Topic 8 Transformational Geometry

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Topic 8 Transformational Geometry

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English/Spanish Vocabulary Audio Online:

English Spanish

compression, p. 364 compreción

congruence transformation, p. 350 transformación de congruencia

dilation, p. 356 dilatación

image, p. 318 imagen

preimage, p. 318 preimagen

reflection, p. 326 reflexión

rigid transformation, p. 318 transformación rígido

stretch, p. 364 estiramiento

rotation, p. 332 rotación

translation, p. 319 translación

VOCABULARY

8-1 Translations

8-2 Reflections

8-3 Rotations

8-4 Symmetry

8-5 Compositions of Rigid

Transformations

8-6 Congruence Transformations

8-7 Dilations

8-8 Other Non-Rigid Transformations

TOPIC OVERVIEW

316 Topic 8 Transformational Geometry

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If You Need Help . . .Vocabulary onlineYou’ll find definitions of math terms in both English and Spanish. All of the terms have audio support.

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The Perplexing Polygon

Look around and you will probably see shapes and patterns everywhere you look. The tiles on a floor are often all the same shape and fit together to form a pattern. The petals on a flower frequently create a repeating pattern around the center of the flower. When you look at snowflakes under a microscope, you’ll notice that they are made up of repeating three-dimensional crystals. Think about this as you watch this 3-Act Math video.

Scan page to see a video for this 3-Act Math Task.

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TEKS (3)(A) Describe and perform transformations of figures in a plane using coordinate notation.

TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Additional TEKS (1)(F), (3)(C), (6)(C)

TEKS FOCUS•Composition of transformations – A composition of transformations

is a combination of two or more transformations. In a composition, you perform each transformation on the image of the preceding transformation.

•Image – the resulting figure in a transformation

•Preimage – the original figure in a transformation

•Rigid transformation – a transformation that preserves distance and angle measures

•Transformation – a function, or mapping, that results in a change in the position, shape, or size of a figure

•Translation – a transformation that maps all points of a figure the same distance in the same direction.

•Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated

•Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.

VOCABULARY

You can change the position of a geometric figure so that the angle measures and the distance between any two points of a figure stay the same.

ESSENTIAL UNDERSTANDING

A transformation is a function that maps every point of a figure, called the preimage, onto its image. A transformation may be described with arrow notation (S). Prime notation (′) is sometimes used to identify image points. In the diagram below, K ′ is the image of K.

Notice that you list corresponding points of the preimage and image in the same order, as you do for corresponding points of congruent figures.

Key Concept Transformations

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8-1 Translations

318 Lesson 8-1 Translations

A translation can be performed as a composition of a horizontal and a vertical translation. In the diagram at the right, each point of ABCD is translated 4 units right and 2 units down. So each (x, y) pair in ABCD is mapped to (x + 4, y - 2). You can use the function notation T64, -27 (ABCD) = A′B′C′D′ to describe this translation, where 4 represents the horizontal translation of each point of the figure and -2 represents the vertical translation.

Key Concept Translation in the Coordinate Plane

T64, -27 (x, y) = (x + 4, y - 2)(x, y) S (x + 4, y - 2)

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B moves 4 unitsright and2 units down.


Problem 1

Identifying a Rigid Transformation

Does the transformation below appear to be a rigid transformation? Explain.

No, a rigid transformation preserves both distance and angle measure. In this transformation, the distances between the vertices of the image are not the same as the corresponding distances in the preimage.

Preimage Image


What must be true about a rigid transformation?In a rigid transformation, the image and the preimage must preserve distance and angle measures.

A translation is a transformation that maps all points of a figure the same distance in the same direction.

You write the translation that maps △ABC onto △A′B′C′ using the function notation T (△ABC) = △A′B′C′. A translation is a rigid transformation with the following properties.

If T (△ABC) = △A′B′C′, then• AA′ = BB′ = CC′• AB = A′B′, BC = B′C′, AC = A′C′• m∠A = m∠A′, m∠B = m∠B′, m∠C = m∠C′

Key Concept Translation

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A�C�


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Problem 3

What does the rule tell you about the direction each point moves?-2 means that each point moves 2 units left. -5 means that each point moves 5 units down.

Naming Images and Corresponding Parts

In the diagram, EFGH u E′F′G′H′.

A What are the images of jF and jH?

∠F ′ is the image of ∠F . ∠H ′ is the image of ∠H .

B What are the pairs of corresponding sides?

EF and E′F ′ FG and F ′G′

EH and E′H′ GH and G′H′

TEKS Process Standard (1)(F)

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How do you identify corresponding points?Corresponding points have the same position in the names of the preimage and image. You can use the statement EFGH S E′F′G′H′.

Problem 2

Finding the Image of a Translation

What are the vertices of T*−2, −5+(△PQR)? Graph the image of △PQR.

Identify the coordinates of each vertex. Use the coordinate rule T6-2, -57 (x, y) = (x - 2, y - 5) to find the coordinates of each vertex of the image.

T6-2, -57(P) = (2 - 2, 1 - 5), or P′(0, -4).

T6-2, -57(Q) = (3 - 2, 3 - 5), or Q′(1, -2).

T6-2, -57(R) = (-1 - 2, 3 - 5), or R′(-3, -2).

To graph the image of △PQR, first graph P′, Q′, and R′. Then draw P′Q′, Q′R′, and R′P′.

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Writing a Rule to Describe a Translation

What is a coordinate rule that describes the translation that maps PQRS onto P′Q′R′S′?

Use one pair of corresponding vertices to find the change in the horizontal direction x and the change in the vertical direction y. Then use the other vertices to verify.

An algebraic relationship that maps each point of PQRS onto P′Q′R′S′

The coordinates of the vertices of both figures

The translation maps each (x, y) to (x + 8, y - 2). The coordinate rule that describes the translation is T68, -27(x, y) = (x + 8, y - 2).

TEKS Process Standard (1)(D)

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yP(�3, 4)

Use P(�3, 4) andits image P�(5, 2). Horizontal change: 5 � (�3) � 8

x S x � 8

Vertical change: 2 � 4 � �2 y S y � 2


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How do you know which pair of corresponding vertices to use?A translation moves all points the same distance and the same direction. You can use any pair of corresponding vertices.

Problem 4

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1

2

Composing Translations

Chess The diagram at the right shows two moves of the black bishop in a chess game. Where is the bishop in relation to its original position?

Use (0, 0) to represent the bishop’s original position. Write coordinate rules to represent each move.

T64, -47(x, y) = (x + 4, y - 4) The bishop moves 4 squares right and 4 squares down.

T62, 27(x, y) = (x + 2, y + 2) The bishop moves 2 squares right and 2 squares up.

The bishop’s current position is the composition of the two translations.

First, T64, -47(0, 0) = (0 + 4, 0 - 4), or (4, -4).

Then T62, 27(4, -4) = (4 + 2, -4 + 2), or (6, -2).

The bishop is 6 squares right and 2 squares down from its original position.

Problem 5

How can you define the bishop’s original position?You can think of the chessboard as a coordinate plane with the bishop’s original position at the origin.

PRACTICE and APPLICATION EXERCISES

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For additional support whencompleting your homework, go to PearsonTEXAS.com.

Tell whether the transformation appears to be a rigid transformation. Explain.

1. 2. 3.

4. You are a graphic designer for a company that manufactures wrapping paper. Make a design for wrapping paper that involves translations.

5. Analyze Mathematical Relationships (1)(F) Your friend and her parents are visiting colleges. They leave their home in Enid, Oklahoma, and drive to Tulsa, which is 107 mi east and 18 mi south of Enid. From Tulsa, they go to Norman, 83 mi west and 63 mi south of Tulsa. Where is Norman in relation to Enid?


Preimage

Image


Preimage Image


PreimageImage

Scan page for a Virtual Nerd™ tutorial video.


In each diagram, the blue figure is an image of the black figure. (a) Choose an angle or point from the preimage and name its image. (b) List all pairs of corresponding sides.

6. 7. 8.

9. In the diagram at the right, the orange figure is a translation image of the red figure. Write a coordinate rule that describes the translation.

10. Display Mathematical Ideas (1)(G) △MUG has coordinates M(2, -4), U(6, 6), and G(7, 2). A translation maps point M to M′( -3, 6). What are the coordinates of U′ and G′ for this translation?

11. Justify Mathematical Arguments (1)(G) PLAT has vertices P( -2, 0), L( -1, 1), A(0, 1), and T( -1, 0). The translation T62, -37(PLAT) = P′L′A′T′. Show that PP′, LL′, AA′, and TT ′ are all parallel.

12. Analyze Mathematical Relationships (1)(F) If T65, 77(△MNO) = △M′N′O′, what coordinate rule maps △M′N′O′ onto △MNO?

13. Apply Mathematics (1)(A) The diagram at the right shows the site plan for a backyard storage shed. Local law, however, requires the shed to sit at least 15 ft from property lines. Describe how to move the shed to comply with the law.

14. You write a computer animation program to help young children learn the alphabet. The program draws a letter, erases the letter, and makes it reappear in a new location two times. The program uses the following composition of translations to move the letter.

T65, 77(x, y) followed by T6-9, -27(x, y)

Suppose the program makes the letter W by connecting the points (1, 2), (2, 0), (3, 2), (4, 0) and (5, 2). What points does the program connect to make the last W?

15. Connect Mathematical Ideas (1)(F) △ABC has vertices A( -2, 5), B( -4, -1), and C (2, -3). If T64, 27(△ABC) = △A′B′C′, show that the images of the midpoints of the sides of △ABC are the midpoints of the sides of △A′B′C ′.

16. Explain Mathematical Ideas (1)(G) Explain how to use translations to draw a parallelogram.

17. Use the graph at the right. Write three different rules for which the image of △JKL has a vertex at the origin.


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Find a translation that has the same effect as each composition of translations.

18. T62, 57(x, y) followed by T6-4, 97(x, y)

19. T612, 0.57(x, y) followed by T61, -37(x, y)

Copy each graph. Graph the image of each figure under the given translation.

20. T63, 27(x, y) 21. T6-2, 57(x, y)

The blue figure is a translation image of the black figure. Write coordinate rules to describe each translation.

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TEXAS Test Practice

24. △ABC has vertices A( -5, 2), B(0, -4), and C(3, 3). What are the vertices of the image of △ABC after the translation T67, -57(△ABC)?

A. A′(2, -3), B′(7, -9), C′(10, -2) C. A′( -12, 7), B′( -7, 1), C′( -4, 8)

B. A′( -12, -3), B′( -7, -9), C′( -4, -2) D. A′(2, -3), B′(10, -2), C′(7, -9)

25. In △PQR, PQ = 4.5, QR = 4.4, and RP = 4.6. Which statement is true?

F. m∠P + m∠Q 6 m∠R H. ∠R is the largest angle.

G. ∠Q is the largest angle. J. m∠R 6 m∠P

26. ▱ABCD has vertices A(0, -3), B( -4, -2), and D( -1, 1). Point C is in Quadrant II.

a. What are the coordinates of C? b. Is ▱ABCD a rhombus? Explain.


Use With Lesson 8-2 teks (3)(A), (1)(E)

Paper Folding and ReflectionsActivity Lab

In Activity 1, you will see how a figure and its reflection image are related. In Activity 2, you will use these relationships to construct a reflection image.

Step 1 Use a piece of tracing paper and a straightedge. Using less than half the page, draw a large, scalene triangle. Label its vertices A, B, and C.

Step 2 Fold the paper so that your triangle is covered. Trace △ABC using a straightedge.

Step 3 Unfold the paper. Label the traced points corresponding to A, B, and C as A′, B′, and C′, respectively. △A′B′C′ is a reflection image of △ABC. The fold is the reflection line.

1. Use a ruler to draw AA′. Measure the perpendicular distances from A to the fold and from A′ to the fold. What do you notice?

2. Measure the angles formed by the fold and AA′. What are the angle measures?

3. Repeat Exercises 1 and 2 for B and B′and for C and C′. Then, make a conjecture: How is the reflection line related to the segment joining a point and its image?

