
lled
pol
ygon
al n
um
ber
s.T
he
char
tsh
ows
seve
ral
tria
ngu
lar,
squ
are,
and
pen
tago
nal
nu
mbe
rs.T
he
ran
kof
a p
olyg
on n
um
ber
is t
he
nu
mbe
r of
dot
s on
eac
h “
side
”of
th
e ou
ter
poly
gon
.For
exa
mpl
e,th
e pe
nta
gon
aln
um
ber
22 h
as a
ran
k of
4.
Pol
ygon
al n
um
bers
can
be
desc
ribe
d w
ith
for
mu
las.
For
exa
mpl
e,a
tria
ngu
lar
nu
mbe
r T
of r
ank
r ca
n b
e de
scri
bed
by T
��r(
r2�
1)�
.
An
swer
eac
h q
ues
tion
.
1.D
raw
a d
iagr
am t
o fi
nd
the
2.D
raw
a d
iagr
am t
o fi
nd
the
tria
ngu
lar
nu
mbe
r of
ran
k 5.
15pe
nta
gon
al n
um
ber
of r
ank
5.35
3.W
rite
a f
orm
ula
for
a s
quar
e n
um
ber
4.W
rite
a f
orm
ula
for
a p
enta
gon
alS
of r
ank
r.n
um
ber
Pof
ran
k r.
S�
r2P
��r(
3r2�
1)�
5.W
hat
is
the
ran
k of
th
e pe
nta
gon
al6.
Lis
t th
e h
exag
onal
nu
mbe
rs f
or r
anks
nu
mbe
r 70
?7
1 to
5.(
Hin
t:D
raw
a d
iagr
am.)
1,6,
15,2
8,45
Ran
k 1
Tria
ngle
13
610
14
916
15
1222
Squ
are
Pen
tago
n
Ran
k 2
Ran
k 3
Ran
k 4
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-4
9-4
Answers (Lesson 9-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Dila
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
©G
lenc
oe/M
cGra
w-H
ill50
3G
lenc
oe G
eom
etry
Lesson 9-5
Cla
ssif
y D
ilati
on
sA
dil
atio
nis
a t
ran
sfor
mat
ion
in
wh
ich
th
e im
age
may
be
adi
ffer
ent
size
th
an t
he
prei
mag
e.A
dil
atio
n r
equ
ires
a c
ente
r po
int
and
a sc
ale
fact
or,r
.
Let
rre
pres
ent
the
scal
e fa
ctor
of
a di
latio
n.
If | r
| �1,
the
n th
e di
latio
n is
an
enla
rgem
ent.
If | r
| �1,
the
n th
e di
latio
n is
a c
ongr
uenc
e tr
ansf
orm
atio
n.
If 0
| r
| 1,
the
n th
e di
latio
n is
a r
educ
tion.
Dra
w t
he
dil
atio
n i
mag
e of
�
AB
Cw
ith
cen
ter
Oan
d r
�2.
Dra
w O
A� �
� ,O
B��
� ,an
d O
C��
� .L
abel
poi
nts
A�,
B�,
and
C�
so t
hat
OA�
�2(
OA
),O
B�
�2(
OB
),an
d O
C�
�2(
OC
).�
A�B
�C�
is a
dil
atio
n o
f �
AB
C.
Dra
w t
he
dil
atio
n i
mag
e of
eac
h f
igu
re w
ith
cen
ter
Can
d t
he
give
n s
cale
fac
tor.
Des
crib
e ea
ch t
ran
sfor
mat
ion
as
an e
nla
rgem
ent,
con
gru
ence
,or
red
uct
ion
.
1.r
�2
enla
rgem
ent
2.r
��1 2�
red
uct
ion
3.r
�1
con
gru
ence
4.r
�3
enla
rgem
ent
5.r
��2 3�
red
uct
ion
6.r
�1
con
gru
ence
KJ
HF
C
G
M C
P
R V
W
M�
P� R
� V�
W�
CS
TS
�
T�
C
W Y
XZ
C
ST
RS
�T
�
R�
CB
A A�
B�
O
B A
CO
B A
C
A�
B�
C�
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill50
4G
lenc
oe G
eom
etry
Iden
tify
th
e Sc
ale
Fact
or
If y
ou k
now
cor
resp
ondi
ng
mea
sure
men
ts f
or a
pre
imag
ean
d it
s di
lati
on i
mag
e,yo
u c
an f
ind
the
scal
e fa
ctor
.
Det
erm
ine
the
scal
e fa
ctor
for
th
e d
ilat
ion
of
X �Y�
to A�
B�.D
eter
min
e w
het
her
th
e d
ilat
ion
is
an e
nla
rgem
ent,
red
uct
ion
,or
con
gru
ence
tra
nsf
orm
ati
on.
scal
e fa
ctor
�� pr
i em imagae gele
ln eg nt gh th�
��8 4
u un n
i it ts s�
�2
Th
e sc
ale
fact
or i
s gr
eate
r th
an 1
,so
the
dila
tion
is
an e
nla
rgem
ent.
Det
erm
ine
the
scal
e fa
ctor
for
eac
h d
ilat
ion
wit
h c
ente
r C
.Det
erm
ine
wh
eth
er t
he
dil
atio
n i
s an
en
larg
emen
t,re
du
ctio
n,o
r co
ng
ruen
ce t
ran
sfor
ma
tion
.
1.C
GH
Jis
a d
ilat
ion
im
age
of C
DE
F.
2.�
CK
Lis
a d
ilat
ion
im
age
of �
CK
L.
2;en
larg
emen
t1;
con
gru
ence
3.S
TU
VW
Xis
a d
ilat
ion
im
age
4.�
CP
Qis
a d
ilat
ion
im
age
of �
CY
Z.
of M
NO
PQ
R.
�1 2� ;re
du
ctio
n�1 3� ;
red
uct
ion
5.�
EF
Gis
a d
ilat
ion
im
age
of �
AB
C.
6.�
HJ
Kis
a d
ilat
ion
im
age
of �
HJ
K.
1.5;
enla
rgem
ent
1;co
ng
ruen
ce
C
J
K
H
CB
DG
F
EA
C
Z
Y
P
QC
MN
O
PQ
RU
X
ST V
W
CKL
G CD
H
FE
J
A B
X Y
C
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Dila
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 9-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Dila
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
©G
lenc
oe/M
cGra
w-H
ill50
5G
lenc
oe G
eom
etry
Lesson 9-5
Dra
w t
he
dil
atio
n i
mag
e of
eac
h f
igu
re w
ith
cen
ter
Can
d t
he
give
n s
cale
fac
tor.
1.r
�2
2.r
��1 4�
Fin
d t
he
mea
sure
of
the
dil
atio
n i
mag
e M�
��N���o
r of
th
e p
reim
age
M�N�
usi
ng
the
give
n s
cale
fac
tor.
3.M
N�
3,r
�3
4.M
�N�
�7,
r�
21
M�N
��
9M
N�
�1 3�
CO
OR
DIN
ATE
GEO
MET
RYF
ind
th
e im
age
of e
ach
pol
ygon
,giv
en t
he
vert
ices
,aft
era
dil
atio
n c
ente
red
at
the
orig
in w
ith
a s
cale
fac
tor
of 2
.Th
en g
rap
h a
dil
atio
n
cen
tere
d a
t th
e or
igin
wit
h a
sca
le f
acto
r of
�1 2� .
5.J
(2,4
),K
(4,4
),P
(3,2
)6.
D(�
2,0)
,G(0
,2),
F(2
,�2)
Det
erm
ine
the
scal
e fa
ctor
for
eac
h d
ilat
ion
wit
h c
ente
r C
.Det
erm
ine
wh
eth
er t
he
dil
atio
n i
s an
en
larg
emen
t,re
du
ctio
n,o
r co
ng
ruen
ce t
ran
sfor
ma
tion
.Th
e d
ash
edfi
gure
is
the
dil
atio
n i
mag
e.
7.8.
