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Chapter 9: Transformations

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Chapter 9: Transformations Transformations • Lesson 9-1, 9-2, 9-3, and 9-5 Name, draw, and Key Vocabulary recognize figures that have been reflected, • reflection (p. 463) translated, rotated, or dilated. • translation (p. 470) • Lesson 9-4 Identify and create different types • rotation (p. 476) of tessellations. • tessellation (p. 483) Find the magnitude and direction • Lesson 9-6 • dilation (p. 490) of vectors and perform operations on vectors. • vector (p. 498) • Lesson 9-7 Use matrices to perform transformations on the coordinate plane. Transformations, lines of symmetry, and tessellations can be seen in artwork, nature, interior design, quilts, amusement parks, and marching band performances. These geometric procedures and characteristics make objects more visually pleasing. You will learn how mosaics are created by using transformations in Lesson 9-2. 460 Chapter 9 Transformations Dan Gair Photographic/Index Stock Imagery/PictureQuest Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 9. For Lessons 9-1 through 9-5 Graph Points Graph each pair of points. (For review, see pages 728 and 729.) 1. A(1, 3), B(Ϫ1, 3) 2. C(Ϫ3, 2), D(Ϫ3, Ϫ2) 3. E(Ϫ2, 1), F(Ϫ1, Ϫ2) 4. G(2, 5), H(5, Ϫ2) 5. J(Ϫ7, 10), K(Ϫ6, 7) 6. L(3, Ϫ2), M(6, Ϫ4) For Lesson 9-6 Distance and Slope Find mЄA. Round to the nearest tenth. (For review, see Lesson 7-4.) 3 5 2 7. tan A ϭ ᎏᎏ 8. tan A ϭ ᎏᎏ 9. sin A ϭ ᎏᎏ 4 8 3 4 9 15 10. sin A ϭ ᎏᎏ 11. cos A ϭ ᎏᎏ 12. cos A ϭ ᎏᎏ 5 12 17 For Lesson 9-7 Multiply Matrices Find each product. (For review, see pages 752 and 753.) 0 1 5 4 Ϫ1 0 0 Ϫ2 13. ΄΅и ΄΅ 14. ΄΅и ΄΅ 1 Ϫ1 Ϫ5 Ϫ1 1 1 Ϫ2 3 0 1 Ϫ3 4 5 Ϫ1 0 Ϫ1 Ϫ3 Ϫ3 2 15. ΄΅и ΄΅ 16. ΄΅и ΄΅ Ϫ1 0 Ϫ2 Ϫ5 1 0 Ϫ1 3 Ϫ1 Ϫ2 1 Transformations Make this Foldable to help you organize the types of transformations. Begin with one sheet of notebook paper. Fold Cut Cut the Fold a sheet of flap on every notebook paper third line. in half lengthwise. Label reflection Label each tab translation rotation with a vocabulary dilation word from this chapter. Reading and Writing As you read and study the chapter, use each page to write notes and examples of transformations, tessellations, and vectors on the coordinate plane. Chapter 9 Transformations 461 A Preview of Lesson 9-1 Transformations In a plane, you can slide, flip, turn, enlarge, or reduce figures to create new figures. These corresponding figures are frequently designed into wallpaper borders, mosaics, and artwork. Each figure that you see will correspond to another figure. These corresponding figures are formed using transformations. A transformation maps an initial image, called a preimage, onto a final image, called an image. Below are some of the types of transformations. The red lines show some corresponding points. translation reflection A figure can be slid in any direction. A figure can be flipped over a line. preimage preimage image image rotation dilation A figure can be turned around a point. A figure can be enlarged or reduced. preimage preimage image image Exercises Identify the following transformations. The blue figure is the preimage. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Make a Conjecture 11. An isometry is a transformation in which the resulting image is congruent to the preimage. Which transformations are isometries? 462 Investigating Slope-Intercept Form 462 Chapter 9 Transformations Reflections • Draw reflected images. • Recognize and draw lines of symmetry and points of symmetry. Vocabulary are reflections found Where in nature? • reflection • line of reflection On a clear, bright day glacial-fed lakes can • isometry provide vivid reflections of the surrounding • line of symmetry vistas. Note that each point above the water • point of symmetry line has a corresponding point in the image in the lake. The distance that a point lies above the water line appears the same as the distance its image lies below the water. DRAW REFLECTIONS A reflection is a transformation representing a flip of a figure. Figures may be reflected in a point, a line, or a plane. The figure shows a reflection of ABCDE Study Tip B D Reading