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LESSON 9 –4

LESSON 9 –4. Compositions of Transformations. Five-Minute Check (over Lesson 9–3) TEKS Then/Now New Vocabulary Key Concept: Glide Reflection Example 1: Graph a Glide Reflection Theorem 9.1: Composition of Isometries Example 2: Graph Other Compositions of Isometries

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LESSON 9 –4

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  1. LESSON 9–4 Compositions of Transformations

  2. Five-Minute Check (over Lesson 9–3) TEKS Then/Now New Vocabulary Key Concept: Glide Reflection Example 1: Graph a Glide Reflection Theorem 9.1: Composition of Isometries Example 2: Graph Other Compositions of Isometries Theorem 9.2: Reflections in Parallel Lines Theorem 9.3: Reflections in Intersecting Lines Example 3: Reflect a Figure in Two Lines Example 4: Find the Preimage Example 5: Real-World Example: Identify Transformations Concept Summary: Compositions of Translations Lesson Menu

  3. The coordinates of quadrilateral ABCD before and after a rotation about the origin are shown in the table. Find the angle of rotation. A. 90° clockwise B. 90° counterclockwise C. 60° clockwise D. 45° clockwise 5-Minute Check 1

  4. The coordinates of triangle XYZ before and after a rotation about the origin are shown in the table. Find the angle of rotation. A. 180° clockwise B. 270° clockwise C. 90° clockwise D. 90° counterclockwise 5-Minute Check 2

  5. A.B. C.D. Draw the image of ABCD under a 180° clockwise rotation about the origin. 5-Minute Check 3

  6. The point (–2, 4) was rotated about the origin so that its new coordinates are (–4, –2). What was the angle of rotation? A. 180° clockwise B. 120° counterclockwise C. 90° counterclockwise D. 60° counterclockwise 5-Minute Check 4

  7. Targeted TEKS G.3(B) Determine the image or pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane. G.3(C) Identify the sequence of transformations that will carry a given pre-image onto an image on and off the coordinate plane. Also addresses G.3(A). Mathematical Processes G.1(A), G.1(G) TEKS

  8. 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. Then/Now

  9. composition of transformations • glide reflection Vocabulary

  10. Concept

  11. Graph a Glide Reflection 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. Example 1

  12. Graph a Glide Reflection Step 1 translation 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) Example 1

  13. Graph a Glide Reflection 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) Answer: Example 1

  14. 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' Example 1

  15. Concept

  16. 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. Example 2

  17. Graph Other Compositions of Isometries Step 1 translation 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) Example 2

  18. Graph Other Compositions of Isometries 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) Answer: Example 2

  19. ΔJKL has vertices J(2, 3), K(5, 2), and L(3, 0). Graph ΔTUV and its image after a translation along 3, 1and a rotation 180° about the origin. What are the new coordinates of L''? A. (–3, –1) B. (–6, –1) C. (1, 6) D. (–1, –6) Example 2

  20. Concept

  21. Concept

  22. Reflect a Figure in Two Lines 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''. Example 3

  23. Reflect a Figure in Two Lines Step 1 Reflect EFGH in line p. Example 3

  24. Reflect a Figure in Two Lines 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. Example 3

  25. Copy and reflect figure ABC in line s and then line t. Then describe a single transformation that maps ABC onto A''B''C''. A.ABC is reflected across lines and translated down 2 inches. B.ABC is translated down 2 inches onto A''B''C''. C.ABC is translated down 2 inches and reflected across line t. D.ABC is translated down 4 inches onto A''B''C''. Example 3

  26. Find the Preimage Example 4

  27. Find the Preimage B. Determine the preimage given the image and the composition of transformations. Rotation 180° about the origin; reflection in the line x-axis. Example 4

  28. Concept

  29. Identify Transformations A. LANDSCAPING Identify the preimage in the brick pattern. Then identify the sequence of transformations that will carry the preimage onto the images(s).

  30. Identify Transformations Answer: The preimage is one brick. Successive translations and rotations are used to carry the preimage onto the images.

  31. Identify Transformations A. LANDSCAPING Identify the preimage in the stepping stone pattern. Then identify the sequence of transformations that will carry the preimage onto the images(s).

  32. Identify Transformations

  33. A.What transformation must occur to the brick at point M to further complete the pattern shown here? A. The brick must be rotated 180° counterclockwise about point M. B. The brick must be translated one brick width right of point M. C. The brick must be rotated 90° counterclockwise about point M. D. The brick must be rotated 360° counterclockwise about point M. Example 4

  34. B.What transformation must occur to the brick at point M to further complete the pattern shown here? A. The two bricks must be translated one brick length to the right of point M. B. The two bricks must be translated one brick length down from point M. C. The two bricks must be rotated 180° counterclockwise about point M. D. The two bricks must be rotated 90° counterclockwise about point M. Example 4

  35. LESSON 9–4 Compositions of Transformations

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