4;en
larg
emen
t1;
con
gru
ence
tra
nsf
orm
atio
n
CC
G�
F�
D�
G�
F�
D�
x
y
O
J�K
�
P�
J�
K�
P�
x
y
O
CC
©G
lenc
oe/M
cGra
w-H
ill50
6G
lenc
oe G
eom
etry
Dra
w t
he
dil
atio
n i
mag
e of
eac
h f
igu
re w
ith
cen
ter
Can
d t
he
give
n s
cale
fac
tor.
1.r
��3 2�
2.r
��2 3�
Fin
d t
he
mea
sure
of
the
dil
atio
n i
mag
e A�
��T���o
r of
th
e p
reim
age
A�T�
usi
ng
the
give
nsc
ale
fact
or.
3.A
T�
15,r
��3 5�
4.A
T�
30,r
��
�1 6�5.
A�T
��
12,r
��4 3�
A�T
��
9A
�T�
�5
AT
�9
CO
OR
DIN
ATE
GEO
MET
RYF
ind
th
e im
age
of e
ach
pol
ygon
,giv
en t
he
vert
ices
,aft
era
dil
atio
n c
ente
red
at
the
orig
in w
ith
a s
cale
fac
tor
of 2
.Th
en g
rap
h a
dil
atio
n
cen
tere
d a
t th
e or
igin
wit
h a
sca
le f
acto
r of
�1 2� .
6.A
(1,1
),C
(2,3
),D
(4,2
),E
(3,1
)7.
Q(�
1,�
1),R
(0,2
),S
(2,1
)
Det
erm
ine
the
scal
e fa
ctor
for
eac
h d
ilat
ion
wit
h c
ente
r C
.Det
erm
ine
wh
eth
er t
he
dil
atio
n i
s an
en
larg
emen
t,re
du
ctio
n,o
r co
ng
ruen
ce t
ran
sfor
ma
tion
.Th
e d
otte
dfi
gure
is
the
dil
atio
n i
mag
e.
8.�1 3� ;
red
uct
ion
9.2;
enla
rgem
ent
10.P
HO
TOG
RA
PHY
Est
ebe
enla
rged
a 4
-in
ch b
y 6-
inch
ph
otog
raph
by
a fa
ctor
of
�5 2� .W
hat
are
the
new
dim
ensi
ons
of t
he
phot
ogra
ph?
10 in
.by
15 in
.
CC
S�
Q�
R�
S�
Q�
R�
x
y
O
C�
D�
E�
A�
C�
D�
E�
A�
x
y
O
C
C
Pra
ctic
e (
Ave
rag
e)
Dila
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
Answers (Lesson 9-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csD
ilati
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
©G
lenc
oe/M
cGra
w-H
ill50
7G
lenc
oe G
eom
etry
Lesson 9-5
Pre-
Act
ivit
yH
ow d
o yo
u u
se d
ilat
ion
s w
hen
you
use
a c
omp
ute
r?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-5
at
the
top
of p
age
490
in y
our
text
book
.
In a
ddit
ion
to
the
exam
ple
give
n i
n y
our
text
book
,giv
e tw
o ev
eryd
ayex
ampl
es o
f sc
alin
g an
obj
ect,
one
that
mak
es t
he
obje
ct l
arge
r an
d an
oth
erth
at m
akes
it
smal
ler.
Sam
ple
an
swer
:la
rger
:en
larg
ing
ap
ho
tog
rap
h;
smal
ler:
mak
ing
a s
cale
mo
del
of
a bu
ildin
gR
ead
ing
th
e Le
sso
n1.
Eac
h o
f th
e va
lues
of
rgi
ven
bel
ow r
epre
sen
ts t
he
scal
e fa
ctor
for
a d
ilat
ion
.In
eac
hca
se,d
eter
min
e w
het
her
th
e di
lati
on i
s an
en
larg
emen
t,a
red
uct
ion
,or
a co
ngr
uen
cetr
ansf
orm
atio
n.
a.r
�3
enla
rgem
ent
b.r
�0.
5re
du
ctio
nc.
r�
�0.
75re
du
ctio
nd
.r�
�1
con
gru
ence
tra
nsf
orm
atio
ne.
r�
�2 3�re
du
ctio
nf.
r�
��3 2�
enla
rgem
ent
g.r
��
1.01
enla
rgem
ent
h.r
�0.
999
red
uct
ion
2.D
eter
min
e w
het
her
eac
h s
ente
nce
is
alw
ays,
som
etim
es,o
r n
ever
tru
e.If
th
e se
nte
nce
is
not
alw
ays
tru
e,ex
plai
n w
hy.
Fo
r ex
pla
nat
ion
s,sa
mp
le a
nsw
ers
are
giv
en.
a.A
dil
atio
n r
equ
ires
a c
ente
r po
int
and
a sc
ale
fact
or.
alw
ays
b.
A d
ilat
ion
ch
ange
s th
e si
ze o
f a
figu
re.
So
met
imes
;if
th
e d
ilati
on
is a
con
gru
ence
tra
nsf
orm
atio
n,t
he
size
of
the
fig
ure
is u
nch
ang
ed.
c.A
dil
atio
n c
han
ges
the
shap
e of
a f
igu
re.
Nev
er;
all d
ilati
on
s p
rod
uce
sim
ilar
fig
ure
s,an
d s
imila
r fi
gu
res
hav
e th
e sa
me
shap
e.d
.T
he
imag
e of
a f
igu
re u
nde
r a
dila
tion
lie
s on
th
e op
posi
te s
ide
of t
he
cen
ter
from
th
epr
eim
age.
So
met
imes
;th
is is
on
ly t
rue
wh
en t
he
scal
e fa
cto
r is
neg
ativ
e.e.
A s
imil
arit
y tr
ansf
orm
atio
n i
s a
con
gru
ence
tra
nsf
orm
atio
n.
So
met
imes
;th
is is
tru
e o
nly
wh
en t
he
scal
e fa
cto
r is
1 o
r �
1.f.
Th
e ce
nte
r of
a d
ilat
ion
is
its
own
im
age.
alw
ays
g.A
dil
atio
n i
s an
iso
met
ry.
So
met
imes
;th
is is
tru
e o
nly
wh
en t
he
dila
tio
n is
a co
ng
ruen
ce t
ran
sfo
rmat
ion
.h
.T
he
scal
e fa
ctor
for
a d
ilat
ion
is
a po
siti
ve n
um
ber.
So
met
imes
;th
e sc
ale
fact
or
can
be
any
po
siti
ve o
r n
egat
ive
nu
mb
er.
i.D
ilat
ion
s pr
odu
ce s
imil
ar f
igu
res.
alw
ays
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r so
met
hin
g is
to
expl
ain
it
to s
omeo
ne
else
.Su
ppos
e th
at y
our
clas
smat
e L
ydia
is
hav
ing
trou
ble
un
ders
tan
din
g th
e re
lati
onsh
ip b
etw
een
sim
ilar
ity
tran
sfor
mat
ion
san
d co
ngr
uen
ce t
ran
sfor
mat
ion
s.H
ow c
an y
ou e
xpla
in t
his
to
her
?S
amp
le a
nsw
er:
A c
on
gru
ence
tra
nsf
orm
atio
n p
rese
rves
th
e si
ze a
nd
shap
e o
f a
fig
ure
,th
at is
,th
e im
age
is c
on
gru
ent
to t
he
pre
imag
e.A
sim
ilari
ty t
ran
sfo
rmat
ion
pre
serv
es t
he
shap
e o
f a
fig
ure
,th
at is
,th
eim
age
is s
imila
r to
th
e p
reim
age.
A s
imila
rity
tra
nsf
orm
atio
n is
aco
ng
ruen
ce t
ran
sfo
rmat
ion
if t
he
scal
e fa
cto
r is
1 o
r �
1.
©G
lenc
oe/M
cGra
w-H
ill50
8G
lenc
oe G
eom
etry
Sim
ilar
Cir
cles
You
may
be
surp
rise
d to
lea
rn t
hat
tw
o n
onco
ngr
uen
t ci
rcle
s th
at l
ie i
nth
e sa
me
plan
e an
d h
ave
no
com
mon
in
teri
or p
oin
ts c
an b
e m
appe
don
e on
to t
he
oth
er b
y m
ore
than
on
e di
lati
on.
