Transcription of 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. 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 .
2 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. 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.
3 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. 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.
4 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 .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?
5 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). 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.
6 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. 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.
7 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. Graph KLMNand its image under reflection in theline y x. Compare the coordinates of each vertex with the coordinates of its slope of y xis 1. K K is perpendicular to y x, so its slope is 1. From Kto the line y x, move up three units and left three units.
8 From the line y xmove up three units and left three units to K ( 4, 2).K(2, 4) K ( 4, 2)L( 1, 3) L (3, 1)M( 4, 2) M (2, 4)N( 3, 4) N ( 4, 3)Plot the reflected vertices and connect to form the image K L M N . Comparing coordinates shows that (a, b) (b, a).yxNKMLN'M'L'K' AND POINTS OF SYMMETRYSome figures can be folded so thatthe two halves match exactly. The fold is a line of reflection called a .For some figures, a point can be found that is a common point of reflection for allpoints on a figure. This common point of reflection is called a .Lines of SymmetryPoints of SymmetryPPFEach point on the figure musthave an image on the figure fora point of symmetry to point of symmetry is themidpoint of all segmentsbetween the preimage andthe points than Twopoint of symmetryline of symmetry466 Chapter 9 TransformationsUse ReflectionsGOLFA deel and Natalie are playing miniature golf. Adeel says that he read how to use reflections to help make a hole-in-one on mostminiature golf holes.
9 Describe how he should putt the ball to make a Adeel tries to putt the ball directly to the hole, he will strike the border as indicated by the blue line. So, he can mentally reflect the hole in the line that contains the right border. If he putts the ball at the reflected image of the hole, the ball will strike the border, and it will rebound on a path toward the ofreflectionBallHoleImageBallHoleExample 6 Example6 Draw Lines of SymmetryDetermine how many lines of symmetry a square has. Then determine whethera square has point has four lines of square has point symmetry. Pis thepoint of symmetry such that AP PA ,BP PB , CP PC , and so 'B' A'Example7 Example7A Point ofSymmetryA point of symmetry is the intersection of all the segments joining apreimage to an TipLesson 9-1 Reflections467(l)Siede Pries/PhotoDisc, (c)Spike Mafford/PhotoDisc, (r)Lynn Stone Concept CheckGuided PracticeApplication1. Find a counterexampleto disprove the statement A point of symmetry is theintersection of two or more lines of symmetry for a plane ENDEDDraw a figure on the coordinate plane and then reflect it in the line y x.
10 Label the coordinates of the preimage and the image. 3. Identifyfour properties that are preserved in the figure at the right. Draw itsreflected image in line how many lines of symmetry each figure has. Then determine whetherthe figure has point GEOMETRYG raph each figure and its image under the B with endpoints A(2, 4) and B( 3, 3) reflected in the x-axis9. ABCwith vertices A( 1, 4), B(4, 2), and C(0, 3) reflected in the y-axis10. DEF with vertices D( 1, 3), E(3, 2), and F(1, 1) reflected in the origin11. GHIJ with vertices G( 1, 2), H(2, 3), I(6, 1), and J(3, 0) reflected in the line y xNATURED etermine how many lines of symmetry each object has. Thendetermine whether each object has point and ApplyPractice and ApplyRefer to the figure at the right. Name the image of eachfigure under a reflection in:line X Z Y 22. TXZ17. XZY20. YVW23. YUZCopy each figure. Draw the image of each figure under a reflection in line . WXVYZTUm ForExercises15 26, 38, 3929, 34, 404130, 33, 4127, 2831, 32, 404235 37, 44 47 SeeExamples1234567 Extra Practice See page Practice See page Chapter 9 TransformationsCOORDINATE GEOMETRYG raph each figure and its image under the MNPQ with vertices M(2, 3), N(2, 3), P( 2, 3), and Q( 2, 3) in the GHIJ with vertices G( 2, 2), H(2, 0), I(3, 3), and J( 2, 4) in the QRST with vertices Q( 1, 4), R(2, 5), S(3, 2), and T(0, 1) in the with vertices D(4, 0), E( 2, 4), F( 2, 1), and G(4, 3) in the y-axis31.
