Compositions of Transformations LESSON 9–4

Lesson Menu 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

Over Lesson 9–3 5-Minute Check 1 A.90° clockwise B.90° counterclockwise C.60° clockwise D.45° clockwise The coordinates of quadrilateral ABCD before and after a rotation about the origin are shown in the table. Find the angle of rotation.

Over Lesson 9–3 5-Minute Check 2 A.180° clockwise B.270° clockwise C.90° clockwise D.90° counterclockwise The coordinates of triangle XYZ before and after a rotation about the origin are shown in the table. Find the angle of rotation.

Over Lesson 9–3 5-Minute Check 3 Draw the image of ABCD under a 180° clockwise rotation about the origin. A.B. C.D.

Over Lesson 9–3 5-Minute Check 4 A.180° clockwise B.120° counterclockwise C.90° counterclockwise D.60° counterclockwise The point (–2, 4) was rotated about the origin so that its new coordinates are (–4, –2). What was the angle of rotation?

TEKS 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)

Then/Now 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.

Vocabulary composition of transformations glide reflection

Concept

Example 1 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 Graph a Glide Reflection 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)

Example 1 Graph a Glide Reflection Step 2reflection 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 A.R' B.S' C.T' D.U' 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)?

Concept

Example 2 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 Graph Other Compositions of Isometries 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)

Example 2 Graph Other Compositions of Isometries Step 2rotation 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 A.(–3, –1) B.(–6, –1) C.(1, 6) D.(–1, –6) 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''?

Concept

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

Example 3 Reflect a Figure in Two Lines Step 2Reflect 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 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''. Copy and reflect figure ABC in line s and then line t. Then describe a single transformation that maps ABC onto A''B''C''.

Example 4 Find the Preimage

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

Concept

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).

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

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).

Identify Transformations

Example 4 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. A. What transformation must occur to the brick at point M to further complete the pattern shown here?

Example 4 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. B. What transformation must occur to the brick at point M to further complete the pattern shown here?

Compositions of Transformations LESSON 9–4