1.H
ere
is d
iagr
am t
hat
su
gges
ts o
ne
way
to
m
ap a
sm
alle
r ci
rcle
on
to a
lar
ger
one
usi
ng
a di
lati
on.T
he
circ
les
are
give
n.T
he
lin
essu
gges
t h
ow t
o fi
nd
the
cen
ter
for
the
dila
tion
.Des
crib
e h
ow t
he
cen
ter
is f
oun
d.U
se s
egm
ents
in
th
e di
agra
m t
o n
ame
the
scal
e fa
ctor
.
Dra
w t
he
two
co
mm
on
ext
ern
alta
ng
ents
.Th
e p
oin
t w
her
e th
ey in
ters
ect
is t
he
cen
ter
of
a d
ilati
on
th
at m
aps
�A
on
to �
A�;
scal
e fa
cto
r �
�A A�M M�
�.
2.H
ere
is a
not
her
pai
r of
non
con
gru
ent
circ
les
wit
h n
o co
mm
onin
teri
or p
oin
t.F
rom
Exe
rcis
e 1,
you
kn
ow y
ou c
an l
ocat
e a
poin
t of
f to
th
e le
ft o
f th
e sm
alle
r ci
rcle
th
at i
s th
e ce
nte
r fo
r a
dila
tion
map
pin
g �
Con
to �
C�.
Fin
d an
oth
er c
ente
r fo
r an
oth
er d
ilat
ion
that
map
s �
Con
to �
C�.
Mar
k an
d la
bel
segm
ents
to
nam
e th
esc
ale
fact
or.
Fin
d t
he
inte
rsec
tio
n o
f th
e tw
o c
om
mo
n in
tern
al t
ang
ents
;sc
ale
fact
or:
��C C�X X
��
.
X
X�
C�
CO
A �
M�
M
AO
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-5
9-5
Answers (Lesson 9-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Vec
tors
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
©G
lenc
oe/M
cGra
w-H
ill50
9G
lenc
oe G
eom
etry
Lesson 9-6
Mag
nit
ud
e an
d D
irec
tio
nA
vec
tor
is a
dir
ecte
d se
gmen
t re
pres
enti
ng
a qu
anti
ty t
hat
has
bot
h m
agn
itu
de,
or l
engt
h,
and
dir
ecti
on.F
or e
xam
ple,
the
spee
d an
d di
rect
ion
of
an
airp
lan
e ca
n b
e re
pres
ente
d by
a v
ecto
r.In
sym
bols
,a v
ecto
r is
w
ritt
en a
s A
B�
,wh
ere
Ais
th
e in
itia
l po
int
and
Bis
th
e en
dpoi
nt,
or a
s v �
.
A v
ecto
r in
sta
nd
ard
pos
itio
nh
as i
ts i
nit
ial
poin
t at
(0,
0) a
nd
can
be
repr
esen
ted
by t
he
orde
red
pair
for
poi
nt
B.T
he
vect
or
at t
he
righ
t ca
n b
e ex
pres
sed
as v �
��5
,3�.
You
can
use
th
e D
ista
nce
For
mu
la t
o fi
nd
the
mag
nit
ude
| A
B�
| of
a ve
ctor
.You
can
des
crib
e th
e di
rect
ion
of
a ve
ctor
by
mea
suri
ng
the
angl
e th
at t
he
vect
or f
orm
s w
ith
th
e po
siti
vex-
axis
or
wit
h a
ny
oth
er h
oriz
onta
l li
ne.
Fin
d t
he
mag
nit
ud
e an
d d
irec
tion
of
AB�
for
A(5
,2)
and
B(8
,7).
Fin
d th
e m
agn
itu
de.
| AB
| ��
(x2
��
x 1)2
��
(y2
��
y 1)2
��
�(8
�5
�)2
�(
�7
�2
�)2 �
��
34�or
abo
ut
5.8
un
its
To
fin
d th
e di
rect
ion
,use
th
e ta
nge
nt
rati
o.
tan
A�
�5 3�T
he t
ange
nt r
atio
is o
ppos
ite o
ver
adja
cent
.
m�
A�
59.0
Use
a c
alcu
lato
r.
Th
e m
agn
itu
de o
f th
e ve
ctor
is
abou
t 5.
8 u
nit
s an
d it
s di
rect
ion
is
59°.
Fin
d t
he
mag
nit
ud
e an
d d
irec
tion
of
AB�
for
the
give
n c
oord
inat
es.R
oun
d t
o th
en
eare
st t
enth
.
1.A
(3,1
),B
(�2,
3)2.
A(0
,0),
B(�
2,1)
5.4;
158.
2°2.
2;15
3.4°
3.A
(0,1
),B
(3,5
)4.
A(�
2,2)
,B(3
,1)
5;53
.1°
5.1;
348.
7°
5.A
(3,4
),B
(0,0
)6.
A(4
,2),
B(0
,3)
5;23
3.1°
4.1;
166.
0°
x
y
O
B( 8
,7)
A( 5
, 2)
x
y
O
B( 5
, 3)
A( 0
, 0)
V�
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill51
0G
lenc
oe G
eom
etry
Tran
slat
ion
s w
ith
Vec
tors
Rec
all
that
th
e tr
ansf
orm
atio
n (
a,b)
→(a
�2,
b�
3)re
pres
ents
a t
ran
slat
ion
rig
ht
2 u
nit
s an
d do
wn
3 u
nit
s.T
he
vect
or �
2,�
3�is
an
oth
er w
ay t
ode
scri
be t
hat
tra
nsl
atio
n.A
lso,
two
vect
ors
can
be
adde
d:�a
,b�
��c
,d�
��a
�c,
b�
d�.
Th
esu
m o
f tw
o ve
ctor
s is
cal
led
the
resu
ltan
t.
Gra
ph
th
e im
age
of p
aral
lelo
gram
RS
TU
un
der
th
e tr
ansl
atio
n b
y th
e ve
ctor
s m �
��3
,�1�
and
n �
���
2,�
4�.
Fin
d th
e su
m o
f th
e ve
ctor
s.m�
�n�
��3
,�1�
���
2,�
4��
�3 �
2,�
1 �
4��
�1,�
5�T
ran
slat
e ea
ch v
erte
x of
par
alle
logr
am R
ST
Uri
ght
1 u
nit
an
d do
wn
5 u
nit
s.
Gra
ph
th
e im
age
of e
ach
fig
ure
un
der
a t
ran
slat
ion
by
the
give
n v
ecto
r(s)
.
1.�
AB
Cw
ith
ver
tice
s A
(�1,
2),B
(0,0
),2.
AB
CD
wit
h v
erti
ces
A(�
4,1)
,B(�
2,3)
,an
d C
(2,3
);m �
��2
,�3�
C(1
,1),
and
D(�
1,�
1);n�
��3
�3�
3.A
BC
Dw
ith
ver
tice
s A
(�3,
3),B
(1,3
),C
(1,1
),an
d D
(�3,
1);t
he
sum
of
p ��
��2,
1�an
dq�
��5
,�4�
Giv
en m �
��1
,�2�
and
n��
��3,
�4�
,rep
rese
nt
each
of
the
foll
owin
g as
a s
ingl
eve
ctor
.
4.m �
�n�
5.n�
�m�
��2,
�6�
��4,
�2�
x
y
D�
A�
B� C
�
DAB C
x
y
D�
A�
B�
C�
D
A
B
C
x
y A�
B�C
�
A
B
C
x
y
R�S
�
T�
U�
R
S
TU
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Vec
tors
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 9-6)
© Glencoe/McGraw-Hill A18 Glencoe Geometry
Skil
ls P
ract
ice
Vec
tors
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
©G
lenc
oe/M
cGra
w-H
ill51
1G
lenc
oe G
eom
etry
Lesson 9-6
Wri
te t
he
com
pon
ent
form
of
each
vec
tor.
1.2.
�3,4
��5
,�5�
Fin
d t
he
mag
nit
ud
e an
d d
irec
tion
of
RS�
for
the
give
n c
oord
inat
es.R
oun
d t
o th
en
eare
st t
enth
.
3.R
(2,�
3),S
(4,9
)4.
R(0
,2),
S(3
,12)
2�37�
�12
.2,8
0.5°
�10
9�
�10
.4,7
3.3°
5.R
(5,4
),S
(�3,
1)6.