Step 1 On regular paper, draw a simple shape or design made of segments. Use less than half the page. Draw a reflection line near your figure.

Step 2 Use a compass and straightedge to construct a perpendicular to the reflection line through one point of your drawing.

4. Explain how you can use a compass and the perpendicular you drew to find the reflection image of the point you chose.

5. Connect the reflection images for several points of your shape and complete the image. Check the accuracy of the reflection image by folding the paper along the reflection line and holding it up to a light source.

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TEKS (1)(E) Create and use representations to organize, record, and communicate mathematical ideas.

Additional TEKS (1)(D), (1)(G), (3)(C)

TEKS FOCUS

•Line of reflection – See reflection.

•Orientation – the order in which the vertices of the figure appear in either a clockwise or counterclockwise order

•Reflection – A reflection across a line m, called the line of reflection, is a transformation such that if a point A is on line m, then the image of A is itself, and if a point B is not on line m, then m is the perpendicular bisector of BB′.


VOCABULARY

When you reflect a figure across a line, each point of the figure maps to another point the same distance from the line but on the other side. The orientation of the figure reverses.


A reflection across a line m, called the line of reflection, is a transformation with the following properties:• If a point A is on line m, then the image of A is itself

(that is, A′ = A).• If a point B is not on line m, then m is the

perpendicular bisector of BB′.You write the reflection across m that takes △ABC to △A′B′C′ as Rm(△ABC) = △A′B′C′.

A reflection is a rigid transformation with the following properties:• Reflections preserve distance. If Rm(A) = A′, and Rm(B) = B′, then AB = A′B′.

• Reflections preserve angle measure. If Rm(∠ABC) = ∠A′B′C′, then m∠ABC = m∠A′B′C′.

• Reflections map each point of the preimage to one and only one corresponding point of its image.

Rm(A) = A′ if and only if Rm(A′) = A.

Key Concept Reflection Across a Line

The preimage B andits image B’ areequidistant fromthe line of reflection.

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8-2 Reflections

326 Lesson 8-2 Reflections

Reflection across the x-axis Reflection across the y-axis

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2

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Q′(4, −3)

Q(4, 3)

y

x-6 -4 -2

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Q′(−4, 3) Q(4, 3)

Multiply the y-coordinate by 21. Multiply the x-coordinate by 21.

Rx@axis (x, y) = (x, -y) Ry@axis(x, y) = (-x, y)

(x, y) S (x, -y) (x, y) S (-x, y)

Key Concept Reflection in the Coordinate Plane

Problem 1

Reflecting a Point Across a Line

Multiple Choice Point P has coordinates (3, 4). What are the coordinates of Ry = 1(P)?

(3, -4)

(0, 4)

(3, -2)

(-3, -2)

Graph point P and the line of reflection y = 1. P and its reflection image across the line must be equidistant from the line of reflection.

P is 3 units above the line y = 1, so P′ must be 3 units below the line y = 1. The line y = 1 is the perpendicular bisector of PP′ if P′ is (3, -2). The correct answer is C.


Stop when the distances of P and P�to the line of reflection are the same.

Move along the line through P thatis perpendicular to the line of reflection.

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2�2

�2

2

4

O 4

yP

P�

How does a graph help you visualize the problem?A graph shows that y = 1 is a horizontal line, so the line through P that is perpendicular to the line of reflection is a vertical line.

327PearsonTEXAS.com

Problem 3

If Triangle 2 is the image of a reflection, what do you know about the preimage? The preimage has opposite orientation, and lies on the opposite side of the line of reflection.

Problem 2

Graphing a Reflection Image

Graph points A(23, 4), B(0, 1), and C(4, 2). Graph and label Ry@axis(△ABC).

Step 1 Graph △ABC. Show the y-axis as the dashed line of reflection.

Step 2 Find A′, B′, and C′ using the coordinate rule (x, y) S (-x, y).

A(- 3, 4) S A′(3, 4)

B(0, 1) S B′(0, 1)

C(4, 2) S C′(-4, 2)

Locate A′(3, 4), B′(0, 1), and C′(-4, 2) on the coordinate plane. Draw △A′B′C′.



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Writing a Reflection Rule

Each triangle in the diagram is a reflection of another triangle across one of the given lines. How can you describe Triangle 2 by using a reflection rule?

Triangle 2 is the image of a reflection, so find the preimage and the line of reflection to write a rule.

The preimage cannot be Triangle 3 because Triangle 2 and Triangle 3 have the same orientation and reflections reverse orientation.

Check Triangles 1 and 4 by drawing line segments that connect the corresponding vertices of Triangle 2. Because neither line k nor line m is the perpendicular bisector of the segment drawn from Triangle 1 to Triangle 2, Triangle 1 is not the preimage.

Line k is the perpendicular bisector of the segments joining corresponding vertices of Triangle 2 and Triangle 4. So Triangle 2 = Rk(Triangle 4).

TEKS Process Standard (1)(E)

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1

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Point B is located on the line of reflection. How will point B9 relate to the line of reflection? Point B9 will also be on the line of reflection.


Problem 4

Using Properties of Reflections

In the diagram, Rt(G) = G, Rt(H) = J, and Rt(D) = D. Use the properties of reflections to describe how you know that △GHJ is an isosceles triangle.

Since Rt(G) = G, Rt(H) = J, and reflections preserve distance, Rt(GH ) = GJ. So GH = GJ and, by definition, △GHJ is an isosceles triangle.

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Create Representations to Communicate Mathematical Ideas (1)(E) Given points J(1, 4), A(3, 5), and G(2, 1), graph △JAG and its reflection image as indicated.

1. Rx-axis 2. Ry-axis 3. Ry = 2

4. Ry = 5 5. Rx = -1 6. Rx = 2

7. Each figure in the diagram at the right is a reflection of another figure across one of the reflection lines.

a. Write a reflection rule to describe Figure 3. Justify your answer.

b. Write a reflection rule to describe Figure 2. Justify your answer.

c. Write a reflection rule to describe Figure 4. Justify your answer.

8. Apply Mathematics (1)(A) Give three examples from everyday life of objects or situations that show or use reflections.

9. In the diagram at the right, LMNP is a rectangle with LM = 2MN.

a. Copy the diagram. Then sketch R LM (LMNP).

b. What figure results from the reflection? Use properties of reflections to justify your solution.

Copy each pair of figures. Then draw the line of reflection you can use to map one figure onto the other.

10. 11.

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Figure 3

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What do you have to know about △GHJ to show that it is an isosceles triangle?Isosceles triangles have at least two congruent sides.

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12. Explain Mathematical Ideas (1)(G) The following steps explain how to reflect point A across the line y = x.

Step 1 Draw line / through A(5, 1) perpendicular to the line y = x. The slope of y = x is 1, so the slope of line / is 1 # (-1), or -1.

Step 2 From A, move two units left and two units up to y = x. Then move two more units left and two more units up to find the location of A′ on line /. The coordinates of A′ are (1, 5).

a. Copy the diagram. Then draw the lines through B and C that are perpendicular to the line y = x. What is the slope of each line?

b. Ry = x(B) = B′ and Ry = x(C) = C ′. What are the coordinates of B′ and C ′?

c. Graph △A′B′C′.

d. Compare the coordinates of the vertices of △ABC and △A′B′C ′. Make a conjecture about the coordinates of the point P(a, b) reflected across the line y = x.

13. In the diagram R(ABCDE) = A′B′C′D′E′. What is the equation of the line of reflection? Write a coordinate rule that describes this reflection.

14. Use Representations to Communicate Mathematical Ideas (1)(E) The coordinates of the vertices of △FGH are F(2, -1), G(-2, -2), and H(-4, 3). Graph △FGH and Ry = x - 3(△FGH).

15. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) △ABC has vertices A(-3, 5), B(-2, -1), and C(0, 3). Graph Ry = -x(△ABC) and label it.

16. Explain Mathematical Ideas (1)(G) The work of artist and scientist Leonardo da Vinci (1452–1519) has an unusual characteristic. His handwriting is a mirror image of normal handwriting.

a. Write the mirror image of the sentence “Leonardo da Vinci was left-handed.” Use a mirror to check how well you did.

b. Explain why the fact about da Vinci in part (a) might have made mirror writing seem natural to him.

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TEXAS Test Practice

33. What is the reflection image of (a, b) across the line y = -6?

A. (a - 6, b) C. (-12 - a, b)

B. (a, b - 6) D. (a, -12 - b)

34. The diagonals of a quadrilateral are perpendicular and bisect each other. What is the most precise name for the quadrilateral?

F. rectangle H. rhombus

G. parallelogram J. kite

35. Write an indirect proof of the following statement: The hypotenuse of a right triangle is the longest side of the right triangle.

17. Display Mathematical Ideas (1)(G) Recall that when a ray of light hits a mirror, it bounces off the mirror at the same angle at which it hits the mirror. You are installing a security camera. At what point on the mirrored wall should you aim the camera at C in order to view the door at D? Draw a diagram and explain your reasoning.

18. Explain Mathematical Ideas (1)(G) When you reflect a figure across a line, does every point on the preimage move the same distance? Explain.

Find the coordinates of each image.

19. Rx = 1(Q) 20. Ry = -1(P)

21. Ry-axis(S) 22. Ry = 0.5(T)

23. Rx = -3(U ) 24. Rx-axis(V)

Explain Mathematical Ideas (1)(G) Can you form the given type of quadrilateral by drawing a triangle and then reflecting one or more times? Explain.

25. parallelogram 26. isosceles trapezoid

27. kite 28. rhombus

29. rectangle 30. square

31. Show that Ry = x(A) = B for points A(a, b) and B(b, a).

32. Use the diagram at the right. Find the coordinates of each image point.

a. Ry = x(A) = A′

b. Ry = -x(A′) = A″

c. Ry = x(A″) = A′″

d. Ry = -x(A′″) = A″″

e. How are A and A″″ related?

Mirrored wall

CD


O�2

�2

1

3

2 4

y

x

P SV

QU

T


x

y A (1, 3)

y � x

y � �x

2

2

�2

�2

331PearsonTEXAS.com

TEKS (3)(C) Identify the sequence of transformations that will carry a given pre-image onto an image on and off the coordinate plane.


Additional TEKS (1)(D), (1)(F), (3)(A), (6)(C)

TEKS FOCUS

•Angle of rotation – the positive number of degrees that a figure rotates

•Center of rotation – See rotation.

•Rotation – A rotation (turn) of x° about point Q, called the center of rotation, is a transformation such that for any point V, its image is the point V′, where QV′ = QV and m∠VQV′ = x. The image of Q is itself.


VOCABULARY

Rotations preserve distance, angle measures, and orientation of figures.


A rotation of x° about a point Q, called the center of rotation, is a transformation with these two properties:• The image of Q is itself (that is, Q′ = Q).• For any other point V, QV ′ = QV and

m∠VQV ′ = x.The number of degrees a figure rotates is the angle of rotation.

A rotation about a point is a rigid transformation. You write the x° rotation of △UVW about point Q as r(x°, Q)(△UVW) = △U′V′W′. Unless stated otherwise, rotations in this course are counterclockwise.

Key Concept Rotation About a Point

V�

Q�

Q

W�

U�

U

WV

x �

The preimage V andits image V� areequidistant fromthe center of rotation.


8-3 Rotations

332 Lesson 8-3 Rotations

r(90°, O)(x, y) = (-y, x)

(x, y) S (-y, x)

r(180°, O)(x, y) = (-x, -y)

(x, y) S (-x, -y)

r(270°, O)(x, y) = (y, -x)

(x, y) S (y, -x)

r(360°, O)(x, y) = (x, y)

(x, y) S (x, y)

Key Concept Rotation in the Coordinate Plane

2 4 6-2-2

-4-6x

2

4y

O

G(2, 3)G′(-3, 2)

G(2, 3)

G′(-2, -3)

1805 2 4 6-2-4-6

x

2

-2

4y

2 4 6

-2-2-4-6

x

2

4y

2705

G(2, 3)

G′(3, -2)

42 6-2-2

-4-6x

2

4y

G′(2, 3)G(2, 3)

3605

Problem 1

Drawing a Rotation Image

What is the image of r(100°, C)(△LOB)?

