Answers to Exercises
CHAPTER 7 • CHAPTER
7
11.
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.
P
P
ᐉ
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.
or
18.
19. P(a, b), Q(a, b), R(a, b)
20. possible construction:
P
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.
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
CE
by CPCTC.
ASA, and BC
16. (Lesson 7.1)
Number of sides of
regular polygon
3
4
5
6
7
8
...
n
Number of reflectional
symmetries
3
4
5
6
7
8
...
n
Number of rotational
symmetries ( 360°)
3
4
5
6
7
8
...
n
ANSWERS TO EXERCISES
85
Answers to Exercises
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.
LESSON 7.2
1.
y translation
5
x
5
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.
N
2. reflection
y
Cue ball
8
W
E
8 ball
x
–5
Answers to Exercises
S
10. There are two possible points, one on the N
wall and one on the W wall.
–8
3.
reflection
y
N
5
H
W
–4
x
7
T
y
S
11.
–5
4.
E
H''
reflection
N
5
W
E
T
x
5
H
12. by the Minimal Path Conjecture
Proposed freeway
5.
rotation
y
5
5
Mason
x
Perry
13.
14.
86
H'
S
ANSWERS TO EXERCISES
,
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
ANSWERS TO EXERCISES
87
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:
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
′O
′A
N ′′
A
H
′
A′
H ′′
O
N
′′
′H
O
Answers to Exercises
′N
8. possible answer:
Center of rotation
17. possible answer:
A
9.
18a.
18b.
a b
d e
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°.
88
ANSWERS TO EXERCISES
O
5
12
B
9
0
12
7
14 11
?
? 11
0 13
?
?
20
c
a
f
d
2a
2b
d–e
3c
0
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. 3.4.6.43.42.6
7. 33.4232.4.3.4
8. 3632.4.12
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.
15. Answers will vary.
1
16. y 2x 4
y
4
–3
5
x
–6
17. possible answer: TOT
18.
12.
Answers to Exercises
N
8-ball
W
E
Cue ball
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.
S
ANSWERS TO EXERCISES
89
LESSON 7.5
1. Answers will vary.
2. The dual is a 5354 tessellation.
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
3.
y
Answers to Exercises
8
4. Yes. The four angles of the quadrilateral
will be around each point of intersection in the
tessellation.
a
a
c
c
5.
b
b
b
a
c
c
a
b a
90
b a
c
b
c b a
a
c
b
c b a
ANSWERS TO EXERCISES
c
a
c b
5
–2
x
LESSON 7.6
1.
2.
3.
4.
5.
6.
7.
Answers will vary.
Answers will vary.
Answers will vary.
regular hexagons
squares or parallelograms
squares or parallelograms
2
12. y 3x 3; the slope is the opposite sign.
y
5
–10
10
x
13. 3.4.6.4 4.6.12
440 rev 2 28 ft 1 min 1290 ft/s
14. 1 min 60 s
1 rev
8.
9. Answers will vary.
10. Answers will vary.
11.
B
E
A
S
ANSWERS TO EXERCISES
91
Answers to Exercises
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.
LESSON 7.7
1. equilateral triangles.
2. regular hexagons.
3.
Answers to Exercises
4.
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
and BB
. Because these segments are both
AA
perpendicular to the reflection line, they are
is parallel to
parallel to each other. Note that if AB
the reflection line, quadrilateral AABB 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.
5. Answers will vary.
6. Answers will vary.
7. sample design:
8. False; they must bisect each other in a
parallelogram.
92
ANSWERS TO EXERCISES
14b.
31
5 6
0
4
128
108
28 15
0
? ? 13?
3
5
9
8 7
?
? 6 0 ?
?
9 2
2
?
1 10
29
30
50
LESSON 7.8
1. parallelograms
2. parallelograms
3.
5. Answers will vary.
6. Answers will vary.
7. Circumcenter is (3, 4); orthocenter is (10, 8).
8.
9.
4.
10.
Answers to Exercises
ANSWERS TO EXERCISES
93
USING YOUR ALGEBRA SKILLS 7
1
1. y 6x
Answers to Exercises
2. y 2x 2
3. Centroid is 2, 23; orthocenter is (0, 5).
94
ANSWERS TO EXERCISES
4. Centroid is (4, 0); orthocenter is (3, 0).
4
5. 1, 3
6. (1, 1)
7. (5, 8)
22.
CHAPTER 7 REVIEW
T
H
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.
Answers to Exercises
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. 3632.4.3.4; 2-uniform
20. 4.82; semiregular
1
21. y 2x
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.
y
x
ANSWERS TO EXERCISES
95