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Lesson Menu Five-Minute Check (over Lesson 9–3) NGSSS 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

Over Lesson 9–3 A.A B.B C.C D.D 5-Minute Check 1 A.90° counterclockwise B.90° clockwise 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 A.A B.B C.C D.D 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 A.A B.B C.C D.D 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 A.A B.B C.C D.D 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?

NGSSS MA.912.G.2.4 Apply transformations to polygons to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons. MA.912.G.2.6 Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane.

Then/Now 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.

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:

A.A B.B C.C D.D 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:

A.A B.B C.C D.D 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.

A.A B.B C.C D.D 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 Describe Transformations A. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown.

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

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

Example 4 Describe Transformations Step 3The 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 1Two bricks are copied and translated 1 brick length to the right.

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

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

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

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

A.A B.B C.C D.D 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?

A.A B.B C.C D.D 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 brick 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?

Concept

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