R(1
,5),
S(�
4,�
6)
�73�
�8.
5,20
0.6°
�14
6�
�12
.1,2
45.6
°
Gra
ph
th
e im
age
of e
ach
fig
ure
un
der
a t
ran
slat
ion
by
the
give
n v
ecto
r(s)
.
7.�
AB
Cw
ith
ver
tice
s A
(�4,
3),B
(�1,
4),
8.tr
apez
oid
wit
h v
erti
ces
T(�
4,�
2),
C(�
1,1)
;t��
�4,�
3�R
(�1,
�2)
,S(�
2,�
3),Y
(�3,
�3)
;a��
�3,1
�an
d b�
��2
,4�
Fin
d t
he
mag
nit
ud
e an
d d
irec
tion
of
each
res
ult
ant
for
the
give
n v
ecto
rs.
9.y �
��7
,0�,
z��
�0,6
�10
.b��
�3,2
�,c�
���
2,3�
�85�
�9.
2,40
.6°
�26�
�5.
1,78
.7°
S�
Y�
T�
R�
SY
TR
x
y
O
C�
B�
A�
CB
A
x
y
O
x
y
O
B( –
2, 2
)
D( 3
, –3)
x
y
OC
( 1, –
1)
E( 4
, 3)
©G
lenc
oe/M
cGra
w-H
ill51
2G
lenc
oe G
eom
etry
Wri
te t
he
com
pon
ent
form
of
each
vec
tor.
1.2.
�6,4
��5
,�8�
Fin
d t
he
mag
nit
ud
e an
d d
irec
tion
of
FG�
for
the
give
n c
oord
inat
es.R
oun
d t
o th
en
eare
st t
enth
.
3.F
(�8,
�5)
,G(�
2,7)
4.F
(�4,
1),G
(5,�
6)
6�5�
�13
.4,6
3.4°
�13
0�
�11
.4,3
22.1
°
Gra
ph
th
e im
age
of e
ach
fig
ure
un
der
a t
ran
slat
ion
by
the
give
n v
ecto
r(s)
.
5.�
QR
Tw
ith
ver
tice
s Q
(�1,
1),R
(1,4
),6.
trap
ezoi
d w
ith
ver
tice
s J
(�4,
�1)
,T
(5,1
);s �
���
2,�
5�K
(0,�
1),L
(�1,
�3)
,M(�
2,�
3);c�
��5
,4�
and
d��
��2,
1�
Fin
d t
he
mag
nit
ud
e an
d d
irec
tion
of
each
res
ult
ant
for
the
give
n v
ecto
rs.
7.a�
���
6,4�
,b��
�4,6
�8.
e��
��4,
�5�
,f��
��1,
3�
2�26�
�10
.2,1
01.3
°�
29��
5.4,
201.
8°
AV
IATI
ON
For
Exe
rcis
es 9
an
d 1
0,u
se t
he
foll
owin
g in
form
atio
n.
A je
t be
gins
a f
light
alo
ng a
pat
h du
e no
rth
at 3
00 m
iles
per
hour
.A w
ind
is b
low
ing
due
wes
tat
30
mile
s pe
r ho
ur.
9.F
ind
the
resu
ltan
t ve
loci
ty o
f th
e pl
ane.
abo
ut
301.
5 m
ph
10.F
ind
the
resu
ltan
t di
rect
ion
of
the
plan
e.ab
ou
t 5.
7°w
est
of
du
e n
ort
h
L�M
�
J�K
�
LM
JK
x
y
O
T�
R�
Q�
T
R
Qx
y
O
x
y
O
K( –
2, 4
)
L(3,
–4)
x
y
O
A( –
2, –
2)
B( 4
, 2)
Pra
ctic
e (
Ave
rag
e)
Vec
tors
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
Answers (Lesson 9-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csV
ecto
rs
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
©G
lenc
oe/M
cGra
w-H
ill51
3G
lenc
oe G
eom
etry
Lesson 9-6
Pre-
Act
ivit
yH
ow d
o ve
ctor
s h
elp
a p
ilot
pla
n a
fli
ght?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-6
at
the
top
of p
age
498
in y
our
text
book
.
Wh
y do
pil
ots
ofte
n h
ead
thei
r pl
anes
in
a s
ligh
tly
diff
eren
t di
rect
ion
fro
mth
eir
dest
inat
ion
?S
amp
le a
nsw
er:
to c
om
pen
sate
fo
r th
e ef
fect
of
the
win
d s
o t
hat
th
e re
sult
will
be
that
th
e p
lan
e w
ill a
ctu
ally
fly
in t
he
corr
ect
dir
ecti
on
Rea
din
g t
he
Less
on
1.S
upp
ly t
he
mis
sin
g w
ords
or
phra
ses
to c
ompl
ete
the
foll
owin
g se
nte
nce
s.
a.A
is
a d
irec
ted
segm
ent
repr
esen
tin
g a
quan
tity
th
at h
as b
oth
mag
nit
ude
an
d di
rect
ion
.
b.
Th
e le
ngt
h o
f a
vect
or i
s ca
lled
its
.
c.T
wo
vect
ors
are
para
llel
if
and
only
if
they
hav
e th
e sa
me
or
dire
ctio
n.
d.
A v
ecto
r is
in
if
it
is d
raw
n w
ith
in
itia
l po
int
at t
he
orig
in.
e.T
wo
vect
ors
are
equ
al i
f an
d on
ly i
f th
ey h
ave
the
sam
e an
d th
e
sam
e.
f.T
he
sum
of
two
vect
ors
is c
alle
d th
e .
g.A
vec
tor
is w
ritt
en i
n
if i
t is
exp
ress
ed a
s an
ord
ered
pai
r.
h.
Th
e pr
oces
s of
mu
ltip
lyin
g a
vect
or b
y a
con
stan
t is
cal
led
.
2.W
rite
eac
h v
ecto
r de
scri
bed
belo
w i
n c
ompo
nen
t fo
rm.
a.a
vect
or i
n s
tan
dard
pos
itio
n w
ith
en
dpoi
nt
(a,b
)�a
,b�
b.
a ve
ctor
wit
h i
nit
ial
poin
t (a
,b)
and
endp
oin
t (c
,d)
�c�
a,d
�b
�c.
a ve
ctor
in
sta
nda
rd p
osit
ion
wit
h e
ndp
oin
t (�
3,5)
��3,
5�d
.a
vect
or w
ith
in
itia
l po
int
(2,�
3) a
nd
endp
oin
t (6
,�8)
�4,�
5�e.
a��
b�if
a��
��3,
5�an
d b�
��6
,�4�
�3,1
�f.
5u�if
u��
�8,�
6��4
0,�
30�
g.�
�1 3� v�if
v��
��15
,24�
�5,�
8�h
.0.
5u��
1.5v�
if u�
��1
0,�
10�
and
v��
��8,
6���
7,4�
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al t
erm
is
to r
elat
e it
to
a te
rm y
ou a
lrea
dykn
ow.Y
ou le
arne
d ab
out
scal
e fa
ctor
sw
hen
you
stud
ied
sim
ilari
ty a
nd d
ilati
ons.
How
is t
heid
ea o
f a
scal
arre
late
d to
sca
le f
acto
rs?
Sam
ple
an
swer
:A s
cala
ris
th
e te
rm u
sed
for
a co
nst
ant
(a s
pec
ific
real
nu
mb
er)
wh
en w
ork
ing
with
vec
tors
.A v
ecto
rh
as b
oth
mag
nit
ud
e an
d d
irec
tio
n,w
hile
a s
cala
r is
just
a m
agn
itu
de.
