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Five-Minute Check (over Lesson 9–3) 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: Real-World Example: Describe Transformations Concept Summary: Compositions of Translations Lesson Menu

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

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

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

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

Draw compositions of reflections in parallel and intersecting lines. You drew reflections, translations, and rotations. (Lessons 9–1, 9–2, and 9–3) Draw glide reflections and other compositions of isometries in the coordinate plane. Draw compositions of reflections in parallel and intersecting lines. Then/Now

composition of transformations glide reflection Vocabulary

Concept

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

Step 1 translation along 5, 0 (x, y) → (x + 5, y) 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

Step 2 reflection in the x-axis (x, y) → (x, –y) B'(2, 4) → B''(2, –4) 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

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

Concept

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

Step 1 translation along –1 , 5 (x, y) → (x + (–1), y + 5) 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

Step 2 rotation 180 about the origin (x, y) → (–x, –y) 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

ΔJKL has vertices J(2, 3), K(5, 2), and L(3, 0) 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

Concept

Concept

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

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

Step 2 Reflect E'F'G'H' in line q. 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

A. ABC is reflected across lines and translated down 2 inches. 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

Describe Transformations A. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown. Example 4

Step 1 A brick is copied and translated to the right one brick length. Describe Transformations Step 1 A brick is copied and translated to the right one brick length. Example 4

Describe Transformations Step 2 The brick is then rotated 90° counterclockwise about point M, given here. Example 4

Step 3 The new brick is in place. Describe Transformations Step 3 The new brick is in place. Answer: The pattern is created by successive translations and rotations shown above. Example 4

Describe Transformations B. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown. Example 4

Describe Transformations Step 1 Two bricks are copied and translated 1 brick length to the right. Example 4

Describe Transformations Step 2 The two bricks are then rotated 90 clockwise or counterclockwise about point M, given here. Example 4

Step 3 The new bricks are in place. Describe Transformations Step 3 The new bricks are in place. Another transformation is possible. Example 4

Step 1 Two bricks are copied and rotated 90 clockwise about point M. Describe Transformations Step 1 Two bricks are copied and rotated 90 clockwise about point M. Example 4

Step 2 The new bricks are in place. Describe Transformations Step 2 The new bricks are in place. Answer: The pattern is created by successive rotations of two bricks or by alternating translations then rotations. Example 4

A. The brick must be rotated 180° counterclockwise about point M. 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

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

Concept

End of the Lesson