Starter(s) 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

You drew reflections, translations, and rotations. 9.4 Composition of Transformations and Congruence Transformations You drew reflections, translations, and rotations. You proved whether two triangles were congruent. Draw glide reflections and other compositions of isometries in the coordinate plane. Draw compositions of reflections in parallel and intersecting lines. Verify congruence after a congruence transformation. Then/Now

composition of transformations glide reflections preimage image congruence transformation isometry Vocabulary

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

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

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

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

Step 1 translation along –1 , 5 (x, y) → (x + (–1), y + 5) Example 2) 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) Example 2) 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

2) 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

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

Step 1 Reflect EFGH in line p. Example 3) 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. Example 3) 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. 3) 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

Concept

Example 4) Verify Congruence after a Transformation Triangle PQR with vertices P(4, 2), Q(3, –3), and R(5, –2) is a transformation of ΔJKL with vertices J(–2, 0), K(–3, –5), and L(–1, –4). Graph the original figure and its image. Identify the transformation and verify that it is a congruence transformation. Understand You are asked to identify the type of transformation—reflection, translation, or rotation. Then, you need to show that the two figures are congruent. Plan Use the Distance Formula to find the measure of each side. Then show that the two triangles are congruent by SSS. Example 3

Example 4) Verify Congruence after a Transformation Solve Graph each figure. The transformation appears to be a translation 6 units right and 2 units up. Find the measures of the sides of each triangle. Example 3

Example 4) Verify Congruence after a Transformation

Answer: By SSS, ΔJKL  ΔPQR. Example 4) Verify Congruence after a Transformation Answer: By SSS, ΔJKL  ΔPQR. Check Use the definition of a translation. Use a ruler to measure and compare the corresponding sides of the triangles. The corresponding sides are congruent, so the triangles are congruent. Example 3

4) Triangle ABC with vertices A(–1, –4), B(–4, –1), and C(–1, –1) is a transformation of ΔXYZ with vertices X(–1, 4), Y(–4, 1), and Z(–1, 1). Graph the original figure and its image. Identify the transformation and verify that it is a congruence transformation. A. B. C. D. Example 3