Mu
ltip
lyin
g a
vec
tor
by a
po
siti
ve s
cala
r ch
ang
es t
he
mag
nit
ud
e o
f th
eve
cto
r,bu
t n
ot
the
dir
ecti
on
,so
it r
epre
sen
ts a
ch
ang
e in
sca
le.
scal
ar m
ult
iplic
atio
nco
mp
on
ent
formre
sult
ant
dir
ecti
on
mag
nit
ud
est
and
ard
po
siti
on
op
po
site
mag
nit
ud
e
vect
or
©G
lenc
oe/M
cGra
w-H
ill51
4G
lenc
oe G
eom
etry
Rea
din
g M
ath
emat
ics
Man
y qu
anti
ties
in
nat
ure
can
be
thou
ght
of a
s ve
ctor
s.T
he
scie
nce
of
phys
ics
invo
lves
man
y ve
ctor
qu
anti
ties
.In
rea
din
g ab
out
appl
icat
ion
sof
mat
hem
atic
s,as
k yo
urs
elf
wh
eth
er t
he
quan
titi
es i
nvo
lve
only
mag
nit
ude
or
both
mag
nit
ude
an
d di
rect
ion
.Th
e fi
rst
kin
d of
qu
anti
tyis
cal
led
scal
ar.T
he
seco
nd
kin
d is
a v
ecto
r.
Cla
ssif
y ea
ch o
f th
e fo
llow
ing.
Wri
te s
cala
ror
vec
tor.
1.th
e m
ass
of a
boo
ksc
alar
2.a
car
trav
elin
g n
orth
at
55 m
phve
cto
r
3.a
ball
oon
ris
ing
24 f
eet
per
min
ute
vect
or
4.th
e si
ze o
f a
shoe
scal
ar
5.a
room
tem
pera
ture
of
22 d
egre
es C
elsi
us
scal
ar
6.a
wes
t w
ind
of 1
5 m
phve
cto
r
7.th
e ba
ttin
g av
erag
e of
a b
aseb
all
play
ersc
alar
8.a
car
trav
elin
g 60
mph
vect
or
9.a
rock
fal
lin
g at
10
mph
vect
or
10.y
our
age
scal
ar
11.t
he
forc
e of
Ear
th’s
gra
vity
act
ing
on a
mov
ing
sate
llit
eve
cto
r
12.t
he
area
of
a re
cord
rot
atin
g on
a t
urn
tabl
esc
alar
13.t
he
len
gth
of
a ve
ctor
in
th
e co
ordi
nat
e pl
ane
scal
ar
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-6
9-6
Answers (Lesson 9-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Tran
sfo
rmat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-7
9-7
©G
lenc
oe/M
cGra
w-H
ill51
5G
lenc
oe G
eom
etry
Lesson 9-7
Tran
slat
ion
s an
d D
ilati
on
sA
vec
tor
can
be
repr
esen
ted
by t
he
orde
red
pair
�x,
y�or
by t
he
colu
mn
mat
rix
.Wh
en t
he
orde
red
pair
s fo
r al
l th
e ve
rtic
es o
f a
poly
gon
are
plac
ed t
oget
her
,th
e re
sult
ing
mat
rix
is c
alle
d th
e ve
rtex
mat
rix
for
the
poly
gon
.
For
�A
BC
wit
h A
(�2,
2),B
(2,1
),an
d C
(�1,
�1)
,th
e ve
rtex
mat
rix
for
the
tria
ngl
e is
.
For
�A
BC
abov
e,u
se a
mat
rix
to f
ind
th
e co
ord
inat
es o
f th
eve
rtic
es o
f th
e im
age
of �
AB
Cu
nd
er t
he
tran
slat
ion
(x,
y) →
(x�
3,y
�1)
.T
o tr
ansl
ate
the
figu
re 3
un
its
to t
he
righ
t,ad
d 3
to e
ach
x-c
oord
inat
e.T
o tr
ansl
ate
the
figu
re 1
un
it d
own
,add
�1
to e
ach
y-c
oord
inat
e.V
erte
x M
atri
xT
ran
slat
ion
V
erte
x M
atri
xof
�A
BC
Mat
rix
of �
A�B
�C�
��
Th
e co
ordi
nat
es a
re A
�(1,
1),B
�(5,
0),a
nd
C�(
2,�
2).
For
�A
BC
abov
e,u
se a
mat
rix
to f
ind
th
e co
ord
inat
es o
f th
e ve
rtic
esof
th
e im
age
of �
AB
Cfo
r a
dil
atio
n c
ente
red
at
the
orig
in w
ith
sca
le f
acto
r 3.
Sca
leV
erte
x M
atri
xV
erte
x M
atri
xFa
ctor
of �
AB
Cof
�A
�B�C
�
3
�
Th
e co
ordi
nat
es a
re A
�(�
6,6)
,B�(
6,3)
,an
d C
�(�
3,�
3).
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
eac
h f
igu
reu
nd
er t
he
give
n t
ran
slat
ion
s or
dil
atio
ns.
1.�
AB
Cw
ith
A(3
,1),
B(�
2,4)
,C(�
2,�
1);(
x,y)
→(x
�1,
y�
2)A
�(2,
3),B
�(�
3,6)
,C�(
�3,
1)2.
para
llel
ogra
m R
ST
Uw
ith
R(�
4,�
2),S
(�3,
1),T
(3,4
),U
(2,1
);(x
,y)
→(x
�4,
y�
3)
R�(
�8,
�5)
,S�(
�7,
�2)
,T�(
�1,
1),U
�(�
2,�
2)3.
rect
angl
e P
QR
Sw
ith
P(4
,0),
Q(3
,�3)
,R(�
3,�
1),S
(�2,
2);(
x,y)
→(x
�2,
y�
1)P
�(2,
1),Q
�(1,
�2)
,R�(
�5,
0),S
�(�
4,3)
4.�
AB
Cw
ith
A(�
2,�
1),B
(�2,
�3)
,C(2
,�1)
;dil
atio
n c
ente
red
at t
he
orig
in w
ith
sca
lefa
ctor
2A
�(�
4,�
2),B
�(�
4,�
6),C
�(4,
�2)
5.pa
rall
elog
ram
RS
TU
wit
h R
(4,�
2),S
(�4,
�1)
,T(�
2,3)
,U(6
,2);
dila
tion
cen
tere
d at
th
eor
igin
wit
h s
cale
fac
tor
1.5
R�(
6,�
3),S
�(�
6,�
1.5)
,T�(
�3,
4.5)
,U�(
9,3)
�6
6�
3
63
�3
�2
2�
1
21
�1
15
2 1
0�
2
33
3 �
1�
1�
1�
22
�1
2
1�
1
�2
2�
1
21
�1
x
y
B( 2
, 1)
C( –
1, –
1)
A( –
2, 2
)
x
y
Exer
cises
Exer
cises
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
©G
lenc
oe/M
cGra
w-H
ill51
6G
lenc
oe G
eom
etry
Ref
lect
ion
s an
d R
ota
tio
ns
Wh
en y
ou r
efle
ct a
n i
mag
e,on
e w
ay t
o fi
nd
the
coor
din
ates
of
the
refl
ecte
d ve
rtic
es i
s to
mu
ltip
ly t
he
vert
ex m
atri
x of
th
e ob
ject
by
are
flec
tion
mat
rix.
To
perf
orm
mor
e th
an o
ne
refl
ecti
on,m
ult
iply
by
one
refl
ecti
on m
atri
xto
fin
d th
e fi
rst
imag
e.T
hen
mu
ltip
ly b
y th
e se
con
d m
atri
x to
fin
d th
e fi
nal
im
age.
Th
em
atri
ces
for
refl
ecti
ons
in t
he
axes
,th
e or
igin
,an
d th
e li
ne
y�
xar
e sh
own
bel
ow.
Fo
r a
refl
ecti
on
in t
he:
x-ax
isy-
axis
ori
gin
line
y�
x
Mu
ltip
ly t
he
vert
ex
mat
rix
by:
�A
BC
has
ver
tice
s A
(�2,
3),B
(1,4
),an
d
C(3
,0).
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
th
e im
age
of �
AB
Caf
ter
a re
flec
tion
in
th
e x-
axis
.T
o re
flec
t in
th
e x-
axis
,mu
ltip
ly t
he
vert
ex m
atri
x of
�A
BC
by t
he
refl
ecti
on m
atri
x fo
r th
e x-
axis
.
Ref
lect
ion
Mat
rix
Ver
tex
Mat
rix
Ver
tex
Mat
rix
for
x-ax
isof
�A
BC
of �
A’B
’C’
�
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
eac
h f
igu
reu
nd
er t
he
give
n r
efle
ctio
n.
