Symmetry TransformationsYou can make symmetric designs by copying a basic figure to produce a balanced pattern. For example, to construct a design withreflection symmetry, start with pentagon ABCDE and line m below.Reflect ABCDE in line m to get pentagon A9B9C9D9E9.Reflecting a figure in a line is an example of a geometric operation called a A transformation produces a copy, or image,of an original figure in a new position.In this investigation, you will explore the transformations associated with reflection, rotation, and translation symmetry.transformation.AEDCBmABCDEAEDCBmInvestigation 2 Symmetry Transformations 278cmp06se_KH2.qxd 6/8/06 7:22 AM Page 27Problem2.12.1Describing Line ReflectionsTransformations that produce patterns with reflectionsymmetry are called Suppose you start with pentagon ABCDE and line mfrom the previous page.How can you locate the reflection image A9B9C9D9E9without folding, tracing, or using a mirror? Look for a precise way to describe a line reflection as you work through this problem.Describing Line ReflectionsA. Copy pentagon ABCDE, its image, A9B9C9D9E9, and the line ofreflection, m.1. Draw segments connecting each vertex of ABCDE to its image onA9B9C9D9E9. In other words, connect A to A9, B to B9, and so on.2. Use tools for measuring angles and lengths to see how the line ofreflection is related to each segment you drew in part (1).3. Describe the patterns in your measurements from part (2).B. 1. Copy quadrilateral JKLM and line m below. Use what youdiscovered in Question A to draw J9K9L9M9, the image of JKLMunder a reflection in line n. Use only a pencil, a ruler, and an angleruler or protractor. Explain how you located the image.2. Does JKLM have any symmetries? Explain.3. Does the figure made up of both JKLM and its reflection,J9K9L9M9, have any symmetries? Explain.JKMLnline reflections.AEDCBmABCDE28 Kaleidoscopes, Hubcaps, and Mirrors8cmp06se_KH2.qxd 6/8/06 7:22 AM Page 28C. The design below has reflection symmetry. Copy the design. Use only apencil, a ruler, and an angle ruler or protractor to locate the line ofsymmetry. Explain how you found the location of the line.D. Complete this definition: A line reflection in a line m matches eachpoint X on a figure to an image point X9 so that...E. Copy triangle DEF and line O. Notice triangle DEF crosses the line.1. Does triangle DEF have reflection symmetry?2. Draw the image of triangle DEF under a reflection in line O.3. Does the final figure, made of triangle DEF and its image, havereflection symmetry? Explain.F. When you reflect a figure in a line, you can visualize reflecting the entire plane and taking the figure along for the ride. Are any points in the plane unmoved by a reflection? That is, are there any fixed points? Explain.Homework starts on page 36.FEDᐉInvestigation 2 Symmetry Transformations 29For: The Transformation ToolVisit: PHSchool.comWeb Code: apd-52018cmp06se_KH2.qxd 6/8/06 7:22 AM Page 29Problem2.22.2Describing RotationsThe compass star shown at the right has rotationsymmetry. You can turn it around its center point to a position in which it looks identical to the original figure.Such a turn matches each point in the original to an image point on the original figure.The transformation that turns a figure about a point,matching each point to an image point, is called a In this problem, look for a way to describe the relationship between any point X and its image point X9 under a rotation.Describing RotationsA. Copy the compass star above.1. What is the smallest counterclockwise turn (in degrees) that willrotate the star to a new position in which it looks identical to theoriginal?2. Because the original figure has rotation symmetry, the image ofeach point on the original star is also a point on the rotated star.List the pairs of points and their images, matched by the rotation inpart (1).3. Describe the paths the points of the original figure follow as theyare “moved” to the positions of their images.4. How would you describe the relationship among any point X, itsimage, and the center of the compass star? B. Copy the “flag” at the right.1. Does the flag have rotation symmetry? Explain.2. Draw the flag’s image, PQ9R9S9, after a 60° counterclockwiserotation about point P. Use only tools for drawing and measuringsegments and angles. Explain how you located the image points.3. Does the final figure, made up of the original flag and its image,have rotation symmetry?RSPQrotation.ACGHBFDE30 Kaleidoscopes, Hubcaps, and Mirrors8cmp06se_KH2.qxd 6/8/06 7:22 AM Page 30Investigation 2 Symmetry Transformations 314. For which of these rotations about point P will the original flag andits image form a design with rotation symmetry?128 90° 408 458 18085. Can you make a design with rotation symmetry about P thatconsists of the original flag and more than one rotation image? If so,tell what rotations of the original are needed. If not, explain why.C. 1. Point P is outside of rectangle ABCD. Copy the rectangle and point P. Draw the image of ABCD after a 908 counterclockwiserotation about point P. Use only tools for drawing and measuringsegments and angles.2. On your drawing in part (1), use a compass to draw a circle withcenter P and radius PB.3. Describe the path the image of vertex B travels in a 908 rotationabout point P. How is the movement of the image of vertex Asimilar? How is it different?4. What can you say about segments PA and PA9? What can you sayabout segments PB and PB9?5. Find the measures of angles APA9, BPB9, CPC9, and DPD9. Whatcan you conclude?D. Complete this definition: A rotation of d degrees about a point Pmatches any point X with an image point X9 so that...E. When you rotate a figure about a point, you can visualize rotating theentire plane and taking the figure along for the ride. Are any points in the plane unmoved by a rotation? That is, are there any fixed points? Explain.Homework starts on page 36.ADBCP8cmp06se_KH2.qxd 6/8/06 7:22 AM Page 31Problem2.32.3Describing TranslationsStrip patterns and wallpaper designs have translation symmetry. You canslide the designs to new positions where the overall pattern appearsunchanged. The transformation that slides a figure, matching each point toan image point, is called a In this problem, look for a way to describe the relationship between anypoint X and its image point X9 after a translation.Describing TranslationsA. Copy Diagrams 1 and 2, which show polygon GHJKLM