Compositions of Reflections GEOMETRY LESSON 12-4 Pages 657-660 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. 12-4

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. 12-4

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 12-4

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° 36. 51 ° 37. 30° 3 7 12-4

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 12-4

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 12-4

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 63. 123 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 12-4

Compositions of Reflections GEOMETRY LESSON 12-4 66. a. 6 b. 84, 96, 180, 180, 264, 276 67. 15 cm 68. 185 m 69. 5.5 m 12-4