1.�
AB
Cw
ith
A(�
3,2)
,B(�
1,3)
,C(1
,0);
refl
ecti
on i
n t
he
x-ax
is
A�(
�3,
�2)
,B�(
�1,
�3)
,C�(
1,0)
2.�
XY
Zw
ith
X(2
,�1)
,Y(4
,�3)
,Z(�
2,1)
;ref
lect
ion
in
th
e y-
axis
X�(
�2,
�1)
,Y�(
�4,
�3)
,Z�(
2,1)
3.�
AB
Cw
ith
A(3
,4),
B(�
1,0)
,C(�
2,4)
;ref
lect
ion
in
th
e or
igin
A�(
�3,
�4)
,B�(
1,0)
,C�(
2,�
4)
4.pa
rall
elog
ram
RS
TU
wit
h R
(�3,
2),S
(3,2
),T
(5,�
1),U
(�1,
�1)
;ref
lect
ion
in t
he li
ne y
�x
R�(
2,�
3),S
�(2,
3),T
�(�
1,5)
,U�(
�1,
�1)
5.�
AB
Cw
ith
A(2
,3),
B(�
1,2)
,C(1
,�1)
;ref
lect
ion
in
th
e or
igin
,th
en r
efle
ctio
n i
n t
he
lin
ey
�x
A�(
�3,
�2)
,B�(
�2,
1),C
�(1,
�1)
6.pa
rall
elog
ram
RS
TU
wit
h R
(0,2
),S
(4,2
),T
(3,�
2),U
(�1,
�2)
;ref
lect
ion
in
th
e x-
axis
,th
en r
efle
ctio
n i
n t
he
y-ax
is
R�(
0,�
2),S
�(�
4,�
2),T
�(�
3,2)
,U�(
1,2)
�2
13
�3
�4
0�
21
3
34
01
0 0
�1
x
y
A�
B�
AB
C
01
10
�1
0
0�
1�
10
0
11
00
�1
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Tran
sfo
rmat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-7
9-7
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 9-7)
© Glencoe/McGraw-Hill A21 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Tran
sfo
rmat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-7
9-7
©G
lenc
oe/M
cGra
w-H
ill51
7G
lenc
oe G
eom
etry
Lesson 9-7
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
eac
h f
igu
reu
nd
er t
he
give
n t
ran
slat
ion
s.
1.�
ST
Uw
ith
S(6
,4),
T(9
,7),
and
U(1
4,2)
;(x,
y) →
(x�
4,y
�3)
S�(
2,7)
,T�(
5,10
),U
�(10
,5)
2.�
GH
Iw
ith
G(�
5,0)
,H(�
3,6)
,an
d I(
�2,
1);(
x,y)
→(x
�2,
y�
6)
G�(
�3,
6),H
�(�
1,12
),I�
(0,7
)
Use
sca
lar
mu
ltip
lica
tion
to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
each
fig
ure
for
a d
ilat
ion
cen
tere
d a
t th
e or
igin
wit
h t
he
give
n s
cale
fac
tor.
3.�
DE
Fw
ith
D(2
,1),
E(5
,4),
and
F(7
,2);
r�
4
D�(
8,4)
,E�(
20,1
6),F
�(28
,8)
4.qu
adri
late
ral
WX
YZ
wit
h W
(�9,
6),X
(�6,
3),Y
(3,1
2),a
nd
Z(�
6,15
);r
��1 3�
W�(
�3,
2),X
�(�
2,1)
,Y�(
1,4)
,Z�(
�2,
5)
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
eac
h f
igu
reu
nd
er t
he
give
n r
efle
ctio
n.
5.�
MN
Ow
ith
M(�
5,1)
,N(�
2,3)
,an
d O
(2,0
);y-
axis
M�(
5,1)
,N�(
2,3)
,O�(
�2,
0)
6.qu
adri
late
ral
AB
CD
wit
h A
(3,1
),B
(6,�
2),C
(5,�
5),a
nd
D(1
,�6)
;x-a
xis
A�(
3,�
1),B
�(6,
2),C
�(5,
5),D
�(1,
6)
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
eac
h f
igu
reu
nd
er t
he
give
n r
otat
ion
.
7.�
RS
Tw
ith
R(�
2,�
2),S
(�3,
3),a
nd
T(2
,2);
90°c
oun
terc
lock
wis
e
R�(
2,�
2),S
�(�
3,�
3),T
�(�
2,2)
8.�
LM
NP
wit
h L
(3,4
),M
(7,4
),N
(9,�
3),a
nd
P(5
,�3)
;180
°co
un
terc
lock
wis
e
L�(
�3,
�4)
,M�(
�7,
�4)
,N�(
�9,
3),P
�(�
5,3)
©G
lenc
oe/M
cGra
w-H
ill51
8G
lenc
oe G
eom
etry
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
eac
h f
igu
reu
nd
er t
he
give
n t
ran
slat
ion
s.
1.�
KL
Mw
ith
K(�
7,�
3),L
(4,9
),an
d M
(9,�
6);(
x,y)
→(x
�7,
y�
2)
K�(
�14
,�1)
,L�(
�3,
11),
M�(
2,�
4)
2.�
AB
CD
wit
h A
(�4,
3),B
(�2,
8),C
(3,1
0),a
nd
D(1
,5);
(x,y
) →
(x�
3,y
�9)
A�(
�1,
�6)
,B�(
1,�
1),C
�(6,
1),D
�(4,
�4)
Use
sca
lar
mu
ltip
lica
tion
to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
each
fig
ure
for
a d
ilat
ion
cen
tere
d a
t th
e or
igin
wit
h t
he
give
n s
cale
fac
tor.
3.qu
adri
late
ral
HIJ
Kw
ith
H(�
2,3)
,I(2
,6),
J(8
,3),
and
K(3
,�4)
;r�
��1 3�
H� ��2 3� ,
�1 �,
I���
�2 3� ,�
2 �,J
� ���8 3� ,
�1 �,
K� ��
1,�4 3� �
4.pe
nta
gon
DE
FG
Hw
ith
D(�
8,�
4),E
(�8,
2),F
(2,6
),G
(8,0
),an
d H
(4,�
6);r
��5 4�
D�(
�10
,�5)
,E� ��
10,�
5 2� �,F
� ��5 2� ,�1 25 �
�,G�(
10,0
),H
� �5,�
�1 25 ��
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
eac
h f
igu
reu
nd
er t
he
give
n r
efle
ctio
n.
5.�
QR
Sw
ith
Q(�
5,�
4),R
(�1,
�1)
,an
d S
(2,�
6);x
-axi
s
Q�(
�5,
4),R
�(�
1,1)
,S�(
2,6)
6.qu
adri
late
ral
VX
YZ
wit
h V
(�4,
�2)
,X(�
3,4)
,Y(2
,1),
and
Z(4
,�3)
;y�
x
V�(
�2,
�4)
,X�(
4,�
3),Y
�(1,
2),Z
�(�
3,4)
Use
a m
atri
x to
fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
imag
e of
eac
h f
igu
reu
nd
er t
he
give
n r
otat
ion
.
7.�
EF
GH
wit
h E
(�5
�4)
,F(�
3,�
1),G
(5,�
1),a
nd
H(3
,�4)
;90°
cou
nte
rclo
ckw
ise
E�(
4,�
5),F
�(1,
�3)
,G�(
1,5)
,H�(
4,3)
8.qu
adri
late
ral
PS
TU
wit
h P
(�3,
5),S
(2,6
),T
(8,1
),an
d U
(�6,
�4)
;270
°co
un
terc
lock
wis
e
P�(
5,3)
,S�(
6,�
2),T
�(1,
�8)
,U�(
�4,
6)
9.FO
RES
TRY
A r
esea
rch
bot
anis
t m
appe
d a
sect
ion
of
fore
sted
lan
d on
a c
oord
inat
e gr
idto
kee
p tr
ack
of e
nda
nge
red
plan
ts i
n t
he
regi
on.T
he
vert
ices
of
the
map
are
A(�
2,6)
,B
(9,8
),C
(14,
4),a
nd
D(1
,�1)
.Aft
er a
mon
th,t
he
bota
nis
t h
as d
ecid
ed t
o de
crea
se t
he
rese
arch
are
a to
�3 4�of
its
ori
gin
al s
ize.
