Chapter 9: Transformations

1 460 Chapter 9 TransformationsTransformations reflection (p. 463) translation (p. 470) rotation (p. 476) tessellation (p. 483) dilation (p. 490) vector (p. 498)Key Vocabulary Lesson 9-1, 9-2, 9-3, and 9-5 Name, draw, andrecognize figures that have been reflected,translated, rotated, or dilated. Lesson 9-4 Identify and create different typesof tessellations. Lesson 9-6 Find the magnitude and directionof vectors and perform operations on vectors. Lesson 9-7 Use matrices to performtransformations on the coordinate Gair Photographic/Index Stock Imagery/PictureQuest Transformations , lines of symmetry, and tessellations can be seen in artwork, nature, interior design, quilts, amusement parks, and marching band performances.

2 These geometric procedures and characteristics make objects more visually will learn how mosaics are created by using Transformations in Lesson Chapter 9 TransformationsChapter 9 Transformations461 TransformationsMake this Foldable to help you organize the types of Transformations . Begin with one sheet ofnotebook each tabwith a vocabularyword from theflap on everythird a sheet ofnotebook paperin half lengthwise. Reading and WritingAs you read and study the Chapter , use each page towrite notes and examples of Transformations , tessellations, and vectors on thecoordinate SkillsTo be successful in this Chapter , you ll need to masterthese skills and be able to apply them in problem-solving situations.

3 Reviewthese skills before beginning Chapter Lessons 9-1 through 9-5 Graph PointsGraph each pair of points.(For review, see pages 728 and 729.) (1, 3), B( 1, 3) ( 3, 2), D( 3, 2) ( 2, 1), F( 1, 2) (2, 5), H(5, 2) ( 7, 10), K( 6, 7) (3, 2), M(6, 4)For Lesson 9-6 Distance and SlopeFind m A. Round to the nearest tenth.(For review, see Lesson 7-4.) A 34 A 58 A 23 A 45 A 192 A 1157 For Lesson 9-7 Multiply MatricesFind each product.(For review, see pages 752 and 753.)13. 14. 15. 16. 21 3 2 3 1 130 1 10514 5 3 2100 1 230 201 114 15 51 101 FoldLabelCutChapter 9 Transformations461462 Investigating slope -Intercept Form462 Chapter 9 TransformationsA Preview of Lesson 9-1In a plane, you can slide, flip, turn, enlarge, or reduce figures to create new corresponding figures are frequently designed into wallpaper borders,mosaics, and artwork.

4 Each figure that you see will correspond to another corresponding figures are formed using Transformations . Amaps an initial image, called a preimage, onto a final image,called an image. Below are some of the types of Transformations . The red linesshow some corresponding figure can be slid in any figure can be flipped over a figure can be turned around a figure can be enlarged or the following Transformations . The blue figure is the a isometryis a transformation in which the resulting image is congruent to the Transformations are isometries?DRAW REFLECTIONSAis a transformation representing a flip of a figure. Figures may be reflected in a point, a line, or a figure shows a reflection of ABCDEin line m.

5 Note that the segment connecting a point and its image is perpendicular to line mand is bisected by line m. Line mis called the for ABCDEand its image A B C D E . Because Elies on the line of reflection, its preimage and image are the same is possible to reflect a preimage in a point. In the figure below, polygonUVWXYZis reflected in point that Pis the midpoint of each segment connecting a point with its P P U , V P PP V ,W P P W , X P P X ,Y P P Y , Z P P Z When reflecting a figure in a line or in a point, the image is congruent to the preimage. Thus, a reflection is a congruence transformation, or an . That is, reflections preserve distance, angle measure, betweenness of points, and collinearity. In the figure above, polygon UVWXYZ polygon U V W X Y Z.

6 IsometryY'W'Z'U'V'X'YXWPZUV line of reflectionA'B'C'D'E'ABCDE mreflectionReflectionsLesson 9-1 Reflections463 Vocabulary reflection line of reflection isometry line of symmetry point of symmetry Draw reflected images. Recognize and draw lines of symmetry and points of a clear, bright day glacial-fed lakes canprovide vivid reflections of the surroundingvistas. Note that each point above the waterline has a corresponding point in the imagein the lake. The distance that a point liesabove the water line appears the same asthe distance its image lies below the Glusic/PhotoDisc Reading MathA , A , A , and so on namecorresponding points forone or TipCorrespondingCorresponding SidesAnglesU V U V UVW U V W V W V W VWX V W X W X W X WXY W X Y X Y X Y XYZ X Y Z Y Z Y Z YZU Y Z U U Z U Z ZUV Z U V Look BackTo review congruencetransformations, seeLesson TipWhereWhereare reflections found in nature?