Step 1Draw CO. Use a protractor to draw a 100° angle with vertex C and side CO.

Step 2Use a compass to construct CO′ ≅ CO.

Step 3Locate B9 and L9 in a similar manner.

Step 4Draw △L′O′B′.


L

O

C

B


L

O

C

100�

B


L

O

C

O�

B


L

O

C

B�

L�O�

B

hsm11gmse_0903_t08073.aihsm11gmse_0903_t08074.ai

L

O

C

B�

L�O�

B

How do you use the definition of rotation about a point to help you get started?You know that O and O9 must be equidistant from C and that m∠OCO′ must be 100.

333PearsonTEXAS.com

Problem 3

Problem 2

Drawing Rotations in a Coordinate Plane

PQRS has vertices P(1, 1), Q(3, 3), R(4, 1), and S(3, 0). What is the graph of r(90°, O)(PQRS)?

Find and graph the image of each vertex. Use the coordinate rule that describes a 90° rotation about the origin: r(90°, O)(x, y) = (-y, x).

P′ = r(90°, O)(1, 1) = (-1, 1)

Q′ = r(90°, O)(3, 3) = (-3, 3)

R′ = r(90°, O)(4, 1) = (-1, 4)

S′ = r(90°, O)(3, 0) = (0, 3)

Next, connect the vertices to graph P′Q′R′S′.


geom12_se_ccs_c09l03_t04.ai

2 4 6

�2

�4 �2�6x

4y

R

SO

P

Q

P� (�1, 1)

Q� (�3, 3)

R� (�1, 4)S� (0, 3)

Using Properties of Rotations

In the diagram, WXYZ is a parallelogram, and T is the midpoint of the diagonals. How can you use the properties of rotations to show that the lengths of the opposite sides of the parallelogram are equal?

Because T is the midpoint of the diagonals, XT = ZT and WT = YT. Since W and Y are equidistant from T, and the measure of ∠WTY = 180, you know that r(180°, T)(W) = Y . Similarly, r(180°, T)(X) = Z .

You can rotate every point on WX in this same way, so r(180°, T)(WX) = YZ .

Likewise, you can map WZ to YX with r(180°, T)(WZ) = YX .

Because rotations are rigid transformations and preserve distance, WX = YZ and WZ = YX .geom12_se_ccs_c09l03_t05.ai

W Z

YX

T

How do you know where to draw the vertices on the coordinate plane?Use the rules for rotating a point and apply them to each vertex of the figure. Then graph the points and connect them to draw the image.

What do you know about rotations that can help you show that opposite sides of the parallelogram are equal?You know that rotations are rigid transformations, so if you show that the opposite sides can be mapped to each other, then the side lengths must be equal.



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For Exercises 1 and 2, use the graph below.

1. Graph r(90°, O)(FGHJ). 2. Graph r(270°, O)(FGHJ).

3. The coordinates of △PRS are P(-3, 2), R(2, 5), and S(0, 0). Use a coordinate rule to find the coordinates of the vertices of r(270°, O)(△PRS).

4. Create Representations to Communicate Mathematical Ideas (1)(E) Draw △LMN with vertices L(2, -1), M(6, -2), and N(4, 2). Find the coordinates of the vertices after a 90° rotation about the origin and about each of the points L, M, and N.

5. Explain Mathematical Ideas (1)(G) If you are given a figure and a rotation image of the figure, how can you find the center and angle of rotation?


4�4�6

�4

2

4y

O

F (0, 3) J (3, 2)

H (1, �4)

G (�4, 1)x

6

Identifying a Sequence of Transformations

You are rearranging the furniture in your living room. Identify a sequence of translations and rotations that will move the sofa from its location in the southwest corner of the floor plan to a new location in the northeast corner, facing west.

Step 1 Rotate the sofa 180° about the point marked by the black dot.

Step 2 Translate the sofa north until it is against the northwest corner.

Step 3 Translate the sofa east until it is against the northeast corner.

N

180º

N

Problem 4

Could you do these steps in a different order?Yes. For instance, you could translate the sofa before rotating it.


335PearsonTEXAS.com

6. Display Mathematical Ideas (1)(G) The Millenium Wheel, also known as the London Eye, contains 32 observation cars. Determine the angle of rotation that will bring Car 3 to the position of Car 18.

7. Explain Mathematical Ideas (1)(G) For center of rotation P, does an x° rotation followed by a y° rotation give the same image as a y° rotation followed by an x° rotation? Explain.

8. Describe how a series of rotations can have the same effect as a 360° rotation about a point X.

9. Create Representations to Communicate Mathematical Ideas (1)(E) Graph A(5, 2). Graph B, the image of A for a 90° rotation about the origin O. Graph C, the image of A for a 180° rotation about O. Graph D, the image of A for a 270° rotation about O. What type of quadrilateral is ABCD? Explain.

Point O is the center of the regular nonagon shown at the right.

10. Analyze Mathematical Relationships (1)(F) Describe a rotation that maps H to C.

11. Evaluate Reasonableness (1)(B) Your friend says that AB is the image of ED for a 120° rotation about O. What is wrong with your friend’s statement?

Copy each figure and point P. Draw the image of each figure for the given rotation about P. Use prime notation to label the vertices of the image.

12. 60° 13. 90° 14. 180°

15. In the diagram at the right, the figures are congruent. Identify a sequence of transformations that will carry Figure 1 to Figure 2.

16. V′W′X′Y′ has vertices V′(-3, 2), W′(5, 1), X′(0, 4), and Y′(-2, 0). If r(90°, O)(VWXY) = V′W′X′Y′, what are the coordinates of VWXY?

17. A Ferris wheel is drawn on a coordinate plane so that the first car is located at the point (30, 0). What are the coordinates of the first car after a rotation of 270° about the origin?

Car 3

Car 18

Car 3

Car 18

A B

C

DO

EF

G

H

I



B

A

D

P


R

P

E

T C


DP

R

B

Figure 1

Figure 2


Connect Mathematical Ideas (1)(F) Use the diagram at the right. TQNV is a rectangle. M is the midpoint of the diagonals.

18. Can you use the properties of rotations to show that the lengths of the diagonals are equal? Explain.

19. Can you use properties of rotations to conclude that the diagonals of TQNV bisect the angles of TQNV? Explain.

20. Apply Mathematics (1)(A) Symbols are used in dictionaries to help users pronounce words correctly. The symbol is called a schwa. It is used in dictionaries to represent neutral vowel sounds such as a in ago, i in sanity, and u in focus. What transformation maps a to a lowercase e?

21. A classmate says that the puzzle piece shown can fit into both Location A and Location B using only a sequence of translations and rotations. Is the classmate correct? Explain your reasoning by identifying a sequence of transformations that will carry the piece onto Locations A and B in the puzzle.

22. Use Representations to Communicate Mathematical Ideas (1)(E) Draw a bird’s-eye view of one room in your house, labeling the four cardinal directions (north, south, east, and west). Draw a second bird’s-eye view with one piece of furniture moved to a new location in the room. Identify a sequence of transformations that will carry the piece of furniture from its initial location to its new location.


NM

T

V

Q

A B

TEXAS Test Practice

23. What is the image of (1, -6) after a 90° counterclockwise rotation about the origin?

A. (6, 1) B. (-1, 6) C. (-6, -1) D. (-1, -6)

24. The costume crew for your school musical makes aprons like the one shown. If blue ribbon costs $1.50 per foot, what is the cost of ribbon for six aprons?

F. $15.75 H. $42.00

G. $31.50 J. $63.00

25. Use the following statement: If two lines are parallel, then the lines do not intersect.

a. What are the converse, inverse, and contrapositive of the statement?

b. What is the truth value of each statement you wrote in part (a)? If a statement is false, give a counterexample.

hsm11gmse_0903_t14047

18 in.

24 in.

5 in.

5 in.

5 in.

5 in.

337PearsonTEXAS.com

TEKS (3)(D) Identify and distinguish between reflectional and rotational symmetry in a plane figure.

TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Additional TEKS (1)(D), (1)(E)

TEKS FOCUS

•Line of symmetry – See reflectional symmetry.

•Line symmetry – See reflectional symmetry.

•Point symmetry – A figure has point symmetry if it has 180° rotational symmetry.

•Reflectional symmetry – A figure has reflectional symmetry, or line symmetry, if there is a reflection for which the figure is its own image. The line of reflection is called the line of symmetry. It divides the figure into congruent halves.

•Rotational symmetry – A figure has rotational symmetry if there is a rotation of 180° or less for which the figure is its own image. The angle of rotation for rotational symmetry is the smallest angle needed for the figure to rotate onto itself.

•Number sense – the understanding of what numbers mean and how they are related

VOCABULARY

Some figures appear unchanged after a reflection across a line or a rotation about a point. Such figures are said to have symmetry .


A figure has line symmetry or reflectional symmetry if there is a reflection for which the figure is its own image. The line of reflection is called a line of symmetry. It divides the figure into congruent halves.

A figure has rotational symmetry if there is a rotation of 180° or less for which the figure is its own image. The angle of rotation for rotational symmetry is the smallest angle needed for the figure to rotate onto itself.

A figure with 180° rotational symmetry also has point symmetry. Each segment joining a point and its 180° rotation image passes through the center of rotation.

A square, which has both 90° and 180° rotational symmetry, also has point symmetry.

Key Concept Types of Symmetry

1205

1805

8-4 Symmetry

338 Lesson 8-4 Symmetry

Problem 2

Problem 1

Identifying Lines of Symmetry

How many lines of symmetry does a regular hexagon have? Select a tool (such as geoboards, pencil and paper, or geometry software) that will help you draw a diagram of a regular hexagon.

TEKS Process Standard (1)(C)

How do you identify rotational symmetry?Look for a possible center point. Think about the angles formed by joining preimage-image pairs to the center. All these angles must be congruent for the figure to have rotational symmetry.

725

Identifying Rotational Symmetry

Does the figure appear to have rotational symmetry? If so, what is the angle of rotation?

A There is no center point about which the triangle will rotate onto itself. This figure does not have rotational symmetry.

B The star has rotational symmetry. The angle of rotation is 72°.

A regular hexagon has six lines of symmetry.

Use a pencil and paper to draw a diagram of a regular hexagon. Look for the ways the hexagon will reflect across a line onto itself.

Count the lines of symmetry.

The hexagon reflects onto itself across each line that passes through the midpoints of a pair of parallel sides.

The hexagon also reflects onto itself across each diagonal that passes through the center of the hexagon.

339PearsonTEXAS.com

Problem 3

Distinguishing Between Rotational and Reflectional Symmetry

Does the plane figure appear to have rotational symmetry, reflectional symmetry, neither, or both? Explain your reasoning.

A

Both

A regular pentagon looks the same after being rotated 72° about its center. So a regular pentagon has rotational symmetry.

There are five lines shown that divide the pentagon in half so that one half is the same as the other. So a regular pentagon has reflectional symmetry.

B

Neither

The letter R does not look the same after being rotated less than 180° about its center. So the letter R does not have rotational symmetry.

There are no lines that divide the letter R in half so that one half is the mirror image of the other. So the letter R does not have reflectional symmetry.

C

Reflectional Symmetry

The letter M does not look the same after being rotated less than 180° about its center. So the letter M does not have rotational symmetry.

There is one line that divides the letter M in half so that one half is the mirror image of the other. So the letter M has reflectional symmetry.

D

Rotational Symmetry

The letter S looks the same after being rotated 180° about its center. So the letter S has rotational symmetry.

There are no lines that divide the letter S in half so that one half is the mirror image of the other. So the letter S does not have reflectional symmetry.

R

MM S S

How can you tell if afigure has rotationalsymmetry?A figure has rotational symmetry if there is a rotation of 180° or less for which the figure is unchanged.



PRACTICE and APPLICATION EXERCISESON

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1. Display Mathematical Ideas (1)(G) Use the letters of the alphabets below. English: ABCDEFGHIJKLMNOPQRSTUVWXYZ

Greek: ABGDEZHQIKLMNJOPRSTYFXCV

Type of Symmetry

LanguageHorizontal

LineVertical

Line Point

English

Greek

Alphabet Symmetry

a. Copy the table. Classify the letters of the alphabets. You will list some letters in more than one category.

b. Which alphabet has more symmetrical letters? Explain.

