compositions of reflections

(For help, go to Lessons 12-1 and 12-2.)

Given points R(–1, 1), S(–4, 3), and T(–2, 5), draw RST and its reflection image in each line.

1. the y-axis 2. the x-axis 3. y = 1

Draw RST described above and its translation image for each translation vector.

4. 0, –3 5. 4, 0 6. 2, –5

7. Copy the figure. Draw images of the

figure for a reflection in DG and for the

translation vector FG.

GEOMETRY LESSON 12-4GEOMETRY LESSON 12-4Compositions of Reflections

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1.

2.3.

4. 5. 6.

7.

Solutions


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12.4 Composition of Functions

Another way to define translations and rotations:A translation or rotation is a composition of reflections.

Translation:A reflection across two parallel lines.

The length of translation is equal to TWICE the distance between the parallel line.

Rotation:A reflection across two intersecting lines.

The angle of rotation is equal to TWICE the measure of the angle of intersection.

Judging by appearances, is one figure a translation image or rotation image of the other? Explain.

The figures appear to be congruent, and their orientations are the same.

Corresponding sides of the figures appear to be parallel.

This suggests that one figure is a translation image of the other and not a rotation image.


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What is the most amount of reflections required to create an isometry?

The Fundamental Theorem of Isometries:In a plane, one of two congruent figures can be mapped onto the other by AT MOST a composition of three reflections.

Find the image of the figure for a reflection in line and then in line m.

First, find the reflection image in line . It no longer looks like a 4.


Then, find the image of the first reflection in line m. The final image is a translation of the original figure.

The arrow shows the direction and distance of the translation.

The arrow is perpendicular to lines and m with length equal to twice the distance from to m.

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The letter D is reflected in line x and then in line y. Describe

the resulting rotation.

Find the image of D through a reflection in line x.


The composition of two reflections in intersecting lines is a rotation. The center of rotation is the point where the lines intersect, and the angle is twice the angle formed by the intersecting lines.

So the letter D is rotated 86° clockwise, or 274° counterclockwise, with the center of rotation at point A.

Find the image of the reflection through another reflection in line y.

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A glide reflection is a composition of a translation and a reflection in a line parallel to the translation vector.

A

B

C

D

A’

B’

C’

D’

First, translate ABC by 0, 2.

(–4, 5) (–4 + 0, 5 + 2), or (–4, 7)

(6, 2) (6 + 0, 2 + 2), or (6, 4)

(0, 0) (0 + 0, 0 + 2), or (0, 2)

ABC has vertices A(–4, 5), B(6, 2), and C(0, 0). Find the

image of ABC for a glide reflection where the glide is 0, 2 and the

reflection line is x = 1.


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Then, reflect the translated image in the line x = 1.

The glide reflection image A B C has vertices A (6, 7), B (–4, 4), and C (2, 2).

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(continued)

Tell whether orientations are the same or opposite. Then

classify the isometry.

The segment connecting the horizontal segments of the letter Z slopes up from the left to the right.

The segment connecting the horizontal segments of the figure on the right slopes down from left to right.

So the figure on the right and Z have opposite orientations. It is a reflection of Z across a vertical line.


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The orientation of the figure to the left of the dashed line is opposite the figure to the right of the dashed line. The isometry is a reflection.

Use the diagram for Exercises 1–3.

1. Find the image of L for a reflection in line a and then in line b.

2. Find the image of L for a reflection in line a and then in line c.


3. Describe the rotation in Exercise 2.

5. Name the four types of isometries.

4. PQ has endpoints P(4, 15) and Q(–6, 10). Find the image of PQ for a glide reflection where the glide is 0,–8 and the reflection line is x = 0.

180° rotation with center of rotation at the intersection of lines a and c

Check that students’ images have endpoints P (–4, 7) and Q (6, 2).

glide reflection, reflection, rotation, translation

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GEOMETRY LESSON 12-4GEOMETRY 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.

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Compositions of Reflections


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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13.

14.

15.

16.

17.

18. opp.; reflection19. opp.; glide reflection

20. same; translation21. same; rotation

22. same; rotation23. same; translation

24. opp.; reflection25. opp.; glide reflection

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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

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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.

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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°

37

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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 = 443. 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.

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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 XYX Y .

50. XX || YY and XXYY , so XX Y Y is a . Therefore, XYX Y .

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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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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

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66. a. 6

b. 84, 96, 180, 180, 264, 276

67. 15 cm

68. 185 m

69. 5.5 m

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Use the diagram for Exercises 1–3.

1. Find the image of L for a reflection in line a and then in line b.

2. Find the image of L for a reflection in line a and then in line c.


3. Describe the rotation in Exercise 2.

5. Name the four types of isometries.

4. PQ has endpoints P(4, 15) and Q(–6, 10). Find the image of PQ for a glide reflection where the glide is 0,–8 and the reflection line is x = 0.

180° rotation with center of rotation at the intersection of lines a and c

Check that students’ images have endpoints P (–4, 7) and Q (6, 2).

glide reflection, reflection, rotation, translation

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compositions of reflections

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