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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


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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


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LESSON 7.5


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




4. regular hexagons

5. squares or parallelograms

6. squares or parallelograms

7.

8.



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


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LESSON 7.7

1. equilateral triangles.

2. regular hexagons.

3.

4.



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.



7. Circumcenter is (3, 4); orthocenter is (10, 8).

8.

9.

10.


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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.



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


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