Identify whether each figure appears to have rotational symmetry, reflectional symmetry, neither, or both. If it has reflectional symmetry, sketch the figure and the line(s) of symmetry. If it has rotational symmetry, tell the angle of rotation.

2. 3. 4.

5. 6. 7.

8. 9. 10.

11. 12. 13.


341PearsonTEXAS.com

Select Tools to Solve Problems (1)(C) Determine how many lines of symmetry each type of quadrilateral has. Select a tool, such as a geoboard or pencil and paper, to help you solve the problem. Include a sketch to support your answer.

14. rhombus 15. kite

16. square 17. parallelogram

18. If you stack the letters of MATH vertically, you can find a vertical line of symmetry. Find two other words for which this is true.

19. Connect Mathematical Ideas (1)(F) A quadrilateral with vertices at (1, 5) and (22, 23) has point symmetry about the origin. Show that the quadrilateral is a parallelogram.

Tell what type(s) of symmetry each figure appears to have. For reflectional symmetry, sketch the figure and the line(s) of symmetry. For rotational symmetry, tell the angle of rotation.

20. 21.

22. Explain Mathematical Ideas (1)(G) Is the line that contains the bisector of an angle also a line of symmetry of the angle? Explain.

23. Explain Mathematical Ideas (1)(G) Is the line that contains the bisector of an angle of a triangle also a line of symmetry of the triangle? Explain.

24. The equation 1010 - 1 = 0 , 83

83 is not only true, but also symmetrical (horizontally). Write four other equations or inequalities that are both true and symmetrical.

Analyze Mathematical Relationships (1)(F) A figure that has a vertex at (3, 4) has the given line of symmetry. Tell the coordinates of another vertex of the figure.

25. the y-axis

26. the x-axis

27. the line y = x

Use Representations to Communicate Mathematical Ideas (1)(E) Graph each equation and describe its symmetry.

28. y = x2 29. y = (x + 2)2

30. y = x3 31. y = |x|


TEXAS Test Practice

42. What is the smallest angle, in degrees, through which you can rotate a regular hexagon onto itself?

43. You place a sprinkler so that it is equidistant from three rose bushes at points A, B, and C. How many feet is the sprinkler from A?

44. △STU has vertices S(1, 2), T(0, 5), and U(28, 0). What is the x-coordinate of S after a 270° rotation about the origin?

45. The diagonals of rectangle PQRS intersect at O. PO = 2x - 5 and OR = 7 - x. What is the length of QS?

3 yd4 yd

A

B

C

For each three-dimensional figure, draw a net that has point symmetry and a net that has 1, 2, or 4 lines of symmetry. (A net is a two-dimensional diagram that you can fold to form a three-dimensional figure.)

32. 33.

34. Do all regular polygons have rotational symmetry? Explain your reasoning.

35. Do all regular polygons have point symmetry? Explain your reasoning.

36. Use a straightedge to copy the rhombus at the right.

a. How many lines of symmetry does the rhombus have?

b. Draw all the lines of symmetry.

37. Do all parallelograms have reflectional symmetry? Explain your reasoning.

Apply Mathematics (1)(A) Describe the types of symmetry, if any, of each logo.

38. 39.

40. 41.

Square pyramid


343PearsonTEXAS.com


TEKS (1)(G) Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Additional TEKS (1)(D), (1)(E), (3)(B)

TEKS FOCUS

•Glide reflection – the composition of a translation (a glide) and a reflection across a line parallel to the direction of translation

•Justify – explain with logical reasoning. You can justify a mathematical argument.

•Argument – a set of statements put forth to show the truth or falsehood of a mathematical claim

VOCABULARY

You can express all rigid transformations as compositions of reflections.


The composition of two or more rigid transformations is a rigid transformation.

Theorem 8-1

There are only four kinds of rigid transformations.


Translation

Orientations are the same. Orientations are opposite.

Rotation Reflection Glide Reflection

R RRRRR RR

R

Orientations are the same. Orientations are opposite.

Key Concept Classification of Rigid Transformations

A composition of reflections across two parallel lines is a translation.

You can write this composition as (Rm ∘ R/)(△ABC) = △A″B″C″ or Rm(R/(△ABC)) = △A″B″C″.

AA″, BB″, and CC″ are all perpendicular to lines / and m.

Theorem 8-2 Reflections Across Parallel Lines


A�

B�

C�

A B

C

A� B�

C��

m

8-5 Compositions of Rigid Transformations

344 Lesson 8-5 Compositions of Rigid Transformations

Problem 1

Composing Reflections Across Parallel Lines

What is (Rm ∘ RO)( J)? What is the distance of the resulting translation?

As you do the two reflections, keep track of the distance moved by a point P of the preimage.

The red arrow shows the translation. The total distance P moved is 2 # AB. Because <AB

># /, AB is the distance between / and m. The distance of the translation is twice the

distance between / and m.


m

�

J


PP�

P�

AB

Step 1 Reflect J across �. PA � AP�, so PP� � 2AP�.

m

�

P moved a total distance of 2AP� � 2P�B, or 2AB.

J J J

Step 2 Reflect the image across m. P�B � BP�, so P�P� � 2P�B.

A composition of reflections across two intersecting lines is a rotation.

You can write this composition as (Rm ∘ R/)(△ABC) = △A″B″C″ or Rm(R/(△ABC)) = △A″B″C″.

The figure is rotated about the point where the two lines intersect—in this case, point Q.

Theorem 8-3 Reflections Across Intersecting Lines


A�

B�C�

A

Q

B

m�

CA�

B�

C�

How do you know that PA = AP′, P′B = BP ″, and <AB

> # O?

All three statements are true by the definition of reflection across a line.

345PearsonTEXAS.com

Problem 3

Problem 2

Composing Reflections Across Intersecting Lines

Lines O and m intersect at point C and form a 70° angle. What is (Rm ∘ RO)(J)? What are the center of rotation and the angle of rotation for the resulting rotation?

After you do the reflections, follow the path of a point P of the preimage.

J is rotated clockwise about the intersection point of the lines. The center of rotation is C. You know that m∠2 + m∠3 = 70. You can use the definition of reflection to show that m∠1 = m∠2 and m∠3 = m∠4. So m∠1 + m∠2 + m∠3 + m∠4 = 140. The angle of rotation is 140° clockwise.

TEKS Process Standard (1)(G)


m

C

�

J 70�


m

C

12 3 4

P

P�

P�

�

J

Step 1 Reflect J across �.

Step 2 Reflect the image across m.

Step 3 Draw the angles formed by joining P, P�, and P� to C.

J

J

Finding a Glide Reflection Image

Coordinate Geometry What is (Rx = 0 ∘ T60, −57)(△TEX)?


x

y

O�4 �2

2

1X

ET

First use the translation rule to translate △TEX . Then reflect the translation image of each vertex across the line of reflection.

The image of △TEX for the glide reflection

•Theverticesof△TEX•Thetranslationrule•Thelineofreflection


x

y

O�4

2

2 4X

E

E�

T�

X�

T

Use the translation ruleT<0, �5> (�TEX ) to move �TEX down 5 units.

Reflect the image of �TEX across the line x � 0.

How do you show that mj1 = mj2?If you draw PP′ and label its intersection point with line / as A, then PA = P′A andPP′ # /. So, by the Converse of the Angle Bisector Theorem, m∠1 = m∠2.


Problem 4


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Determining Preimages Under Rigid Transformations

If (Rx@axis ∘ T<5, 0>) (△ABC) = △A″B″C″, then what are the coordinates of A, B, and C? Graph △ABC, △A′B′C′, and△A″B″C″.

The coordinates of △A″B″C″ are A″(3, -1), B″(4, -4), and C″(1, -2). △A″B″C″ is a transformation of △ABC, where △ABC was translated 5 units right and then reflected across the x-axis. You can determine the graph of △ABC by performing the transformations in reverse.

Step 1 Reflect △A″B″C″ across the x-axis to find the vertices of △A′B′C′, the pre-image of △A″B″C″ before the reflection.

The vertices of △A′B′C′ are A′(3, 1), B′(4, 4), and C′(1, 2).

Step 2 Translate △A′B′C′ 5 units left to find the vertices of △ABC, the pre-image of △A′B′C′ before the translation 5 units right.

The vertices of △ABC are A(-2, 1), B(-1, 4), and C(-4, 2).


y x

-2

-4

2 4O A″

B″

C ″

y4

2

O 42-2-2

-4

-4x

A″

B″

C ″

B′B

A A′

C′C

Identify each mapping as a translation, a reflection, a rotation, or a glide reflection. Write the rule for each translation, reflection, rotation, or glide reflection. For glide reflections, write the rule as a composition of a translation and a reflection.

1. △ABC S △EDC 2. △MNP S △EDC 3. △EDC S △PQM

4. △JLM S △MNJ 5. △PQM S △KJN 6. △HGF S △KJN

7. △ROS was reflected across the y-axis, then reflected across the x-axis, and then translated 2 units right. The resulting triangle has vertices at R‴(8, -3), O‴(4, 5), and S‴(-3, -6). What are the coordinates of R, O, and S?


x

y

O

4

2

1 7

�2

�4

A

B

C

D

EF

G

H

I

J

K N

M P

L Q

How can you find the preimage of a figure that was translated 5 units to the right?You can perform the translation in reverse by translating the figure 5 units to the left.


347PearsonTEXAS.com

Display Mathematical Ideas (1)(G) Find the image of each letter after the transformation Rm ∘ RO. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

8. 9.

10. 11.

Graph △PNB and its image after the given transformation.

12. (Ry = 3 ∘ T62, 07)(△PNB)

13. (Rx = 0 ∘ T60, -37)(△PNB)

14. (Ry = x ∘ T6-1, 17)(△PNB)

15. Analyze Mathematical Relationships (1)(F) Let A′ be the point (1, 5). If (Ry = 1 ∘ T63, 07)(A) = A′, then what are the coordinates of A?

Create Representations to Communicate Mathematical Ideas (1)(E) Graph the preimage of each triangle in the coordinate plane before the given composition of transformations.

16. (Rx = 0 ∘ T<0, 4>)(△DEF) = △D″E″F″ 17. (Ry = 0 ∘ r<180°, O>)(△PQR) = △P″Q″R″


Mm

�


Tm

�


Lm

85�

C

�


Nm

75�

C

�


x

y

O

BN

P

2

2

�2

y4

2

O 42-2-2

-4

-4xF″

D″

E″ y4

2

O 42-2-2

-4

-4x

R″

P ″

Q″


TEXAS Test Practice

25. What is (Rx = 0 ∘ T<-12, -6>)(11, -5)?

A. (1, -11) B. (-1, 11) C. (1, 11) D. (-1, -11)

26. ABCD is a rectangular window divided into 12 panes of glass. E, F, G, and H are midpoints of AB, BC, CD, and AD, respectively. Which statement must be true?

F. The quadrilateral panes are squares.

G. The quadrilateral panes are rhombuses.

H. The triangular panes are all congruent.

J. The triangular panes are right triangles.

27. A triangle has side lengths 7 in., 9 in., and x in. Which inequality must be true?

A. 7 6 x 6 9 B. -2 6 x 6 9 C. 2 6 x 6 16 D. 7 6 x 6 16

28. △ABC and △HIG are acute triangles such that △ABC ≅ △HIG. BL and IT are altitudes of the two triangles. Is BL ≅ IT ? Justify your answer.

hsm11gmse_0906_t14039

A

H

D G C

E B

F

Describe the rigid transformation that maps the black figure onto the blue figure.

18. 19.

