Answers to Exercises11.
12. reflectional symmetry
13. 4-fold rotational and reflectional symmetry
14. reflectional symmetry
15. 7-fold symmetry: possible answers are F or J.
9-fold symmetry: possible answers are E or H.
Basket K has 3-fold rotational symmetry but not
reflectional symmetry.
16. See table below; n, n
17.
18.
19. P(�a, b), Q(�a, �b), R(a, �b)
20. possible construction:
21. 50th figure: 154 (50 shaded, 104 unshaded);
nth figure: 3n � 4 (n shaded, 2(n � 2) unshaded)
22. 46
23. It is given that �1 � �2, and �2 � �3
because of the Vertical Angles Conjecture, so
�1 � �3. Segment DC is congruent to itself.
�DCE and �DCB are both right angles, so they
are congruent. Therefore, �DCB � �DCE by
ASA, and BC� � CE� by CPCTC.
P
, or
P
ANSWERS TO EXERCISES 85
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CHAPTER 7 • CHAPTER CHAPTER 7 • CHAPTER
LESSON 7.1
1. Rigid; reflected, but the size and shape do not
change.
2. Nonrigid; the shape changes.
3. Nonrigid; the size changes.
4. 5.
6.
7. possible answer: a boat moving across the water
8. possible answer: a Ferris wheel
9a. Sample answer: Fold the paper so that the
images coincide, and crease.
9b. Construct a segment that connects
two corresponding points. Construct the
perpendicular bisector of that segment.
10a. Extend the three horizontal segments onto
the other side of the reflection line. Use your
compass to measure lengths of segments and
distances from the reflection line.
10b.
P
��
7
Number of sides of 3 4 5 6 7 8 . . . nregular polygon
Number of reflectional 3 4 5 6 7 8 . . . n
symmetries
Number of rotational 3 4 5 6 7 8 . . . n
symmetries (� 360°)
16. (Lesson 7.1)
86 ANSWERS TO EXERCISES
LESSON 7.2
1.
2. reflection
3.
4.
5.
x
y
5
5
5
5
rotation
x5
y
5
5
5
reflection
x7
y
5
–5
reflection
–4
y
x–5
8
–8
x
y
5
5
translation
6. Rules that involve x or y changing signs,
or switching places, produce reflections.
If both x and y change signs, the rule produces a
rotation. Rules that produce translations involve a
constant being added to the x and/or y terms.
�5, 0� is the translation vector for Exercise 1.
7. (x, y) → (x, �y)
8. (x, y) → (�x, � y)
9.
10. There are two possible points, one on the N
wall and one on the W wall.
11.
12. by the Minimal Path Conjecture
13.
14.
,
Perry
Mason
Proposed freeway
H H'
H''
T
N
S
W E
T
W
H
N
E
S
S
8 ball
Cue ball
EW
N
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15. possible answer: HIKED
16. one, unless it is equilateral, in which case it
has three
17. two, unless it is a square, in which case it has
four
18.
19. sample construction:
20. sample construction:
21. false; possible counterexample: trapezoid
with two right angles
22. false; possible counterexample: isosceles
trapezoid
ANSWERS TO EXERCISES 87
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88 ANSWERS TO EXERCISES
LESSON 7.3
1. �10, 10�2. A 180° rotation. If the centers of rotation differ,
rotate 180° and add a translation.
3a. 20 cm
3b. 20 cm, but in the opposite direction
4a. 80° counterclockwise
4b. 80° clockwise
5. 180°
6. 3 cm
7. possible answer:
8. possible answer:
9.
10. Two reflections across intersecting lines yield
a rotation. The measure of the angle of rotation is
twice the measure of the angle between the lines of
reflection, or twice 90°, or 180°.
Center of rotation
NO A
O′′
A′′
O′A′
H
H′
N′
H′′
N′′
11. Answers may vary. Possible answer: reflection
across the figure’s horizontal axis and 60°
clockwise rotation.
12.
13.
14. Sample answer: Draw a figure on an overhead
transparency and then project the image onto a
screen.
15. possible answers: rotational: playing card,
ceiling fan, propeller blade; reflectional: human
body, backpack
16. one: yes; two: no; three: yes
17. possible answer:
18a. � � � � � � � �18b.
� � � � � � � ��?
�? �?
�? �?
�? 3c
02b
d–e
a
�d
c
f
b
e
a
d
11
13
�14
0�?
�? �?
�? 11
20
�5
�12
A O B
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�9 0
12 �7
2a 3b 4c
0 d f
LESSON 7.4
1. Answers will vary. 2. Answers will vary.
3. 33.42 4. 34.6
5. 32.4.3.4 6. 3.4.6.4�3.42.6
7. 33.42�32.4.3.4 8. 36�32.4.12
9a. The dual of a square tessellation is a square
tessellation.
9b. The dual of a hexagon tessellation is a triangle
tessellation.
9c. If tessellation A is the dual of tessellation B,
then tessellation B is the dual of tessellation A.
10. The dual is a 34�38 tessellation of isosceles
right triangles.
11.
12.
13. A ring of ten pentagons fits around a decagon,
and another decagon can fit into any two of the
pentagons. But another ring of pentagons around
the second decagon doesn’t leave room for a third
decagon.
