8cmp06se KH2 qxd 6 8 06 7 22 AM Page 27 Symmetry Transformations You can make symmetric designs by copying a basic figure to produce a balanced pattern For example to construct a design with reflection symmetry start with pentagon ABCDE and line m below m A B C D E Reflect ABCDE in line m to get pentagon A9B9C9D9E9 m A A B B C E C D D E Reflecting a figure in a line is an example of a geometric operation called a transformation 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 Investigation 2 Symmetry Transformations 27 8cmp06se KH2 qxd 6 8 06 7 22 AM Page 28 2 1 Describing Line Reflections Transformations that produce patterns with reflection symmetry are called line reflections m A A Suppose you start with pentagon ABCDE and line m from the previous page B How can you locate the reflection image A9B9C9D9E9 without folding tracing or using a mirror Look for a precise way to describe a line reflection as you work through this problem C C D E Problem 2 1 Describing Line Reflections A Copy pentagon ABCDE its image A9B9C9D9E9 and the line of reflection m 1 Draw segments connecting each vertex of ABCDE to its image on A9B9C9D9E9 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 of reflection 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 you discovered in Question A to draw J9K9L9M9 the image of JKLM under a reflection in line n Use only a pencil a ruler and an angle ruler or protractor Explain how you located the image n K J L M 2 Does JKLM have any symmetries Explain 3 Does the figure made up of both JKLM and its reflection J9K9L9M9 have any symmetries Explain 28 Kaleidoscopes Hubcaps and Mirrors B D E 8cmp06se KH2 qxd 6 8 06 7 22 AM Page 29 C The design below has reflection symmetry Copy the design Use only a pencil a ruler and an angle ruler or protractor to locate the line of symmetry Explain how you found the location of the line D Complete this definition A line reflection in a line m matches each point X on a figure to an image point X9 so that E Copy triangle DEF and line O Notice triangle DEF crosses the line D E F 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 have reflection 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 For The Transformation Tool Visit PHSchool com Web Code apd 5201 Homework starts on page 36 Investigation 2 Symmetry Transformations 29 8cmp06se KH2 qxd 6 8 06 7 22 AM Page 30 2 2 Describing Rotations The compass star shown at the right has rotation symmetry 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 rotation In this problem look for a way to describe the relationship between any point X and its image point X9 under a rotation A H B G C F D E Problem 2 2 Describing Rotations A Copy the compass star above 1 What is the smallest counterclockwise turn in degrees that will rotate the star to a new position in which it looks identical to the original 2 Because the original figure has rotation symmetry the image of each 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 in part 1 3 Describe the paths the points of the original figure follow as they are moved to the positions of their images 4 How would you describe the relationship among any point X its image and the center of the compass star R 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 counterclockwise rotation about point P Use only tools for drawing and measuring segments and angles Explain how you located the image points S Q 3 Does the final figure made up of the original flag and its image have rotation symmetry P 30 Kaleidoscopes Hubcaps and Mirrors 8cmp06se KH2 qxd 6 8 06 7 22 AM Page 31 4 For which of these rotations about point P will the original flag and its image form a design with rotation symmetry 128 90 408 458 1808 5 Can you make a design with rotation symmetry about P that consists 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 counterclockwise rotation about point P Use only tools for drawing and measuring segments and angles A B D C P 2 On your drawing in part 1 use a compass to draw a circle with center P and radius PB 3 Describe the path the image of vertex B travels in a 908 rotation about point P How is the movement of the image of vertex A similar How is it different 4 What can you say about segments PA and PA9 What can you say about segments PB and PB9 5 Find the measures of angles APA9 BPB9 CPC9 and DPD9 What can you conclude D Complete this definition A rotation of d degrees about a point P matches any point X with an image point X9 so that E When you rotate a figure about a point you can visualize rotating the entire 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 Investigation 2 Symmetry Transformations 31 8cmp06se KH2 qxd 6 8 06 7 22 AM Page 32 2 3 Describing Translations Strip patterns and wallpaper designs have translation symmetry You can slide the designs to new positions where the overall pattern appears unchanged The transformation that slides a figure matching each point to an image point is called a translation In this problem look for a way to describe …