20. Describe a glide reflection that maps the black R to the blue R.

Use the given points and lines. Graph AB and its image A″B ″ after a reflection first across O1 and then across O2. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

21. A(2, 4) and B(3, 1); /1: x-axis; /2: y-axis

22. A( -4, -3) and B( -4, 0); /1: y = x; /2: y = -x

23. A(6, -4) and B(5, 0); /1: x = 6; /2: x = 4

24. Connect Mathematical Ideas (1)(F) Does an x° rotation about a point P followed by a reflection across a line / give the same image as a reflection across / followed by an x° rotation about P? Explain.


x

y

O1

3

�3

�1�3


x

y

O 1

2

4

�2

�2�4


RR

349PearsonTEXAS.com

TEKS (6)(C) Apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles.

TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

Additional TEKS (1)(D), (1)(G), (3)(A), (3)(B), (3)(C)

TEKS FOCUS

•Congruent – Two figures are congruent if and only if there is a sequence of one or more rigid transformations that maps one figure onto the other.

•Congruence transformation – a transformation in which an original figure and its image are congruent

•Analyze – closely examine objects, ideas, or relationships to learn more about their nature

VOCABULARY

You can use compositions of rigid transformations to understand congruence.


Two figures are congruent if and only if there is a sequence of one or more rigid transformations that maps one figure onto the other. This is a second way to define congruence.

Key Concept Congruent Figures

Problem 1

Identifying Corresponding Sides and Angles

The composition (Rn ∘ r(90°, P))(LMNO) ∙ GHJK is shown at the right. Since LMNO maps to GHJK by a sequence of rigid transformations, the figures are congruent.

A Which angle pairs have equal measures?

Because compositions of rigid transformations preserve angle measure, corresponding angles have equal measures.

m∠L = m∠G, m∠M = m∠H , m∠N = m∠J , and m∠O = m∠K

B Which sides have equal lengths?

By definition, rigid transformations preserve distance. So corresponding side lengths have equal measures.

LM = GH, MN = HJ, NO = JK, and LO = GK



n

P H

G

J

K

L M

NO

How can you use the properties of rigid transformations to find equal angle measures and equal side lengths?Rigid transformations preserve angle measure and distance, so identify corresponding angles and corresponding side lengths.


350 Lesson 8-6 Congruence Transformations

Problem 3

Problem 2

Identifying Congruent Figures

Which pairs of figures in the grid are congruent? For each pair, what is a sequence of rigid transformations that maps one figure to the other?

Figures are congruent if and only if there is a sequence of rigid transformations that maps one figure to the other. So, to find congruent figures, look for sequences of translations, rotations, and reflections that map one figure to another.

Because r(180°, O)(△DEF) = △LMN , the triangles are congruent. Because (T6-1, 57 ∘ Ry@axis)(ABCJ) = WXYZ ,

the trapezoids are congruent. Because T6-2, 97(HG) = PQ, the line segments are congruent.

geom12_se_ccs_c09l05_t03_updated.ai

y

xO�2 2

2

�6 6

6

�4

�2

�6

4

4

HG

D

E

F

A B

CJ

L

M

N

Y

X WQ

P

Z

Identifying Congruence Transformations

In the diagram at the right, △JQV @ △EWT. What is a congruence transformation that maps △JQV onto △EWT ?

Identify the corresponding parts and find a congruence transformation that maps the preimage to the image. Then use the vertices to verify the congruence transformation.

A sequence of rigid transformations that maps △JQV onto △EWT

The coordinates of the vertices of the triangles

Because △EWT lies above △JQV on the plane, a translation can map △JQV up on the plane. Also, notice that △EWT is on the opposite side of the y-axis and has the opposite orientation of △JQV. This suggests that the triangle is reflected across the y-axis.

It appears that a translation of △JQV up 5 units followed by a reflection across the y-axis maps △JQV to △EWT . Verify by using the coordinates of the vertices.

T60, 57(x, y) = (x, y + 5)

T60, 57(J) = (2, 4)

Ry@axis(2, 4) = (-2, 4) = E


y

x

T

E

Q

J

V

WO

2

�4

�2

�4

4�2

4


y

x

T

E

Q

J

V

WO

2

4 2

2

4

4

4

Does one rigid transformation count as a sequence?Yes. It is a sequence of length 1.

continued on next page ▶

351PearsonTEXAS.com

Problem 4

continuedProblem 3

Next, verify that the sequence maps Q to W and V to T.

T60, 57(Q) = (1, 1) T60, 57(V) = (5, 2)

Ry@axis(1, 1) = (-1, 1) = W Ry@axis(5, 2) = (-5, 2) = T

So the congruence transformation Ry@axis ∘ T60, 57 maps △JQV onto △EWT . Note that there are other possible congruence transformations that map △JQV onto △EWT .

Verifying the SAS Postulate

Given: ∠J ≅ ∠P, PA ≅ JO, FP ≅ SJ

Prove: △JOS ≅ △PAF

Step 1 Translate △PAF so that points A and O coincide.

Step 2 Because PA ≅ JO, you can rotate △PAF about point A so that PA and JO coincide.

Step 3 Reflect △PAF across PA. Because reflections preserve angle measure and distance, and because ∠J ≅ ∠P and FP ≅ SJ , you know that the reflection maps ∠P to ∠J and FP to SJ . Since points S and F coincide, △PAF coincides with △JOS.

There is a congruence transformation that maps △PAF onto △JOS, so △PAF ≅ △JOS.

Proof



S

O

J

P

F A


O

S

J

A

P

F


P

A

F

O

J

S


OA

FS

PJ

How do you show that the two triangles are congruent?Find a congruence transformation that maps one onto the other.



LINE

HO

M E W O RK


For each coordinate grid, identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image.

1. 2. 3.

4. Apply Mathematics (1)(A) Artists frequently use congruence transformations in their work. The artworks shown below are called tessellations. What types of congruence transformations can you identify in the tessellations?

a. b.

Analyze Mathematical Relationships (1)(F) Find a congruence transformation that maps △LMN to △RST .

5. 6.

7. Verify the ASA Postulate for triangle congruence by using congruence transformations.

Given: EK ≅ LH Prove: △EKS ≅ △HLA

∠E ≅ ∠H

∠K ≅ ∠L


x

y

B

J

T

V

Q

E

YL

G

2 4�4

�4

4


G

A

D

F

C

R

y

xO�4 �2

�2

�4

4

4


x

y

FA

E

K

S

T

B

D

M

I

C

2

�4

4

�4


x

y

L

M N

S

R

T

O 2

2

�4

4

x

yL

M

N

SR

TO 2 4

2

-4 -2

4

Proof


L

H

A

E K

S


353PearsonTEXAS.com

8. Justify Mathematical Arguments (1)(G) Verify the AAS Postulate for triangle congruence by using congruence transformations.

Given: ∠I ≅ ∠V Prove: △NVZ ≅ △CIQ

∠C ≅ ∠N

QC ≅ NZ

9. If two figures are ________________, then there is a sequence of rigid transformation that maps one figure onto the other.

10. The graph at the right shows two congruent isosceles triangles. What are four different rigid transformations that map the top triangle onto the bottom triangle?

11. Prove the statements in parts (a) and (b) to show that congruence in terms of transformations is equivalent to the criteria for triangle congruence you learned in Topic 4.

a. If there is a congruence transformation that maps △ABC to △DEF , then corresponding pairs of sides and corresponding pairs of angles are congruent.

b. In △ABC and △DEF , if corresponding pairs of sides and corresponding pairs of angles are congruent, then there is a congruence transformation that maps △ABC to △DEF .

12. Apply Mathematics (1)(A) Cookie makers often use cookie cutters so that the cookies all look the same. The baker fills a cookie sheet for baking as shown. What types of congruence transformations can you use to show that the cookies are congruent to one another?

13. Use congruence transformations to prove the Isosceles Triangle Theorem.

Given: FG ≅ FH

Prove: ∠G ≅ ∠H

Proof


ZV

N

C

I Q

x

y

O 2 4�4 �2


Proof

Proof


F

H

G

TEXAS Test Practice

14. In △FGH and △XYZ , ∠G and ∠Y are right angles. FH ≅ XZ and GH ≅ YZ . If GH = 7 ft and XY = 9 ft, what is the area of △FGH in square inches?

15. A classmate says that a certain regular polygon has 50° rotational symmetry. Explain your classmate’s error.


Use With Lesson 8-7

Exploring DilationsActivity Lab

teks (3)(A), (1)(E)

In this activity, you will explore the properties of dilations. A dilation is defined by a center of dilation and a scale factor.

To dilate a segment by a scale factor n with center of dilation at the origin, you measure the distance from the origin to each point on the segment. The diagram at the right shows the dilation of GH by the scale factor 3 with center of dilation at the origin. To locate the dilation image of GH , draw rays from the origin through points G and H. Then, measure the distance from the origin to G. Next, find the point along the same ray that is 3 times that distance. Label the point G′. Now dilate the endpoint H similarly. Draw G′H′.

1. Graph RS with R(1, 4) and S(2, -1). What is the length of RS?

2. Graph the dilations of the endpoint of RS by scale factor 2 and center of dilation at the origin. Label the dilated endpoints R′ and S′.

3. What are the coordinates of R′ and S′?

4. Graph R′S′.

5. What is R′S′?

6. How do the lengths of RS and R′S′ compare?

7. Graph the dilation of RS by scale factor 12 with center of dilation at the origin. Label the dilation R″S″.

8. What is R″S″?

9. How do the lengths of R′S′ and R″S″ compare?

10. What can you conjecture about the length of a line segment that has been dilated by scale factor n?

11. Draw a line on a coordinate grid that does not pass through the origin. Use the method in Activity 1 to construct several dilations of the line you drew with different scale factors (not equal to 1). Make a conjecture relating the slopes of the original line and the dilations.

12. On a new coordinate grid, draw a line through the origin. What happens when you try to construct a dilation of this line? Explain.

11y

G�

H�H

G

geo12_se_ccs_c09l06a_patches.ai

O

2

6

6

8

�2

4

4

x

22

355PearsonTEXAS.com


TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Additional TEKS (1)(F), (1)(G), (3)(B)

TEKS FOCUS

•Center of dilation – See dilation.

•Dilation – A dilation with center of dilation C and scale factor n, where n 7 0, is a transformation that maps a point R to R′ in such a way that R′ is on CR

> and

CR′ = n # CR. The image of C is itself.

•Enlargement – A dilation is an enlargement if the scale factor n is greater than 1.

•Non-rigid transformation – a transformation in the plane that does not necessarily preserve distance or angle measure

•Ratio – A ratio is a comparison of two quantities by division. You can write the ratio of two numbers a and b, where b ≠ 0, in three ways: ab, a : b, or a to b.

•Reduction – A dilation is a reduction if the scale factor n is between 0 and 1.

•Scale factor of a dilation – the ratio of the distances from the center of dilation to an image point and to its preimage point.

•Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated


VOCABULARY

You can use a scale factor to make a larger or smaller copy of a figure.


A dilation with center of dilation C and scale factor n, n 7 0, can be written as D(n, C). A dilation is a transformation with the following properties.

• The image of C is itself (that is, C′ = C).

• For any other point R, R′ is on CR> and CR′ = n # CR, or n = CR′

CR .

• Dilations preserve angle measure.

Key Concept Dilation


C� � C

P�

PQ�Q

R�R

CR� � n � CRS

8-7 Dilations

356 Lesson 8-7 Dilations

Problem 1

Finding a Scale Factor

Multiple Choice Is D(n, X)(△XTR) = △X′T′R′ an enlargement or a reduction? What is the scale factor n of the dilation?

enlargement; n = 2 reduction; n = 13

enlargement; n = 3 reduction; n = 3

The image is larger than the preimage, so the dilation is an enlargement. Use the ratio of the lengths of corresponding sides to find the scale factor.

n = X′T ′XT = 4 + 8

4 = 124 = 3

△X ′T ′R′ is an enlargement of △XTR, with a scale factor of 3. The correct answer is B.

Why is the scale factor not 4

12, or 13?The scale factor of a dilation always has the image length (or the distance between a point on the image and the center of dilation) in the numerator.


R T

T�R�

4

8

X� � X

For a dilation centered at the origin, you can find the image of a point P(x, y) by multiplying the coordinates of P by the scale factor n. The coordinate rule for a dilation of scale factor n with center of dilation at the origin can be written as shown below.

