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Warm Up

Apply Compositions of Transformations. Warm Up. Lesson Presentation. Lesson Quiz. (–4, 1). ANSWER. (–2, 3). ANSWER. Warm-Up. Find the image of (2, 3) under each transformation. 1. translation ( x, y ) → ( x – 6, y – 2). 2. reflection in the y -axis. (2, 9). ANSWER. (–3, 2).

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Warm Up

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  1. Apply Compositions of Transformations Warm Up Lesson Presentation Lesson Quiz

  2. (–4, 1) ANSWER (–2, 3) ANSWER Warm-Up Find the image of (2, 3) under each transformation. 1.translation (x, y) → (x – 6, y – 2) 2.reflection in the y-axis

  3. (2, 9) ANSWER (–3, 2) ANSWER Warm-Up Find the image of (2, 3) under each transformation. 3.reflection in the line y = 6 4.rotation 90º about the origin

  4. Begin by graphing ABC. Then graph A′B′C′ after a translation 12 units left. Finally, graph A′′B′′C′′ after a reflection in the x-axis. Example 1 The vertices of ABCare A(3, 2), B(6, 3), and C(7, 1). Find the image of ABCafter the glide reflection. Translation:(x, y) → (x –12, y) Reflection: in the x-axis SOLUTION

  5. 2. In Example 1, describe a glide reflection from A′′B′′C′′ to ABC. (x, y)  (x + 12, y) followed by a reflection inx. ANSWER ANSWER A(–3, –2), B(–6, –1), C(–7, –3) Guided Practice 1. Suppose ABCin Example 1 is translated 4 units down, then reflected in the y-axis. What are the coordinates of the vertices of the image?

  6. The endpoints of RSare R(1, –3) and S(2, –6). Graph the image of RSafter the composition. STEP 1 Graph RS. STEP 2 Reflect RSin the y-axis. R′S′ has endpoints R′(–1, –3) and S′(–2, –6). Rotate R′S′ 90 about the origin. R′′S′′ has endpoints R′′(3, –1) andS′′(6, –2). STEP 3 Example 2 Reflection: in the y-axis Rotation: 90° about the origin SOLUTION

  7. In the diagram, a reflection in line kmaps GH to G′H′. A reflection in line m maps G′H′to G′′H′′. Also, HB = 9 andDH′′ = 4. a. Name any segments congruent to each segment: HG, HB, and GA ~ ~ ~ ~ a. HG H′G′ ,andHG H′′G′′. HB H′B. GA G′A . Example 3 SOLUTION

  8. In the diagram, a reflection in line kmaps GH to G′H′. A reflection in line m maps G′H′to G′′H′′. Also, HB = 9 andDH′′ = 4. b. Yes, AC = BDbecause GG′′ and HH′′ are perpendicular to both kand m,so BDand ACare opposite sides of a rectangle. Example 3 b. Does AC = BD? Explain. SOLUTION

  9. In the diagram, a reflection in line kmaps GH to G′H′. A reflection in line m maps G′H′to G′′H′′. Also, HB = 9 andDH′′ = 4. c.What is the length of GG′′ ? c. By the properties of reflections, H′B = 9 and H′D = 4. Theorem 9.5 implies that GG′′ = HH′′ = 2 BD, so the length of GG′′ is 2(9 + 4), or 26 units. Example 3 SOLUTION

  10. 3.Graph RSfrom Example 2. Do the rotation first, followed by the reflection. Does the order of the transformations matter? Explain. ANSWER Yes; the resulting segment R′′ S ′′is not the same. Guided Practice

  11. ANSWER They are opposite sides of a parallelogram. Guided Practice 4.In Example 3, part (c), explain how you know that GG′′ = HH′′.

  12. ANSWER Translation Guided Practice Use the figure below for Exercises 5 and 6. The distance between line k and line mis 1.6 centimeters. 5. The preimage is reflected in line k , then in line m. Describe a single transformation that maps the blue figure to the green figure.

  13. 6. What is the distance between Pand P′′? If you draw PP′ , what is its relationship with line k? Explain. ANSWER 3.2 cm; They are perpendicular. Guided Practice Use the figure below for Exercises 5 and 6. The distance between line k and line mis 1.6 centimeters.

  14. Example 4 In the diagram, the figure is reflected in line k.The image is then reflected in line m. Describe a single transformation that maps Fto F′′. SOLUTION The measure of the acute angle formed between lines kand mis 70° . So, by Theorem 9.6, a single transformation that maps Fto F′′ is a 140° rotation about point P. You can check that this is correct by tracing lines kand mand point F, then rotating the point 140° .

  15. ANSWER A rotation of 160°about point P Guided Practice 7. In the diagram at the right, the preimage is reflected in line k, then in line m. Describe a single transformation that maps the blue figure onto the green figure.

  16. ANSWER 38° Guided Practice 8. A rotation of 76° maps Cto C′. To map Cto C′ using two reflections, what is the angle formed by the intersecting lines of reflection?

  17. Translation: (x, y) (x, y –2) ANSWER Lesson Quiz 1. The endpoints ofABareA(3, 2) andB(1,4). Graph the image AB after the glide reflection. Reflection: in the linex = 1

  18. 2. The vertices of MNK are M(1, 1), N(2, 3), are K(0, 2). Graph the image of MNK after the composition of the reflection followed by the rotation. Reflection: in the y-axis Rotation: 180° about the origin. ANSWER Lesson Quiz

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