14.
15. Answers will vary.
16. y � ��12
�x � 4
17. possible answer: TOT
18.
S
EW
N
8-ball
Cue ball
y
x
4
–6
5–3
y
x
4
–6
5–3
ANSWERS TO EXERCISES 89
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90 ANSWERS TO EXERCISES
LESSON 7.5
1. Answers will vary.
2. The dual is a 53�54 tessellation.
3.
4. Yes. The four angles of the quadrilateral
will be around each point of intersection in the
tessellation.
5.a c
acbb a c
acbb a c
b
ac
b a c
acbb a c
acbb
By the Triangle Sum Conjecture, a � b � c � 180°.
Around each point, we have 2(a � b � c) �2 � 180° � 360°. Therefore, a triangle will fill the
plane edge to edge without gaps or overlaps. Thus, a
triangle can be used to create a monohedral tiling.
6. three ways
7.
8. y � �2x � 3
y
x
8
–25
y
x
8
–25
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LESSON 7.6
1. Answers will vary.
2. Answers will vary.
3. Answers will vary.
4. regular hexagons
5. squares or parallelograms
6. squares or parallelograms
7.
8.
9. Answers will vary.
10. Answers will vary.
11.
S
E
A
B
12. y � ��23
�x � 3; the slope is the opposite sign.
13. 3.4.6.4�4.6.12
14. �4140
mrienv
� � � �16m0
isn
� � 1290 ft/s
15. Possible explanations:
15a. true; The kite diagonal between vertex angles
is the perpendicular bisector of the other diagonal;
in a square, diagonals would bisect each other
15b. False; it could be an isosceles trapezoid.
15c. False; it could be a rectangle.
15d. true; Parallel lines cut off congruent arcs of a
circle, so inscribed angles (the base angles of the
trapezoid) are congruent.
2� � 28 ft�
1 rev
y
x
5
10–10
ANSWERS TO EXERCISES 91
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92 ANSWERS TO EXERCISES
LESSON 7.7
1. equilateral triangles.
2. regular hexagons.
3.
4.
5. Answers will vary.
6. Answers will vary.
7. sample design:
8. False; they must bisect each other in a
parallelogram.
9. true
10. true
11. False; it could be a kite or an isosceles
trapezoid.
12. The path would be �14
� of Earth’s circumference,
approximately 6280 miles, which will take
126 hours, or around 5�14
� days.
13a. Using the Reflection Line Conjecture, the
line of reflection is the perpendicular bisector of
AA��and BB��. Because these segments are both
perpendicular to the reflection line, they are
parallel to each other. Note that if AB� is parallel to
the reflection line, quadrilateral AA�B�B will be a
rectangle instead of a trapezoid.
13b. Yes; it has reflectional symmetry, so legs and
base angles are congruent.
13c. greatest: near each of the acute vertices;
least: at the intersection of the diagonals (where A,
C, and B� become collinear and A�, C, and B
become collinear)
14a. � � � � � � �
14b. � � � � � � �30
�50
13
�?�?
�? �2
�? 3
�5
8
12
�?
�? �?
�?
7
0
2
8
6
�9
�6
4
5
0
3
1
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108 9
�28 150
29�1 10
LESSON 7.8
1. parallelograms
2. parallelograms
3.
4.
5. Answers will vary.
6. Answers will vary.
7. Circumcenter is (3, 4); orthocenter is (10, 8).
8.
9.
10.
ANSWERS TO EXERCISES 93
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94 ANSWERS TO EXERCISES
USING YOUR ALGEBRA SKILLS 7
1. y � ��16
�x
2. y � �2x � 2
3. Centroid is �2, �23
��; orthocenter is (0, 5).
4. Centroid is (4, 0); orthocenter is (3, 0).
5. �1, �43
��6. (�1, �1)
7. (5, �8)
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CHAPTER 7 REVIEW
1. true 2. true
3. true 4. true
5. true 6. true
7. False; a regular pentagon does not create a
monohedral tessellation and a regular hexagon
does.
8. true 9. true
10. False; two counterexamples are given in
Lesson 7.5.
11. False; any hexagon with one pair of opposite
sides parallel and congruent will create a
monohedral tessellation.
12. This statement can be both true and false.
13. 6-fold rotational symmetry
14. translational symmetry
15. Reflectional; color arrangements will vary, but
the white candle must be in the middle.
16. The two towers are not the reflection (or
even the translation) of each other. Each tower
individually has bilateral symmetry. The center
portion has bilateral symmetry.
17. Answers will vary.
18. Answers will vary.
19. 36�32.4.3.4; 2-uniform
20. 4.82; semiregular
21. y � �12
�x
y
x
22.
23. Use a grid of squares. Tessellate by translation.
24. Use a grid of equilateral triangles. Tessellate by
rotation.
25. Use a grid of parallelograms. Tessellate by
glide reflection.
26. Yes. It is a glide reflection for one pair of sides
and midpoint rotation for the other two sides.
27. No.Because the shape is suitable for glide
reflection,the rows of parallelograms should
alternate the direction in which they lean (row 1
leans right,row 2 leans left,row 3 leans right,and
so on).
28.
T H
ANSWERS TO EXERCISES 95
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