Dn (x, y) = (nx, ny)

Key Concept Dilations Centered at the Origin

ny

y

y

x

nxxO

OP� � n � OP

P�(nx, ny)

P(x, y)


Key Concept Dilations Not Centered at the Origin

The center of a dilation can be any point C(h, k) in the plane. Using a composition of a translation, a dilation centered at the origin, and a second translation, you can write the following coordinate rule for D(n, C).

D(n, C)(x, y) = (n(x - h) + h, n(y - k) + k)

Step 1 Use the translation T6-h, -k7 to move the center of dilation to the origin.

Step 2 Use the dilation Dn to dilate by scale factor n.

Step 3 Use the translation T6h, k7 to move the center of dilation back to (h, k).

(x, y) S (x - h, y - k)

(x - h, y - k) S (n(x - h), n(y - k))

(n(x - h), n(y - k)) S (n(x - h) + h, n(y - k) + k)

P′

Q′

R′C(h, k)

P(x, y)

R

Q

x

y

O

(n(x − h) + h, n(y − k) + k)

357PearsonTEXAS.com

Problem 3

Problem 2

Finding a Dilation Image

What are the coordinates of the vertices of D2(△PZG)? Graph the image of △PZG.

Identify the coordinates of each vertex. The center of dilation is the origin and the scale factor is 2, so use the coordinate rule D2(x, y) = (2x, 2y).

D2(P) = (2 # 2, 2 # (-1)), or P′(4, -2).

D2(Z) = (2 # (-2), 2 # 1), or Z′(-4, 2).

D2(G) = (2 # 0, 2 # (-2)), or G′(0, -4).

To graph the image of △PZG, graph P′, Z′, and G′. Then draw △P′Z′G′.



y

�4 �2 4OxZ

PG

Composing Rigid Transformations, Including a Dilation

Determine the image of △FRM after a dilation centered at (0, 0) with scale factor 1, composed with a translation 4 units down.

Find the coordinates of the vertices of △F′R′M′ and △F″R″M″.

The coordinate rule that describes the dilation is (x, y) S (1x, 1y).

Use the rule to find F′, R′, and M′.

F(-4, 0) S F′(-4, 0)

R(1, 3) S R′(1, 3)

M(3, 2) S M′(3, 2)

The coordinate rule that describes the translation is (x, y) S (x, y - 4).

Use the rule to find F″, R″, and M″.

F′( -4, 0) S F″( -4, -4)

R′(1, 3) S R″(1, -1)

M′(3, 2) S M″(3, -2)

Draw △FRM, △F′R′M′, and △F″R″M″ on the coordinate plane.

y

xO 2−2

4

−4

R

MF

y

xO−2

−4

R = R′ M = M′

F = F′R″

M″

F ″

Will the vertices of the triangle move closer to (0, 0) or farther from (0, 0)?The scale factor is 2, so the dilation is an enlargement. The vertices will move farther from (0, 0).

How can a dilation be a rigid transformation?If the scale factor of the dilation is 1, then the preimage and the image are congruent.

2y

−4 4Ox

Z′

P′

G′

Z

P

G


Problem 4

Determining the Image of a Dilation Not Centered at the Origin

Use ▱HJMN shown at the right.

A Write a coordinate rule that describes a dilation centered at J with scale factor 12.

You can use the composition of a translation, a dilation, and a second translation to find a coordinate rule for D(1

2, J)(x, y).

The coordinates of J are (-2, -4). Use these coordinates to identify the two translations you need to use in the composition.

Step 1 Translate (x, y) 2 units right and 4 units up to move the center of dilation from J(-2, -4) to (0, 0).

(x, y) S (x + 2, y + 4)

Step 2 Dilate by a scale factor of 12.

(12(x + 2), 12(y + 4)) = (1

2x + 1, 12y + 2) Step 3 Translate 2 units left and 4 units down to move

the center of dilation from (0, 0) back to J(-2, -4).

(12x + 1 - 2, 12y + 2 - 4) = (1

2x - 1, 12y - 2) The coordinate rule that describes the dilation is D(1

2, J)(x, y) = (12x - 1, 12y - 2).

B Graph D(12, J) (HJMN).

You can find the coordinates of the vertices of H′J′M′N′ by applying the coordinate rule you wrote in Part A.

H(-4, -2) S (12(-4) - 1, 12(-2) - 2), or H′(-3, -3)

J(-2, -4) S (12(-2) - 1, 12(-4) - 2), or J′(-2, -4)

M(2, -2) S (12# 2 - 1, 12(-2) - 2), or M′(0, -3)

N(0, 0) S (12# 0 - 1, 12

# 0 - 2), or N′(-1, -2)

Graph H′J′M′N′.


yx

O 2 4−2−2

−4

−4N

MH

J

yx

O 2 4−2

−4

−4N

MH

H′

N′

M′ J = J′

What translation will move the center of dilation J(–2, –4) to the origin (0, 0)?J must move 2 units right and 4 units up to be at (0, 0).

359PearsonTEXAS.com

Problem 6

What does a scale factor of 7 tell you?A scale factor of 7 tells you that the ratio of the image length to the actual length is 7, or image lengthactual length = 7. 1.75 in.1.75 in.

Problem 5

Determining the Image of a Composition of Rigid and Non-Rigid Transformations

△ABC has vertices A(22, 22), B(0, 1), and C(0, 22). Determine the vertices of the image of △ABC after a dilation with scale factor 2 and center of dilation at point A, followed by a translation 5 units to the left. Graph the image.

Step 1 The coordinate rule that describes the dilation is (x, y) S (2x + 2, 2y + 2).

A(-2, -2) S A′(-2, -2)

B(0, 1) S B′(2, 4)

C(0, -2) S C′(2, -2)

Step 2 The coordinate rule that describes the translation is (x, y) S (x - 5, y).

A′(-2, -2) S A″(-7, -2)

B′(2, 4) S B″(-3, 4)

C′(2, -2) S C″(-3, -2)

Ox

2

4y

-8 -4 -2

C9

B9

A

B

CA0 C 0

B0

A9

How do you find the rule for a dilation centered at (22, 22)?You can write the rule of a dilation not centered at the origin using a composition of a translation, a dilation, and a second translation of coordinate (x, y).

Using a Scale Factor to Find a Length

Biology A magnifying glass shows you an image of an object that is 7 times the object’s actual size. So the scale factor of the enlargement is 7. The photo shows an apple seed under this magnifying glass. What is the actual length of the apple seed?

The enlarged length of the apple seed is 1.75 in. Set up an equation to find the actual length of the apple seed.

1.75 = 7 # p image length = scale factor # actual length

0.25 = p Divide each side by 7.

The actual length of the apple seed is 0.25 in.



LINE

HO

M E W O RK


The blue figure is a dilation image of the black figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation.

1. 2. 3.

4. 5. 6.

Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Write a coordinate rule that describes each dilation. Use your rule to find the images of the vertices of △PQR for each dilation. Graph the image.

7. D10 (△PQR) 8. D34 (△PQR) 9. D(3, Q) (△PQR)

Apply Mathematics (1)(A) You look at each object described in Exercises 10–12 under a magnifying glass. Find the actual dimension of each object.

10. The image of a button is 5 times the button’s actual size and has a diameter of 6 cm.

11. The image of an ant is 7 times the ant’s actual size and has a length of 1.4 cm.

12. The image of a capital letter N is 6 times the letter’s actual size and has a height of 1.68 cm.

Find the image of each point for the given dilation.

13. L(-3, 0); D5 (L) 14. N(-4, 7); D(0.2, N) (N) 15. A(-6, 2); D1.5 (A)

A 64


2

4

R


L


M


O

x

y

2

2

4

6

4 6

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x

y

O�4�6 �2

4

6


x

y

O

2

1

P

Q

R

�4

�3


x

y

O

1

2P

Q

R�3


x

y

O

2

4

P R

Q

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51


361PearsonTEXAS.com

Use the graph at the right. Find the vertices of the image of QRTW for the given transformation or composition of transformations.

16. a dilation with scale factor 100

17. a dilation with scale factor 12 centered at (0, 2) followed by a translation 3 units down

18. D(1,O) ∘ r(180°,Q)

19. D(10,T) ∘ Ry-axis

20. The vertices of △S″U″J ″ are S″ (-1,-1), U″ (0, 1), and J″ (1,-1). Suppose the composition of transformations resulting in △S″U″J ″ was a dilation with scale factor 13 centered at point S followed by a translation 4 units up. Determine the coordinates of △SUJ . Then, graph △SUJ .

Display Mathematical Ideas (1)(G) Graph MNPQ and its image M′N′P′Q′ for a dilation with center (0, 0) and the given scale factor.

21. M(1, 3), N(-3, 3), P(-5, -3), Q(-1, -3); 3

22. M(2, 6), N(-4, 10), P(-4, -8), Q(-2, -12); 14

23. Select Tools to Solve Problems (1)(C) Use the dilation command in geometry software or drawing software to create a design that involves repeated dilations, such as the one shown at the right. The software will prompt you to specify a center of dilation and a scale factor. Print your design and color it. Feel free to use other transformations along with dilations.

24. Let / be a line through the origin. Show that Dk(/) = / by showing that if C = (c1, c2) is on /, then Dk(C) is also on /.

25. Let A = (a1, a2) and B = (b1, b2), let A′ = Dk(A) and B′ = Dk(B)

with k ≠ 1, and suppose that <AB

> does not pass through the origin.

a. Show that <AB

> and

<A′B′

> are not the same line.

b. Suppose that a1 ≠ b1. Show that <AB

> is parallel to

<A′B′

> by showing that they

have the same slope.

c. Show that <AB

>}<A′B′

> if a1 = b1.

26. Explain Mathematical Ideas (1)(G) You are given AB and its dilation image A′B′ with A, B, A′, and B′ noncollinear. Explain how to find the center of dilation and scale factor.

27. Explain Mathematical Ideas (1)(G) The diagram at the right shows △LMN and its image △L′M′N′ for a dilation with center P. Find the values of x and y. Explain your reasoning.


x

y

O

2

4

2

W

T

Q

R�1�3



L

L�

M�

N�NPM

2

4

y�(2y � 60)�

x � 3

x


28. Analyze Mathematical Relationships (1)(F) An equilateral triangle has 4-in. sides. Describe its image for a dilation with center at one of the triangle’s vertices and scale factor 2.5.

In the coordinate plane, you can extend dilations to include scale factors that are negative numbers. For Exercises 29 and 30, use △PQR with vertices P(1, 2), Q(3, 4), and R(4, 1).

29. Graph D-3 (△PQR).

30. a. Graph D-1 (△PQR).

b. Explain why the dilation in part (a) may be called a reflection through a point.

31. Use Representations to Communicate Mathematical Ideas (1)(E) A flashlight projects an image of rectangle ABCD on a wall so that each vertex of ABCD is 3 ft away from the corresponding vertex of A′B′C′D′. The length of AB is 3 in. The length of A′B′ is 1 ft. How far from each vertex of ABCD is the light?

32. Determine the image of △TRI after a rotation of 180° around T composed with a dilation with scale factor 1.

33. Under a dilation, what scale factor will preserve congruence?

B

A D

CA9

B9

C9

D9

y

xO 2−2

4

−4

I

R

T

TEXAS Test Practice

34. A dilation maps △CDE onto △C′D′E′. If CD = 7.5 ft, CE = 15 ft, D′E′ = 3.25 ft, and C′D′ = 2.5 ft, what is DE?

A. 1.08 ft B. 5 ft C. 9.75 ft D. 19 ft

35. You want to prove indirectly that the diagonals of a rectangle are congruent. As the first step of your proof, what should you assume?

F. A quadrilateral is not a rectangle.

G. The diagonals of a rectangle are not congruent.

H. A quadrilateral has no diagonals.

J. The diagonals of a rectangle are congruent.

36. Which word can describe a kite?

A. equilateral B. equiangular C. convex D. scalene

363PearsonTEXAS.com



Additional TEKS (1)(A), (1)(D), (1)(F), (3)(B), (3)(C)

TEKS FOCUS

•Compression – a transformation that decreases the distance between corresponding points of a figure and a line

•Stretch – a transformation that increases the distance between corresponding points of a figure and a line


VOCABULARY

You can change the size of a figure in the coordinate plane by multiplying the x- and y-coordinates by different factors. You can compose this type of transformation with the other transformations you have learned.