If t
he
cen
ter
for
the
redu
ctio
n i
s O
(0,0
),w
hat
are
the
coor
din
ates
of
the
new
res
earc
h a
rea?
A��
�3 2� ,�9 2� �,
B��2 47 �
,6�,C
��2 21 �,3
�,D��3 4� ,
��3 4� �
Pra
ctic
e (
Ave
rag
e)
Tran
sfo
rmat
ion
s w
ith
Mat
rice
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-7
9-7
Answers (Lesson 9-7)
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csTr
ansf
orm
atio
ns
wit
h M
atri
ces
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-7
9-7
©G
lenc
oe/M
cGra
w-H
ill51
9G
lenc
oe G
eom
etry
Lesson 9-7
Pre-
Act
ivit
yH
ow c
an m
atri
ces
be
use
d t
o m
ake
mov
ies?
Rea
d th
e in
trod
uct
ion
to
Les
son
9-7
at
the
top
of p
age
506
in y
our
text
book
.
•W
hat
kin
d of
tra
nsf
orm
atio
n s
hou
ld b
e u
sed
to m
ove
a po
lygo
n?
refl
ecti
on
,tra
nsl
atio
n,o
r ro
tati
on
•W
hat
kind
of
tran
sfor
mat
ion
shou
ld b
e us
ed t
o re
size
a p
olyg
on?
dila
tio
n
Rea
din
g t
he
Less
on
1.W
rite
a v
erte
x m
atri
x fo
r ea
ch f
igu
re.
a.�
AB
Cb
.par
alle
logr
am A
BC
D
2.M
atch
eac
h t
ran
sfor
mat
ion
fro
m t
he
firs
t co
lum
n w
ith
th
e co
rres
pon
din
g m
atri
x fr
omth
e se
con
d or
th
ird
colu
mn
.In
eac
h c
ase,
the
vert
ex m
atri
x fo
r th
e pr
eim
age
of a
fig
ure
is m
ult
ipli
ed o
n t
he
left
by
one
of t
he
mat
rice
s be
low
to
obta
in t
he
imag
e of
th
e fi
gure
.A
ll r
otat
ion
s li
sted
are
cou
nte
rclo
ckw
ise
thro
ugh
th
e or
igin
.(S
ome
mat
rice
s m
ay b
e u
sed
mor
e th
an o
nce
or
not
at
all.)
a.re
flec
tion
ove
r th
e y-
axis
vii.
v.b
.90
°ro
tati
onvi
iic.
refl
ecti
on o
ver
the
lin
e y
�x
iiiii
.vi
.d
.27
0°ro
tati
onv
e.re
flec
tion
ove
r th
e or
igin
iiii
i.vi
i.f.
180°
rota
tion
iig.
refl
ecti
on o
ver
the
x-ax
isvi
iiv
.vi
ii.
h.
360°
rota
tion
i
Hel
pin
g Y
ou
Rem
emb
er3.
How
can
you
rem
embe
r or
qu
ickl
y fi
gure
ou
t th
e m
atri
ces
for
the
tran
sfor
mat
ion
s in
Exe
rcis
e 2?
Vis
ual
ize
or
sket
ch h
ow t
he
“un
it p
oin
ts”
(1,0
) o
n t
he
x-ax
is a
nd
(0,1
) o
n t
he
y-ax
is a
re m
oved
by
the
tran
sfo
rmat
ion
.Wri
te t
he
ord
ered
pai
rfo
r th
e im
age
po
ints
in a
2 �
2 m
atri
x,w
ith t
he
coo
rdin
ates
of
the
imag
e o
fth
e x-
axis
un
it p
oin
t in
th
e fir
st c
olu
mn
an
d t
he
coo
rdin
ates
of
the
imag
e o
fth
e y-
axis
un
it p
oin
t in
th
e se
con
d c
olu
mn
.
0�
1 1
0
0�
1 �
10
10
0�
10
1 1
0
�1
0
01
�1
0
0�
1
0
1 �
10
10
01
�4
�1
41
0
33
03
�3
�1
1
4�
4
BC
AD
x
y
O
B
C
A
x
y
O
©G
lenc
oe/M
cGra
w-H
ill52
0G
lenc
oe G
eom
etry
Vec
tor
Add
itio
nV
ecto
rs a
re p
hys
ical
qu
anti
ties
wit
h m
agn
itu
de a
nd
dire
ctio
n.F
orce
an
d ve
loci
ty a
re t
wo
exam
ples
.We
wil
l in
vest
igat
e ad
din
g ve
ctor
qu
anti
ties
.Th
e su
m o
f tw
o ve
ctor
s is
cal
led
are
sult
ant
vect
oror
just
th
e re
sult
ant.
Tw
o se
par
ate
forc
es,o
ne
mea
suri
ng
20 u
nit
s an
d t
he
oth
er m
easu
rin
g 40
un
its,
act
on a
n
obje
ct.I
f th
e an
gle
bet
wee
n t
he
forc
es i
s 50
°,fi
nd
th
em
agn
itu
de
and
dir
ecti
on o
f th
e re
sult
ant
forc
e.
Fir
st,t
he
vect
ors
mu
st b
e re
arra
nge
d by
pla
cin
g th
e ta
il o
f th
e 20
-un
it v
ecto
r at
th
e h
ead
of t
he
40-u
nit
vec
tor.
Sin
ce t
hes
e ve
ctor
s ar
e n
ot p
erpe
ndi
cula
r,th
e h
oriz
onta
l an
d ve
rtic
alco
mpo
nen
ts o
f on
e of
th
e ve
ctor
s m
ust
be
fou
nd.
Usi
ng
trig
onom
etry
,th
e h
oriz
onta
l co
mpo
nen
t m
ust
be
(20
cos
50°)
u
nit
s an
d th
e ve
rtic
al c
ompo
nen
t m
ust
be
(20
sin
50°
) u
nit
s.R
epla
cin
g th
e 20
-un
it v
ecto
r w
ith
th
ese
com
pon
ents
,we
can
n
ow f
orm
tw
o ve
ctor
s pe
rpen
dicu
lar
and
use
th
e P
yth
agor
ean
T
heo
rem
to
fin
d th
e re
sult
ant.
r2�
(40
�20
cos
50°
)2�
(20
sin
50°
)2
r2�
(52.
9)2
�(1
5.3)
2
r2�
3032
.5r2
�55
.1
tan
O�� 40
2 �02s 0in
c5 os0°50
°�
�0.
2898
m�
O�
16
Th
eref
ore,
the
resu
ltan
t fo
rce
is 5
5.1
un
its
dire
cted
16°
from
th
e 40
-un
it f
orce
.
Sol
ve.R
oun
d a
ll a
ngl
e m
easu
res
to t
he
nea
rest
deg
ree.
Rou
nd
all
oth
er m
easu
res
to t
he
nea
rest
ten
th.
1.A
pla
ne
flie
s du
e w
est
at 2
50 k
ilom
eter
s pe
r h
our
wh
ile
the
win
d bl
ows
sou
th a
t 70
kilo
met
ers
per
hou
r.F
ind
the
plan
e’s
resu
ltan
t ve
loci
ty.
259.
6 km
/h,1
6�so
uth
of
wes
t
2.A
pla
ne
flie
s ea
st f
or 2
00 k
m,t
hen
60°
sou
th o
f ea
st f
or 8
0 km
.Fin
d th
e pl
ane’
s di
stan
cean
d di
rect
ion
fro
m i
ts s
tart
ing
poin
t.
249.
8 km
,16�
sou
th o
f ea
st
3.O
ne
forc
e of
100
un
its
acts
on
an
obj
ect.
An
oth
er f
orce
of
80 u
nit
s ac
ts o
n t
he
obje
ct a
t a
40°
angl
e fr
om t
he
firs
t fo
rce.
Fin
d th
e re
sult
ant
forc
e on
th
e ob
ject
.
169.