7 Are reflections found in nature?Reflections can also occur in the coordinate Chapter 9 TransformationsReflecting a Figure in a Line Draw the reflected image of quadrilateral DEFGin line 1 Since Dis on line m, Dis its own segments perpendicular to line mfrom E, F, and 2 Locate E , F , and G so that line mis the perpendicular bisector of E E , F F , and G G . Points E , F , and G are the respective images of E, F, and 3 Connect vertices D, E , F , and G .Since points D, E , F , and G are the images of points D, E, F, and Gunderreflection in line m, then quadrilateral DE F G is the reflection of quadrilateralDEFGin line 'G'GF'EDFmExample1 Example1 Reflection in the x-axisCOORDINATE GEOMETRYQ uadrilateral KLMNhas vertices K(2, 4), L( 1, 3),M( 4, 2), and N( 3, 4).

8 Graph KLMNand its image under reflection in the x-axis. Compare the coordinates of each vertex with the coordinates of its the vertical grid lines to find a corresponding point for each vertex so that the x-axis is equidistant from each vertex and its (2, 4) K (2, 4)L( 1, 3) L ( 1, 3)M( 4, 2) M ( 4, 2)N( 3, 4) N ( 3, 4)Plot the reflected vertices and connect to form theimage K L M N . The x-coordinates stay the same, but the y-coordinates are opposite. That is, (a, b) (a, b).yxONKMLN'M'L'K'Example2 Example2 ReadingMathematicsThe expression K(2, 4) K (2, 4) canbe read as point Kismapped to new locationK . This means that point K in the imagecorresponds to point Kin the TipReflection in the y-axisCOORDINATE GEOMETRYS uppose quadrilateral KLMN from Example 2 isreflected in the y-axis.

9 Graph KLMN and its image under reflection in the the coordinates of each vertex with the coordinates of its the horizontal grid lines to find a corresponding point for each vertex so that the y-axis is equidistant from each vertex and its (2, 4) K ( 2, 4)L( 1, 3) L (1, 3)M( 4, 2) M (4, 2)N( 3, 4) N (3, 4)Plot the reflected vertices and connect to form theimage K L M N . The x-coordinates are opposite and the y-coordinates are the same. That is, (a, b) ( a, b).yxONKMLN'M'L'K'Example3 Example3 Reflections in the Coordinate PlaneReflectionx-axisy-axisoriginy xPreimage to(a, b) (a, b)(a, b) ( a, b)(a, b) ( a, b)(a, b) (b, a)ImageHow to findMultiply theMultiply theMultiply both Interchange the coordinatesy-coordinate by by by and ( 3, 2)B'(2, 3)A(1, 3)A'(3, 1)yxOB'( 3, 1)A'( 3, 2)A(3, 2)B(3, 1)yxOA'( 3, 2)B'( 1, 2)A(3, 2)B(1, 2)yxOB'( 3, 1)B( 3, 1)A'(2, 3)A(2, 3)Lesson 9-1 Reflections465 Reflection in the OriginCOORDINATE GEOMETRYS uppose quadrilateral KLMN from Example 2 isreflected in the origin.

10 Graph KLMNand its image under reflection in theorigin. Compare the coordinates of each vertex with the coordinates of its K K passes through the origin, use the horizontal and vertical distances from Kto the origin to find thecoordinates of K . From Kto the origin is 4 units up and 2 units left. K is located by repeating that pattern from the origin. Four units up and 2 units left yields K ( 2, 4).K(2, 4) K ( 2, 4)L( 1, 3) L (1, 3)M( 4, 2) M (4, 2)N( 3, 4) N (3, 4)Plot the reflected vertices and connect to form the image K L M N . Comparingcoordinates shows that (a, b) ( a, b).yxONKMLN'M'L'K'Example4 Example4 Reflection in the Line y = xCOORDINATE GEOMETRYS uppose quadrilateral KLMN from Example 2 isreflected in the line y x.