A stretch is a transformation that increases the distance between corresponding points of a figure and a line. A compression is a transformation that decreases the distance between corresponding points of a figure and a line.

Horizontal Stretch(x, y) S (ax, y), where a 7 1

Ox

y

Ox

y

Vertical Stretch(x, y) S (x, by), where b 7 1

Ox

y

Ox

y

Horizontal Compression(x, y) S (ax, y), where 0 6 a 6 1

Ox

y

Ox

y

Vertical Compression(x, y) S (x, by), where 0 6 b 6 1

Ox

y

Ox

y

Key Concept Other Non-Rigid Transformations


364 Lesson 8-8 Other Non-Rigid Transformations

Describing a Non-Rigid Transformation

Is △STW S △S′T′W′ a vertical compression or a vertical stretch? Write a coordinate rule that maps △STW to △S′T′W′.

Since the image appears to be shorter than the preimage, this transformation is a vertical compression.

Compare corresponding vertical distances for the figures to determine the vertical compression factor that maps △STW to △S′T′W′.

b = S′T′ST = 2

8 = 14

So △S′T′W′ is a vertical compression of △STW by a factor of 14. The coordinate rule that describes the transformation is (x, y) S (x, 14 y).

-2-2

-4

-4x

2

4y

S W

T

T9

S9 W9

O

Why aren’t T9W9 and TW used to find the vertical compression factor? TW and T9W9 are not vertical distances, so it would be more difficult to use them.

Problem 1

How can you tell the difference between a compression and a stretch? In a stretch, at least one coordinate is multiplied by a factor greater than 1. In a compression, at least one coordinate is multiplied by a factor less than 1.

Performing a Stretch

Quadrilateral EFGH has vertices E(22, 2), F(2, 2), G(2, 22), and H(22, 22). What are the coordinates of the vertices of the image of EFGH after the transformation (x, y) S (3x, 2y)? Graph the image of EFGH.

You can think of this transformation as a composition of a horizontal stretch and a vertical stretch. The horizontal stretch factor is 3. The vertical stretch factor is 2.

Use the coordinate rule (x, y) S (3x, 2y) to find the coordinates of the images of the vertices.

E(-2, 2) S E′(-6, 4)

F(2, 2) S F′(6, 4)

G(2, -2) S G′(6, -4)

H(-2, -2) S H′(-6, -4)

Graph EFGH and E′F′G′H′.


�4

O

�2

E� F�

G�H�

2 4 6x

�2

E F

GH

2

4

y

�4�6

Problem 2

365PearsonTEXAS.com

Determining the Preimage of a Composition of Non-Rigid Transformations

The vertices of △P″Q″R″ are P″(-1, -1), Q″(0, 1), and R″(1, -1). △PQR maps to

△P″Q″R″ through a dilation with scale factor 13 centered at (0, -1) followed by the

compression (x, y) S (12 x, y). Determine the coordinates of the vertices of △PQR.

Then graph △PQR.

First, reverse the transformation (x, y) S (12 x, y) by multiplying the x-coordinate of

each vertex of △P″Q″R″ by 2. The vertices of △P′Q′R′ are P′(-2, -1), Q′(0, 1), and R′(2, -1).

Then reverse the dilation with scale factor 13 centered at (0, -1) by translating to bring the center of dilation to the origin, dilating with scale factor 3, and translating to bring the center of dilation back to (0, -1). This sequence of transformations gives the coordinate rule (x, y) S (6x, 3(y + 1) - 1). The vertices of △PQR are P(-6, -1), Q(0, 5), and R(6, -1).

O 2 4

4

6x

�2

y

�4�6P′ R′P R

Q

P″ R″

Q″ = Q′

How can you find the coordinates of the vertices of the preimage?To find the coordinates for the preimage, multiply the x-coordinate of the image by the reciprocal of the given horizontal compression factor.

Problem 3

Problem 4

Determining the Image of a Composition of Non-Rigid Transformations

△ABC has vertices A(22, 22), B(0, 1), and C(0, 22). Determine the vertices of the image of △ABC after a dilation with scale factor 2 centered at the origin, followed by the horizontal stretch (x, y) S (2x, y). Graph the image.

Step 1 The coordinate rule that describes the dilation is (x, y) S (2x, 2y).

A(-2, -2) S A′(-4, -4)

B(0, 1) S B′(0, 2)

C(0, -2) S C′(0, -4)

Step 2 Apply the rule (x, y) S (2x, y) for the stretch to the image of the dilation.

A′(-4, -4) S A″(-8, -4)

B′(0, 2) S B″(0, 2)

C′(0, -4) S C″(0, - 4)

O 2xB

B¿ = B–

C¿ = C–A¿A–

CA

2y

-8 -6

Does the stretch affect the points B′ and C′? No, because B′ and C′ are on the y-axis.


Identifying a Sequence of Transformations

An architect’s plan for a city park is shown on the coordinate grid at the left below. The mayor of the city asks that the swimming pool be 50, longer, but not wider, and wants to move it to the other end of the park. Describe a sequence of transformations that will move the pool to the outlined location on the architect’s plan.

The pool is a rectangle with vertices A(1, 2), B(1, 6), C(4, 6), and D(4, 2). ABCD will need to be vertically stretched and then translated horizontally and vertically to be mapped to EFGH.

Step 1 Stretch ABCD using the rule (x, y) S (x, 32 y) A(1, 2) S A′(1, 3)

B(1, 6) S B′(1, 9)

C(4, 6) S C′(4, 9)

D(4, 2) S D′(4, 3)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

1

2

3

4

5

6

7

A

B C

D

0 2

2

4

6

8

4 6 8 10 12 14

14

12

10

F G

E H

A9

B9 C9

D9

0 2

2

4

6

8

4 6 8 10 12 14

14

12

10

Step 2 Rectangle EFGH appears to be translated 5 units up and 10 units to the right from A′B′C′D′. Find the coordinates using the rule (x, y) S (x + 10, y + 5).

A′(1, 3) S E(11, 8)

B′(1, 9) S F(11, 14)

C′(4, 9) S G(14, 14)

D′(4, 3) S H(14, 8)

The pool can be moved to the outlined location through a stretch followed by a translation.

TEKS Process Standard (1)(A)

How can you determine the vertical stretch factor?The vertical stretch factor is the ratio of the vertical length of the outlined location, which is 6 units, to the length of the original pool, which is 4 units. The vertical stretch factor is 32.

Problem 5

367PearsonTEXAS.com


ONLINE

HO

M E W O RK


Find the vertices of each figure’s image after the given transformation. Then graph the image.

1. (x, y) S (2x, 4y) 2. (x, y) S (2x, 12 y) 3. (x, y) S (12 x, 13 y)

4. Quadrilateral ABCD has vertices A(4, 3), B(4, −3), C(−4, −3), and D(−4, 3).

a. Find a coordinate rule that describes a stretch that, when applied to ABCD, results in an image that is a square. Explain your reasoning.

b. Find a coordinate rule that describes a compression that, when applied to ABCD, results in an image that is a square. Explain your reasoning.

5. Create Representations to Communicate Mathematical Ideas (1)(E) A classmate dilates a figure in the coordinate plane by a scale factor greater than 1 and then compresses the resulting figure vertically by a factor between 1 and 0.

a. Write a coordinate rule that describes the dilation.

b. Write a coordinate rule that describes the vertical compression.

Find the vertices of the image of each figure after the given composition of transformations. Then graph the image.

6. a dilation with scale factor 12 centered at the origin, followed by the transformation

(x, y) S (12x, y)

7. a dilation with scale factor 12 centered at point N, followed by a vertical stretch with factor 3

O 2

2

4 x

yM

NL

O x�2y

�4N L

M

O 2

�2

2

x�2

yA B

CD O 2x

�2

�2

2

yQ R

ST

O 2x

�2

�2

2

yH

L K

J



△A″B″C″ is a transformation of △ABC. Determine the coordinates for △ABC before each composition of transformations. Then graph △ABC.

8. a dilation with scale factor 3 centered at the origin, followed by the transformation

(x, y) S (23x, y)

9. a dilation with scale factor 3 centered at point C, followed by a horizontal compression with

factor 12

Ox

�2

yA″

B″C″

O 2

�2

4xy

A″

B″C″

10. Analyze Mathematical Relationships (1)(F) A rectangle in the coordinate plane is dilated by a scale factor of 3 and then stretched horizontally by a factor of 2. Explain how to find the coordinates of the vertices of the preimage if you know the coordinates of the vertices of the image.

Describe a sequence of transformations that maps quadrilateral EFGH to quadrilateral E′F′G′H′.

11.

O

2

4

6

4x

�2

y

�4

G′

F′

E′

H = H′

F

E G

12.

O

2

4

6

4x

-2

y

-4

G′

F′

E′

H′

F

H

GE

13. Apply Mathematics (1)(A) A rancher’s plan to expand some stables is shown on the coordinate grid. The rancher plans to make the stables larger and move them across the ranch. Describe a sequence of transformations that will move the stables to the outlined location on the rancher’s plan.

14. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) △PQR has vertices P(-2, -2), Q(2, -1), and R(-4, -3), and △P‴Q‴R‴ has vertices P‴(-3, -8), Q‴(5, -4), and R‴(-7, -12). A dilation with scale factor 4, followed by a second transformation, followed by a horizontal compression with factor 12 maps △PQR to △P‴Q‴R‴. Describe the second transformation using both words and a coordinate rule.

0 2

2

4

6

8

4 6 8 10 12 14

14

12

10

369PearsonTEXAS.com

TEXAS Test Practice

17. Which transformation maps △ABC to △A′B′C′?

A. (x, y) S (23 x, 23 y)

B. (x, y) S (13 x, 13 y)

C. (x, y) S (23 x, 13 y)

D. (x, y) S (13 x, 23 y)

18. Which sequence of transformations does not preserve congruence?

F. a dilation followed by a rotation

G. a reflection followed by a rotation

H. a translation followed by a translation

J. a translation followed by a reflection

19. If ∠1 and ∠2 are vertical angles, which of the following statements must be true?

A. m∠1 6 m∠2 C. m∠1 + m∠2 = 90

B. m∠1 = m∠2 D. m∠1 + m∠2 = 180

20. Explain how to write a coordinate proof to show that two lines in the coordinate plane are perpendicular.

O 2

2

4x

yA

BC

A′

B′C′

15. A computer game programmer is testing a transformations game. Describe a possible sequence of transformations the programmer could use to map Figure A onto Figure B. Are the figures congruent? Explain.

12:30 PM

Game:Galactic Grid

Difficulty:Medium

Tester:GamerGal8

2

2

4

4

6

6

0

A

B

16. Explain Mathematical Ideas (1)(G) A sequence of a rigid transformation followed by a non-rigid transformation is applied to a non-rectangular figure in the coordinate plane, with the result that the angle measures in the preimage and the final image are equal. What does this tell you about the rigid transformations in the sequence? What does it tell you about the non-rigid transformations? Explain your reasoning.


TOPIC VOCABULARY

•angleofrotation, p. 332

• centerofdilation, p. 356

• centerofrotation, p. 332

• compositionoftransformations, p. 318

• compression, p. 364

• congruencetransformation, p. 350

• congruent, p. 350

•dilation, p. 356

•enlargement, p. 356

•glidereflection, p. 344

• image, p. 318

• lineofreflection, p. 326

• lineofsymmetry, p. 338

• linesymmetry, p. 338

•non-rigidtransformation, p. 356

•orientation, p. 326

•pointsymmetry, p. 338

•preimage, p. 318

• ratio, p. 356

• reduction, p. 356

• reflection, p. 326

• reflectionalsymmetry, p. 338

• rigidtransformation, p. 318

• rotation, p. 332

• rotationalsymmetry, p. 338

• scalefactorofadilation, p. 356

• stretch, p. 364

• transformation, p. 318

• translation, p. 319

Check Your UnderstandingChoose the correct term to complete each sentence.