Math in line m. Note that the segment connecting C a point and its image is perpendicular to A’, A”, A’”, and so on name A corresponding points for line m and is bisected by line m. Line m is E m one or more called the line of reflection for ABCDE and E' Ј Ј Ј Ј Ј transformations. its image A B C D E . Because E lies on the A' line of reflection, its preimage and image C' are the same point. B' D' It is possible to reflect a preimage in a point. In the figure below, polygon UVWXYZ is reflected in point P. Y Note that P is the midpoint X W of each segment connecting V' U' a point with its image. Z P Z' UෆPෆ PෆUෆෆЈ, VෆPෆ Х PPෆVෆෆЈ, U V ෆෆ Х ෆෆෆЈ ෆෆ Х ෆෆෆЈ W' X' WP PW , XP PX , ෆෆ Х ෆෆෆЈ ෆෆ Х ෆෆෆЈ Y' YP PY , ZP PZ Study Tip When reflecting a figure in a line Corresponding Corresponding or in a point, the image is congruent Sides Angles Look Back to the preimage. Thus, a reflection is To review congruence Є Х Є Ј Ј Ј a congruence transformation, or an UෆෆV Х UෆෆЈVෆෆЈ UVW U V W transformations, see Ј Ј Є Х Є Ј Ј Ј isometry. That is, reflections preserve VෆWෆ VෆෆWෆෆ VWX V W X Lesson 4-3. ෆෆ ෆෆЈෆෆЈ Є Х Є Ј Ј Ј distance, angle measure, betweenness WX W X WXY W X Y ෆෆ ෆෆЈෆෆЈ Є Х Є Ј Ј Ј of points, and collinearity. In the figure XY X Y XYZ X Y Z ෆෆ ෆෆЈෆෆЈ Є Х Є Ј Ј Ј above, polygon UVWXYZ Х polygon YZ Y Z YZU Y Z U UෆZෆ UෆෆЈZෆෆЈ ЄZUV Х ЄZЈUЈVЈ UЈVЈWЈXЈYЈZЈ. Lesson 9-1 Reflections 463 Robert Glusic/PhotoDisc Example 1 Reflecting a Figure in a Line Draw the reflected image of quadrilateral DEFG in line m. Step 1 Since D is on line m, D is its own reflection. F G' m Draw segments perpendicular to line m from E, F, and G. G Step 2 EЈ FЈ GЈ m E Locate , , and so that line is the D perpendicular bisector of EෆEෆෆЈ, FෆFෆෆЈ, and GෆGෆෆЈ. F' Points EЈ, FЈ, and GЈ are the respective E' images of E, F, and G. Step 3 Connect vertices D, EЈ, FЈ, and GЈ. Since points D, EЈ, FЈ, and GЈ are the images of points D, E, F, and G under reflection in line m, then quadrilateral DEЈFЈGЈ is the reflection of quadrilateral DEFG in line m. Reflections can also occur in the coordinate plane. Example 2 Reflection in the x-axis Study Tip COORDINATE GEOMETRY Quadrilateral KLMN has vertices K(2, Ϫ4), L(Ϫ1, 3), Reading M(Ϫ4, 2), and N(Ϫ3, Ϫ4). Graph KLMN and its image under reflection in the Mathematics x-axis. Compare the coordinates of each vertex with the coordinates of its image. The expression K(2, Ϫ4) → K’(2, 4) can Use the vertical grid lines to find a corresponding N' y K' be read as “point K is point for each vertex so that the x-axis is equidistant mapped to new location from each vertex and its image. M L K’.” This means that point K’ in the image K(2, Ϫ4) → KЈ(2, 4) L(Ϫ1, 3) → LЈ(Ϫ1, Ϫ3) corresponds to point K O x in the preimage. M(Ϫ4, 2) → MЈ(Ϫ4, Ϫ2) N(Ϫ3, Ϫ4) → NЈ(Ϫ3, 4) M' L' N Plot the reflected vertices and connect to form the K image KЈLЈMЈNЈ. The x-coordinates stay the same, but the y-coordinates are opposite. That is, (a, b) → (a, Ϫb). Example 3 Reflection in the y-axis COORDINATE GEOMETRY Suppose quadrilateral KLMN from Example 2 is reflected in the y-axis. Graph KLMN and its image under reflection in the y-axis. Compare the coordinates of each vertex with the coordinates of its image. Use the horizontal grid lines to find a corresponding y point for each vertex so that the y-axis is equidistant L L' from each vertex and its image. M M' Ϫ → Ј Ϫ Ϫ Ϫ → Ј K(2, 4) K ( 2, 4) L( 1, 3) L (1, 3) O x M(Ϫ4, 2) → MЈ(4, 2) N(Ϫ3, Ϫ4) → NЈ(3, Ϫ4) Plot the reflected vertices and connect to form the NKK' N' image KЈLЈMЈNЈ. The x-coordinates are opposite and the y-coordinates are the same. That is, (a, b) → (Ϫa, b). 464 Chapter 9 Transformations Example 4 Reflection in the Origin COORDINATE GEOMETRY Suppose quadrilateral KLMN from Example 2 is reflected in the origin. Graph KLMN and its image under reflection in the origin. Compare the coordinates of each vertex with the coordinates of its image. ෆෆෆЈ Since KK passes through the origin, use the horizontal K' y N' and vertical distances from K to the origin to find the L coordinates of KЈ. From K to the origin is 4 units up M and 2 units left. KЈ is located by repeating that pattern from the origin.
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