3 u
nit
s,18
�fr
om
th
e 10
0-u
nit
fo
rce
O
20
40 �
20
cos
50�
20 s
in 5
0�r
20
20 c
os 5
0�
20 s
in 5
0�
50�
O
20 u
nits
40 u
nits
50�
O
20 u
nits
40 u
nits
50�
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
9-7
9-7
Exam
ple
Exam
ple
Answers (Lesson 9-7)
© Glencoe/McGraw-Hill A23 Glencoe Geometry
Chapter 9 Assessment Answer Key Form 1 Form 2APage 521 Page 522 Page 523
(continued on the next page)
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
A
B
C
A
B
D
D
C
B
D
A
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
C
C
C
B
D
B
A
D
A
120�counterclockwiseor 240� clockwise
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
C
A
D
D
A
B
C
C
D
B
© Glencoe/McGraw-Hill A24 Glencoe Geometry
Chapter 9 Assessment Answer KeyForm 2A (continued) Form 2BPage 524 Page 525 Page 526
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
C
B
D
B
D
A
B
A
C
C
��35
�, �45
��
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
D
C
C
C
A
C
B
A
A
C
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
B
B
B
D
A
D
B
A
B
D
��9, �12�
© Glencoe/McGraw-Hill A25 Glencoe Geometry
Chapter 9 Assessment Answer KeyForm 2CPage 527 Page 528
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
(5, �2)
1
No, it is not areflection in
line b.
W�(3, �2),X�(�8, 2)
A�(1, 3) B�(5, 1)
(1, �3)
No, each �measure is 150,which is not afactor of 360.
uniform
reduction
x
y
O
A�
B�
C�
A
B
C
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
3
�34
�
203.2�
10
�7, 4�
yes
(�4, 7)
E�(9, �1),F �(2, �5), G�(5, 2)
A�(0, 2), B�(�3, 6),C�(�5, 0)
about 401.1 mph,about 4.3� south
of due west
M�
N�
O�M
N
OC
© Glencoe/McGraw-Hill A26 Glencoe Geometry
Chapter 9 Assessment Answer KeyForm 2DPage 529 Page 530
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
(3, 6)
2
Yes, it is reflectedover both lines a
and b.
U �(�1, 0), V �(2, 3)
C �(�4, 0),D�(�4, 3)
Q �(�1, 2)
No, each anglemeasure is 156,which is not afactor of 360.
uniform,semi-regular
enlargement
x
y
OR�
P�
Q�
P
QR
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
35
�43
�
213.7�
13
�9, �6�
yes
(�13, �4)
J �(1, �1), K�(12, 3),L�(4, �8)
D�(�2, �1),E�(�1, �6),F�(3, �2)
about 302.7 mph,about 7.6� east of
due south
C�
D�
E�
C
D
E
G
© Glencoe/McGraw-Hill A27 Glencoe Geometry
Chapter 9 Assessment Answer KeyForm 3Page 531 Page 532
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Sample answers:distance measure,
betweenness ofpoints, � measure,
collinearity
C(1, 13), D(�4, 9)
order: 20,magnitude: 18�
R �(3, �5), S �(8, �7),T �(1, �10),
90� clockwise
No, each anglemeasure is 144, whichis not a factor of 360.
not uniform
9
�
AB
C
A� B�
C�
m
x
y
O
V�
T�
U�
T
U
V
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
6.6 in. by 8.4 in.
�65
�
Reflect in the x-axisand the y-axis, in
either order.
333.4�
7.3
�3, 8�
(4 � a, �7 � b)
J �(�2, 5),K �(�8, 1), L�(�2, 3)
G�(5, 4),H�(�1, �1),
I�(�6, 2)
(3, 3�3� )
C
© Glencoe/McGraw-Hill A28 Glencoe Geometry
Chapter 9 Assessment Answer KeyPage 533, Open-Ended Assessment
Scoring Rubric
Score General Description Specific Criteria
• Shows thorough understanding of the concepts ofreflections, translations, rotations, dilations, tessellations,vectors, and matrices.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Figures are accurate and appropriate.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts of reflections,translations, rotations, dilations, tessellations, vectors, andmatrices.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures are mostly accurate and appropriate.• Satisfies all requirements of problems.
• Shows an understanding of the concepts of reflections,translations, rotations, dilations, tessellations, vectors, andmatrices.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures are mostly accurate and appropriate.• Satisfies all requirements of problems.
• Final computation is correct.• No written explanations or work shown to substantiate the
final computation.• Figures may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of the concepts ofreflections, translations, rotations, dilations, tessellations,vectors, and matrices.
• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Figures are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer given.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
© Glencoe/McGraw-Hill A29 Glencoe Geometry
An
swer
s
Chapter 9 Assessment Answer KeyPage 533, Open-Ended Assessment
Sample Answers
1. Students should draw a tessellationwhere they use only one regular polygonand the angle measures at each vertexare congruent and total 360°, as shownin the figure.
2. A rectangular mirror with two lines ofsymmetry, one vertical and one horizontal,through the middle or a spoon with oneline of symmetry down the middle, etc.
3. a.
b. B�(0, 1)
c. A�(�1, 0)
d. The first column of the matrix in parta is the image of (1, 0) and the secondcolumn is the image of (0, 1).
4. The student should draw a tessellationformed by two different regular polygonsas shown in the figure.
5.
Reflect �ABC with A(1, 1), B(2, 3), andC(�1, 2) in the x-axis, rotate 90°counterclockwise about the origin,translate 2 units to the left and 1 unitup, then dilate with center at the originand a scale factor of 2. The image of�ABC has vertices A�(�2, 4), B�(2, 6),and C�(0, 0).
6. CD� has magnitude �41� or �6.4 anddirection 51.3°. In general, if D hascoordinates (a, b) then the magnitude
will be �a2 � b�2� and the direction will
be tan�1��ab
��.
x
y
O C
D
x
y
O C�
B�
A�
CA
B
0 �11 0
In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.
© Glencoe/McGraw-Hill A30 Glencoe Geometry
Chapter 9 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 534 Page 535 Page 536
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
isometry
reflection
rotation
standard position
translation
vector
line of symmetry
dilation
parallel vectors
component form
the fixed point around which thepoints are turned
a constant
a pattern covering aplane by transforming
the same figure(s)with no overlapping
or spaces
1.
2.
3.
4.
5.
Q�(6, 4)
1
The points are notall moved the samedistance or in thesame direction.
(x � 9, y � 4)
x
y
O
Q�
P�
R�
1.
2.
3.
4.
5.
A�(�3, �2)
yes
yes
42
P�Q�
R�R
Q
P
T
Quiz 2Page 206 Quiz 4
Page 206
1.
2.
3.
4.
5.
distance
X�(�4, 12), Y�(20, 28),
Z�(8, �16)
magnitude: �17�,direction: 284°
B
x
y
O
D�
F�
E�D
E
F
1.
2.
3.
4.
5.
Z�(�8, 12), Y �(�4, 16)
M�(�7, 3), N�(�5, 2),O�(�6, 6)
A�(3, �1), B�(�8, 2),C �(�5, 7)
D��1, ��32
��, E��2, �12
��,F���1, �
32
��0 �11 0
© Glencoe/McGraw-Hill A31 Glencoe Geometry
Chapter 9 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 537 Page 538
An
swer
s
Part I
Part II
6.
7.
8.
9.
10.
(7, 1)
D �(�1, 1), E �(�1, 6),F �(4, 6), G�(4, 1)
No, the size hasbeen changed.
B �(�7, 4)
No; measure ofinterior angle � 168.
1.
2.
3.
4.
5.
B
B
B
C
D
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
m�T � 166 andm�S � 14sometimes
4�2�
F��n2
�, b�
23 � x � 113
18
�45� or 6.7
m�S � 49.6,m�V � 48.4,
v � 39.3
no; B�C� ⁄|| D�A�
a � 10, b � 4,m�HJK � 78
�13, �9�
P�
Q�
R�
x
y
O
P
QR
M
© Glencoe/McGraw-Hill A32 Glencoe Geometry
Chapter 9 Assessment Answer KeyStandardized Test Practice
Page 539 Page 541
1.
2.
3.
4.
5.
6.
7. A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D 8. 9.
10. 11.
12.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
13.
14.
15.
m�1 � 77,m�2 � 103
(b � c, a)
E �(7, 6) andF �(�6, �12)
9 6 0 . 1
5 / 7
1 8 6
0 . 8