1. A(n) ? is a change in the position, shape, or size of a figure.

2. A(n) ? is a composition of a translation and a reflection.

3. In a(n) ? , all points of a figure move the same distance in the same direction.

4. A(n) ? is a transformation that preserves distance and angle measure.

Topic 8 Review

8-1 Translations

Exercises 5. a. A transformation maps

ZOWE onto LFMA. Does the transformation appear to be a rigid transformation? Explain.

b. What is the image of ZE? What is the preimage of M?

6. △RST has vertices R(0, -4), S( -2, -1), and T( -6, 1). Graph T6-4, 77(△RST).

7. Write a rule to describe a translation 5 units left and 10 units up.

8. Find a single translation that has the same effect as the following composition of translations. T6-4, 77 followed by T63, 07

hsm11gmse_09cr_t10516.ai

L

A

O

MW

F

E

Z

ExampleWhat are the coordinates of T6-2, 37(5, -9)?

Add -2 to the x-coordinate, and 3 to the y-coordinate.A(5, -9) S (5 - 2, -9 + 3), or A′(3, -6).

Quick ReviewA transformation of a geometric figure is a change in its position, shape, or size.

A translation is a rigid transformation that maps all points of a figure the same distance in the same direction.

In a composition of transformations, each transformation is performed on the image of the preceding transformation.

371PearsonTEXAS.com

8-3 Rotations

Exercises 13. Copy the diagram below. Then draw r(90°, P)(△ZXY).

Label the vertices of the image, using prime notation.

14. What are the coordinates of r(180°, O)(-4, 1)?

15. WXYZ is a quadrilateral with vertices W(3, -1), X(5, 2), Y(0, 8), and Z(2, -1). Graph WXYZ and r(270°, O)(WXYZ).


Y

Z XP

8-2 Reflections

ExercisesGiven points A(6, 4), B( −2, 1), and C(5, 0), graph △ABC and each reflection image.

9. Rx-axis(△ABC)

10. Rx = 4(△ABC)

11. Ry = x(△ABC)

12. Copy the diagram. Then draw Ry-axis (BGHT). Label the vertices of the image, using prime notation.

geom12_se_ccs_c09cr_t03.ai

y

xT G

B

4

H

�2�4

�4

4ExampleUse points P(1, 0), Q(3, −2), and R(4, 0). What is Ry-axis(△PQR)?

Graph △PQR. Find P′, Q′, and R′ such that the y-axis is the perpendicular bisector of PP′, QQ′, and RR′. Draw △P′Q′R′.


yx

�2

5OP� P R

Q� Q

R�

ExampleGHIJ has vertices G(0, 23), H(4, 1), I(21, 2), and J(25, 22). What are the vertices of r(90°, O)(GHIJ)?

Use the rule r(90°, O)(x, y) = (-y, x).

r(90°, O)(G) = (3, 0)

r(90°, O)(H) = (-1, 4)

r(90°, O)(I) = (-2, -1)

r(90°, O)(J) = (2, -5)

Quick ReviewThe diagram shows a reflection across line r. A reflection is a rigid transformation that preserves distance and angle measure. The image and preimage of a reflection have opposite orientations.


r

Quick ReviewThe diagram shows a rotation of x° about point R. A rotation is a rigid transformation in which a figure and its image have the same orientation.


x�R

372 Topic 8 Review

8-5 Compositions of Rigid Transformations

Exercises 21. Sketch and describe the result of

reflecting E first across line / and then across line m.

Each figure is the image of the figure below. Tell whether their orientations are the same or opposite. Then classify the transformation.

22. 23. 24.

25. △TAM has vertices T (0, 5), A(4, 1), and M(3, 6). Find the image of Ry = -2 ∘ T(-4, 0)(△TAM).


m�

E


angle


an

gle


angle


angle

8-4 Symmetry

ExercisesTell what type(s) of symmetry each figure appears to have. If it has reflectional symmetry, sketch the figure and the line(s) of symmetry. If it has rotational symmetry, state the angle of rotation.

16. 17. 18.

19. How many lines of symmetry does an isosceles trapezoid have?

20. What type(s) of symmetry does a square have?

ExampleHow many lines of symmetry does an equilateral triangle have?

An equilateral triangle reflects onto itselfacross each of its three medians. Thetriangle has three lines of symmetry.

Quick ReviewA figure has reflectional symmetry or line symmetry if there is a reflection for which it is its own image.

A figure that has rotational symmetry is its own image for some rotation of 180° or less.

A figure that has point symmetry has 180° rotational symmetry.

ExampleDescribe the result of reflecting P first across line O and then across line m.

A composition of two reflections across intersecting lines is a rotation. The angle of rotation is twice the measure of the acute angle formed by the intersecting lines. P is rotated 100° about C.


P50�

C m

�

Quick ReviewA rigid transformation preserves distance and angle measure. You have learned about translations, reflections, and rotations, which are all rigid transformations. A composition of rigid transformations is also a rigid transformation. All rigid transformations can be expressed as a composition of reflections.

The diagram shows a glide reflection of N. A glide reflection is a rigid transformation in which a figure and its image have opposite orientations.


N N

N

373PearsonTEXAS.com

8-7 Dilations

Exercises 28. The blue figure is a dilation image of the black figure.

The center of dilation is O. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor.


y

x

�2 2

4

4O

Graph the polygon with the given vertices. Then graph its image for a dilation with center (0, 0) and the given scale factor.

29. M( -3, 4), A( -6, -1), T (0, 0), H(3, 2); scale factor 5

30. F( -4, 0), U(5, 0), N( -2, -5); scale factor 12

31. A dilation maps △LMN onto △L′M′N′. LM = 36 ft, LN = 26 ft, MN = 45 ft, and L′M′ = 9 ft. Find L′N′ and M′N′.


Exercises 26. In the diagram at the right,

△LMN ≅ △XYZ. Identify a congruence transformation that maps △LMN onto △XYZ.

27. Graphic designers use some fonts because they have pleasing proportions or are easy to read from far away. The letters p and d above are used on a sign that has a special font. Are the letters congruent? If so, describe a congruence transformation that maps one onto the other. If not, explain why not.


y

xM

L

N

Y

Z

X


p d

ExampleRy-axis(TGMB) = KWAV. What are all of the congruent angles and all of the congruent sides?

A reflection is a congruence transformation, so TGMB ≅ KWAV, and corresponding angles and corresponding sides are congruent.∠T ≅ ∠K , ∠G ≅ ∠W , ∠M ≅ ∠A, and ∠B ≅ ∠V TG = KW , GM = WA, MB = AV, and TB = KV


VK

W

A

y

x

T

G

BM

Quick ReviewTwo figures are congruent if and only if there is a sequence of rigid transformations that maps one figure onto the other.

ExampleThe blue figure is a dilation image of the black figure. The center of dilation is A. Is the dilation an enlargement or a reduction? What is the scale factor?

The image is smaller than the preimage, so the dilation is a reduction. The scale

factor is image length

original length = 22 + 4 = 2

6, or 13.


A

4

2

Quick ReviewThe diagram shows a dilation with center C and scale factor n. Dilations preserve angle measures.


C ana

In the coordinate plane, if the origin is the center of a dilation with scale factor n, then P(x, y) S P′(nx, ny).

374 Topic 8 Review


ExercisesDetermine and graph P′Q′R′S′, the image of PQRS after each transformation or composition of transformations.

y

xO

P Q

S R

32. (x, y) S (2x, y)

33. (x, y) S (x, 12y) 34. a dilation centered at the origin with scale factor 2

followed by the transformation (x, y) S (14x, y)

35. a dilation with scale factor 12 centered at point Q followed by the translation (x, y) S (x - 2, y + 1)

36. If PQRS is the image of EFGH after a dilation of scale factor 13 followed by the transformation (x, y) S (2x, y), what are the coordinates of EFGH?

37. Describe a sequence of transformations that will map PQRS to JKLM with vertices J(-6, 1), K(6, 1), L(6, -2), and M(-6, -2).

Example△ABC has vertices A(-2, -1), B(1, 1), and C(2, -1). What are the coordinates after the transformation (x, y) S (x, 2y)?

A(-2, -1) S A′(-2, -2)B(1, 1) S B′(1, 2)C(2, -1) S C′(2, −2)

y

xO-2 2

2

A C

B

Quick ReviewA horizontal stretch/compression is any transformation (x, y) S (ax, y) for a 7 0.

A vertical stretch/compression is any transformation (x, y) S (x, by) for b 7 0.

A non-rigid transformation that stretches or compresses a figure by different amounts in different directions does not preserve congruence.

375PearsonTEXAS.com

Topic 8 TEKS Cumulative Practice

Multiple ChoiceRead each question. Then write the letter of the correct answer on your paper.

1. In a right triangle, which point lies on the hypotenuse?

A. incenter C. centroid

B. orthocenter D. circumcenter

2. In △LMN, P is the centroid and LE = 24. What is PE?

F. 8 H. 10

G. 9 J. 16

3. What is the sum of the angle measures of a 32-gon?

A. 3200° C. 5400°

B. 3800° D. 5580°

4. The diagonals of rectangle PQRS intersect at H. What is the length of QS?

F. 6 H. 23

G. 12 J. 46

5. The vertices of ▱ABCD are A(1, 7), B(0, 0), C(7, -1), and D(8, 6). What is the perimeter of ▱ABCD?

A. 50 C. 2200

B. 100 D. 2022

6. What type of symmetry does the figure have?

F. 60° rotational symmetry

G. 90° rotational symmetry

H. line symmetry

J. point symmetry

7. Which conditions allow you to conclude that a quadrilateral is a parallelogram?

A. one pair of sides congruent, the other pair of sides parallel

B. perpendicular, congruent diagonals

C. diagonals that bisect each other

D. one diagonal bisects opposite angles

8. Write the horizontal stretch rule that maps P(-1, 2) to P′(-3, 2).

F. (x, y) S (-3x, y)

G. (x, y) S (x, 3y)

H. (x, y) S (3x, y)

J. (x, y) S (-3x, -y)

9. What type of symmetry does the figure have?

A. reflectional symmetry

B. rotational symmetry

C. point symmetry

D. no symmetry

hsm11gmse_09cu_t08646.ai

L

M

F

D

NP

E


S

3x � 54x � 1H

QP

R

hsm16_gmhh_08cp_t012.ai

hsm11gmse_09ct_t10571.ai

376 Topic 8 TEKS Cumulative Practice

10. If you are given a line and a point not on the line, what is the first step to construct the line parallel to the given line through the point?

F. Construct an angle from a point on the line to the given point.

G. Draw a straight line through the given point.

H. Draw a ray from the given point that does not intersect the line.

J. Label a point on the given line, and draw a line through that point and the given point.

11. Which quadrilateral must have congruent diagonals?

A. kite C. parallelogram

B. rectangle D. rhombus

Gridded Response 12. What is the measure of ∠H?

13. What is the area of the square, in square units?

14. In ▱PQRS, what is the value of x?

Constructed Response 15. What is the value of x for which p } q?

16. △DEB has vertices D(3, 7), E(1, 4), and B(-1, 5). In which quadrant(s) is the image of r(270°, O)(△DEB)? Draw a diagram.

17. In △ABC below, AB ≅ CB and BD # AC. Prove that △ABD ≅ △CBD.

18. Is △ABC a right triangle? Justify your answer.

19. LMNO has vertices L( -4, 0), M( -2, 3), N(1, 1), and O( -1, -2). RSTV has vertices R(1, 1), S(3, -2), T(6, 0), and V(4, 3). Graph the two quadrilaterals. Is LMNO ≅ RSTV? If so, write the rule for the congruence transformation that maps LMNO to RSTV. If not, explain why not.


A

B C35�

FG

H

50�


O

y

x

2

�2 4�4

�3


O

Q R

SP

84�

x�

22�


q

�2x � 5��

p

115�


A

B

CD


B

y

x

4

4

O

C

A

�4

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Date post:	17-Mar-2023
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Topic 8 Transformational Geometry

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