1. Compositions of Reflections
GEOMETRY LESSON 12-4
Pages Exercises
1. rotation
2. translation
3. Neither; the figures do not have the same orientation. 4.
F is translated down twice the distance between and m. 5.
M is translated across line m twice the distance between and m.
6.      
T is translated across line m twice the distance between and m.
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2. Compositions of Reflections
GEOMETRY LESSON 12-4
7.        
L is rotated clockwise about 180°. 8.
V is rotated clockwise about 145°.
9.     
N is rotated clockwise about 160°.
10.
11.
12.
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3. Compositions of Reflections
GEOMETRY LESSON 12-4
13.
14.
15.
16.
17.
18. opp.; reflection
19. opp.; glide reflection
20. same; translation
21. same; rotation
22. same; rotation
23. same; translation
24. opp.; reflection
25. opp.; glide reflection
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4. Compositions of Reflections
GEOMETRY LESSON 12-4
26. glide reflection; glide –2, –2, refl. in y = x – 1 to map black figure to blue figure
27. reflection; refl. in
y = –
28. glide reflection; glide 0, 4, refl. in y = 0 to map black figure to blue figure 1 2
29. rotation; 180° about
the pt. 0, , to map
the black figure to blue figure
30. Odd isometries can be expressed as the composition of an odd number of reflections. Even isometries are the composition of an even number of reflections.
31. Check students’ work.
32. Yes; a rotation of x° followed by a rotation of y° is equivalent to a rotation of (x + y)°.
33. No; explanations may vary.
34. 60°
35. 60° °
37. 30° 3 7
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5. Compositions of Reflections
GEOMETRY LESSON 12-4
38. rotation; center C, of rotation 180°
39. glide reflection; 11, 0, y = 0
40. translation; –9, 0
41. reflection; y = 0
42. reflection; x = 4
43. reflection; x = –
44. rotation; center (3, 0), of rotation 180°
45. glide reflection;
0, 4, x = 4
46. translation; –11, –4
47. rotation; center (0, 2), of rotation 180°
48. The glide involves a translation to the right and the reflection is in a line with a positive slope passing between each R.
49–52. Answers may vary. Samples are given.
49. If XY is reflected in line , then is the bis. of XX and YY , so XX || YY and XX YY is an isosc. trap. Therefore XY X Y .
50. XX || YY and XX YY , so XX Y Y is a Therefore, XY X Y . 1 2
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6. Compositions of Reflections
GEOMETRY LESSON 12-4
51. If XY is rotated x° about pt. R, then RX RX and RY RY . Also, m XRY + m YRX = m YRX + m X RY = x, so XRY X RY . So XRY X RY by SAS and XY X Y by CPCTC.
52. Answers may vary. Sample: Since a reflection moves a pt. in the direction to the translation, the order does not matter.
53. No; explanations may vary. Sample: If (1, 1) is reflected over the line y = x and then the x-axis, the image is (1, –1). If the reflections are reversed, the image is (–1, 1).
54. (6, 5)
55. (1, 2)
56. (2, 6)
57. (–3, 1)
58. A
59. H
60. [2] V(–5, 2), T(–4, 0), Y(–1, 3) glided give V (–2, –1), T (–1, –3), Y (2, 0). These vertices reflected over y = –x give V (1, 2), T (3, 1), Y (0, –2).
[1] incorrect method or incorrect answer
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7. Compositions of Reflections
GEOMETRY LESSON 12-4
61 (continued)
[1] composition is partially correct
62. H-180; I-180, O-any rotation; X-180; N-180; S-180; Z-180
64. 90
65. 87
61. [4] a. 
b. Suppose pt. A in F is x units from s. Thus, A reflected across s gives A , x units right of s. A is then PQ – x units left of t. Thus, A reflected across t gives A , PQ – x units right of t. Thus, the total distance traveled is PQ – x + PQ + x = 2PQ.
[3] correct composition, vague explanation
[2] part (a) only
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8. Compositions of Reflections
GEOMETRY LESSON 12-4
66. a. 6
b. 84, 96, 180, 180, 264, 276 cm m m
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