9-4 Compositions of Transformations

1. 9-4 Compositions of Transformations You drew reflections, translations, and rotations. • Draw glide reflections and other compositions of isometries in the coordinate plane. • Draw compositions of reflections in parallel and intersecting lines.

2. Composite Photograph Composite photographs are made by superimposing one or more photographs.

3. Morphing Morphing is a popular special effect in movies. It changes one image into another.

4. Definition When a transformation is applied to a figure, and then another transformation is applied to its image, the result is called a composition of the transformations.

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6. Find a single transformation for a 75° counterclockwise rotation with center (2,1) followed by a 38° counterclockwise rotation with center (2,1) 113° counterclockwise rotation with center (2,1) 38° 75°

7. Find a single transformation equivalent to a translation with vector <−2, 7> followed by a translation with vector <9, 3>. Translation with vector <7, 10>

8. Quadrilateral BGTS has vertices B(–3, 4), G(–1, 3), T(–1 , 1), and S(–4, 2). Graph BGTS and its image after a translation along 5, 0 and a reflection in the x-axis. Step 1translation along 5, 0 (x, y)→(x + 5, y) B(–3, 4) →B'(2, 4) G(–1, 3)→G'(4, 3) S(–4, 2)→S'(1, 2) T(–1, 1)→T'(4, 1) Step 2 reflection in the x-axis (x, y)→(x, –y) B'(2, 4) → B''(2, –4) G'(4, 3)→ G''(4, –3) S'(1, 2)→ S''(1, –2) T'(4, 1)→ T''(4, –1)

9. Quadrilateral RSTU has vertices R(1, –1), S(4, –2), T(3, –4), and U(1, –3). Graph RSTU and its image after a translation along –4, 1and a reflection in the x-axis. Which point is located at (–3, 0)? A.R' B.S' C.T' D.U'

10. Definition An isometry is a transformation that preserves distance. Translations, reflections and rotations are isometries.

11. p. 652 The composition of two or more isometries – reflections, translations, or rotations results in an image that is congruent to its preimage. Glide reflections, reflections, translations, and rotations are the only four rigid motions or isometries in a plane.

12. Two translations equal One translation

13. Two rotations, same center equal One rotation

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15. Reflections over two parallel lines equals One translation

16. Copy and reflect figure EFGH in line p and then line q. Then describe a single transformation that maps EFGH onto E''F''G''H''. Step 1Reflect EFGH in line p. Step 2 Reflect E'F'G'H' in line q. Answer:EFGH is transformed onto E''F''G''H'' by a translation down a distance that is twice the distance between lines p and q.

17. Reflections over two intersection lines equals One rotation

18. Graph Other Compositions of Isometries ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph ΔTUV and its image after a translation along –1 , 5 and a rotation 180° about the origin. Step 1translation along –1 , 5 (x, y) →(x + (–1), y + 5) T(2, –1) → T'(1, 4) U(5, –2)→ U'(4, 3) V(3, –4)→ V'(2, 1) Step 2 rotation 180 about the origin (x, y)→(–x, –y) T'(1, 4) → T''(–1, –4) U'(4, 3)→ U''(–4, –3) V'(2, 1)→ V''(–2, –1)

19. A. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown. Step 1A brick is copied and translated to the right one brick length. Step 2 The brick is then rotated 90° counterclockwise about point M, given here. Step 3 The new brick is in place.

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21. 9-4 Assignment Page 655, 7-8, 13-14, 17, 21-24