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Warm Up

Perform Reflections. Warm Up. Lesson Presentation. Lesson Quiz. 1. The y -axis is the perpendicular bisector of AB . If point A is at (–3, 5) , what is the location of point B ?. (3, 5). ANSWER. ANSWER. –1. Warm-Up.

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Warm Up

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  1. Perform Reflections Warm Up Lesson Presentation Lesson Quiz

  2. 1.The y-axis is the perpendicular bisector of AB . If point A is at (–3, 5), what is the location of point B? (3, 5) ANSWER ANSWER –1 Warm-Up 2.What is the slope of the segment with endpoints at (–7, 4) and (4, –7)?

  3. – 1 –2 6 0 0 1 1 4 7 3.Multiply . 3 2 – 6 –1 7 ANSWER 3 4 Warm-Up

  4. a. In the linen : x = 3 Point Ais 2 units left of n, so its reflectionA′is 2 units right of nat (5, 3). Also, B′ is 2 units left of nat (1, 2), and C′is 1 unit right of n at (4, 1). Example 1 The vertices of ABC are A(1, 3),B(5, 2), and C(2, 1). Graph the reflection of ABC described. SOLUTION

  5. b. In the linem : y = 1 Point Ais 2 units above m, so A′ is 2 units below mat (1, –1). Also, B′ is 1 unit below mat (5, 0). Because point Cis on line m, you know that C = C′. Example 1 The vertices of ABC are A(1, 3),B(5, 2), and C(2, 1). Graph the reflection of ABC described. SOLUTION

  6. ANSWER Guided Practice Graph a reflection of ABC from Example 1 in the given line. 1.y = 4

  7. ANSWER Guided Practice Graph a reflection of ABC from Example 1 in the given line. 2.x = –3

  8. ANSWER Guided Practice Graph a reflection of ABC from Example 1 in the given line. 3.y = 2

  9. The endpoints of FG are F(–1, 2) and G(1, 2). Reflect the segment in the line y = x. Graph the segment and its image. The slope of y = xis 1. The segment from Fto its image, FF ′ , is perpendicular to the line of reflection y = x, so the slope of FF′ will be –1(because 1(–1) = –1). FromF, move 1.5units right and1.5units down toy = x. From that point, move1.5units right and1.5units down to locateF′(2,–1). Example 2 SOLUTION

  10. The slope of GG′ will also be –1. From G, move 0.5 units right and 0.5 units down to y = x. Then move 0.5 units right and 0.5 units down to locate G′(2, 1). Example 2

  11. Reflect FGfrom Example 2 in the line y = –x. Graph FGand its image. (a, b) (–b, –a) F(–1, 2) F ′(–2, 1) G(1, 2) G ′(–2, –1) Example 3 SOLUTION Use the coordinate rule for reflecting in y = –x.

  12. ANSWER Guided Practice 4. Graph ABC with vertices A(1, 3), B(4, 4), and C(3, 1). Reflect ABC in the lines y = –x and y = x. Graph each image.

  13. ANSWER Slope of y = – xis –1. The slope of FF′ is 1. The product of their slopes is –1 making them perpendicular. Guided Practice ′ 5. In Example 3, verify that FF is perpendicular to y = –x.

  14. Parking You are going to buy books. Your friend is going to buy CDs. Where should you park to minimize the distance you both will walk? Example 4

  15. Reflect Bin line mto obtain B′. Then draw AB′ . Label the intersection of AB′ and mas C. Because AB′ is the shortest distance between Aand B′ and BC = B′C, park at point Cto minimize the combined distance, AC + BC, you both have to walk. Example 4 SOLUTION

  16. ANSWER Because A′Bis the shortest distance between A′and Band AC = A′C, park at point Cto minimize the combined distance, AC + BC, you both have to walk. Guided Practice 6. Look back at Example 4. Answer the question by using a reflection of point Ainstead of pointB.

  17. The vertices of DEFare D(1, 2), E(3, 3), and F(4, 0). Find the reflection of DEFin the y-axis using matrix multiplication. Graph DEFand its image. Multiply the polygon matrix by the matrix for a reflection in the y-axis. STEP 1 D E F –1 0 1 3 4 0 1 2 3 0 Polygon matrix Reflection matrix Example 5 SOLUTION

  18. –1(1) + 0(2) –1(3) + 0(3) –1(4) + 0(0) = 0(1) + 1(2) 0(3) + 1(3) 0(4) + 1(0) D′ E′ F′ –1 –3 –4 = 2 3 0 Image matrix Example 5

  19. Example 5 STEP 2 GraphDEFandD′E′F′.

  20. L′(–3, –3), M′(1, –2), and N′(–2, –1) ANSWER Guided Practice The vertices ofLMN areL(–3, 3), M(1, 2),andN(–2, 1). Find the described reflection using matrix multiplication. 7. Reflect LMNin the x-axis.

  21. L′(3, 3), M′(–1, 2), and N′(2, 1) ANSWER Guided Practice The vertices ofLMN areL(–3, 3), M(1, 2),andN(–2, 1). Find the described reflection using matrix multiplication. 8. Reflect LMNin the y-axis.

  22. ANSWER Lesson Quiz Graph the polygon and its reflection in the given line. 1.Quadrilateral with verticesA(–2, 1), B(1, 4), C(3, 1), D(2, –1) overy = 1.

  23. ANSWER Lesson Quiz Graph the polygon and its reflection in the given line. 2.TriangleP(–2, 2), Q(3, 4), R(4, 1) over y = x.

  24. A B C A’ B’ C’ ANSWER 2 –1 0 – 2 1 0 3 4 –1 3 4 –1 Lesson Quiz 3. Use matrix multiplication to find the Image matrix when ABC is reflected in y-axis.

  25. ANSWER ( , 0) 41 8 Lesson Quiz 4.Find thepointCon thex-axis soAC + BC is a minimum for A(2, 5) andB(7, 3).

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