Section 3: Rigid Transformations and Symmetry Topic 1: Introduction to Transformations ...................................................................................................................... 80 Topic 2: Examining and Using Translations ................................................................................................................... 83 Topic 3: Translations of Polygons .................................................................................................................................... 85 Topic 4: Examining and Using Reflections ................................................................................................................... 88 Topic 5: Reflections of Polygons ..................................................................................................................................... 90 Topic 6: Examining and Using Rotations ...................................................................................................................... 93 Topic 7: Rotations of Polygons – Part 1 ......................................................................................................................... 96 Topic 8: Rotations of Polygons – Part 2 ......................................................................................................................... 98 Topic 9: Angle Preserving Transformations ................................................................................................................ 101 Topic 10: Symmetries of Regular Polygons ................................................................................................................ 105 Visit MathNation.com or search "Math Nation" in your phone or tablet's app store to watch the videos that go along with this workbook! 78 Section 3: Rigid Transformations and Symmetry The following Mississippi College- and Career-Readiness Standards for Mathematics will be covered in this section: G-CO.2 - Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not. (e.g., translation versus horizontal stretch) G-CO.3 - Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G-CO.4 - Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.5 - Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. 79 Section 3: Rigid Transformations and Symmetry Section 3: Rigid Transformations and Symmetry There are two main categories of transformations: rigid and Section 3 – Topic 1 non-rigid. Introduction to Transformations A __________________ transformation changes the size of the pre-image. What do you think happens when you transform a figure? A __________________ transformation does not change the size of the pre-image. What are some different ways that you can transform a figure? Write a real-world example of a rigid transformation. Ø In geometry, transformations refer to the ______________ of objects on a coordinate plane. Ø A pre-figure or pre-image is the original object. Write a real-world example of a non-rigid transformation. Ø The prime notation (") is used to represent a transformed figure of the original figure. Consider the graph below, circle the pre-image and box the transformed image. Describe the transformation. There are four common types of transformations: � A rotation turns the shape around a center point. A translation slides the shape in any direction. � A dilation changes the size of an object through an � enlargement or a reduction. A reflection flips the object over a line (as in a mirror �′ image). ’ 80 Section 3: Rigid Transformations and Symmetry In the table below, indicate whether the transformation is rigid Now, identify the transformations shown in the following or non-rigid and justify your answer. graphs and write the names of the transformations in the corresponding boxes under each graph. Transformation Rigid/Non-Rigid Justification Rigid Translation o o Non-Rigid Rigid Reflection o o Non-Rigid o Rigid Rotation o Non-Rigid Rigid Dilation o o Non-Rigid 81 Section 3: Rigid Transformations and Symmetry etsractice rt 1. Consider in the coordinate plane below. 2. Consider the transformations of in the previous problem. a. Write the coordinates of each endpoint, the length of the segment, and the midpoint of the segment. a. Trace the lines and identify the transformations on the graph. Length: ______ units Midpoint: b. Write the coordinates of and after the following transformations. Transformations is translated units up and 3 units to b. What are the and coordinates for each the left. transformation below? Fill in the length and midpoint is rotated clockwise about the of each segment indicated in the chart. origin. Transformation Coordinates Length Midpoint Translation Dilation Rotation Reflection 82 Section 3: Rigid Transformations and Symmetry BEAT THE TEST! Section 3 – Topic 2 Examining and Using Translations 1. Three rays share the same vertex as shown in the coordinate plane below. A translation is a rigid transformation that "slides" an object a fixed distance in a given direction while preserving the ___________ and __________ of the object. Suppose a geometric figure is translated units along the �-axis and units along the �-axis. We use the following Figure A notation to represent the transformation: (� �) (� � ) or (� �) (� � ) Figure B Figure C C Ø (� �) (� � )translates the point (� �) units ______ and units _______. Part A: Which figure represents a reflection across the -axis? Ø What is the algebraic description for a transformation Part B: Which of the following statements are true about that translates the point (� �) units to the left and the figure? Select all that apply. units upward? A rotation of will carry the object onto itself. A reflection of the figure along the -axis carries the figure to Quadrant II. Ø What is the algebraic description for a transformation In Figure A, that translates the point (� �) units to the right and If the vertex of Figure A is translated units downward? , it will carry onto the vertex of Figure B. Figure C is a reflection on the -axis of Figure A. 83 Section 3: Rigid Transformations and Symmetry etsractice rt 1. Transform triangle according to . 3. undergoes the translation , such that Write the coordinates for triangle . and a. What are the values of and ? 2. When the transformation is units performed on point , its image, point , is on the origin. What are the coordinates of ? Justify your answer. units b. Which of the following statements is true? A and have different locations. B and have different shapes. C and have different sizes. D and have different directions. 84 Section 3: Rigid Transformations and Symmetry BEAT THE TEST! Section 3 – Topic 3 Translations of Polygons 1. When the transformation is performed on point , its image is point . What are the Describe the translation of rectangle . coordinates of ? A B C D 2. Consider the following points. and undergoes the translation , such that and Part A: Complete the following algebraic description. The original object and its image are _______________. In other words, the two objects are identical in every respect except for their _____________. Part B: What is the difference between and ? Draw line segments linking a vertex in the original image to the corresponding vertex in the translated image. Make observations about the line segments. 85 Section 3: Rigid Transformations and Symmetry etsractice rt 1. Consider the two right triangles below. 2. is transformed by . a. What is the coordinate of ? b. What is the coordinate of ? c. Show the translation on the coordinate plane below. Rectangle is formed when right triangles and are translated. has vertices at , and . Describe how rectangle ’s location on the coordinate plane is possible with only one translation for and one translation for . 86 Section 3: Rigid Transformations and Symmetry 3. Polygon is the image of polygon after BEAT THE TEST! a translation . 1. Consider the figure below. What are the original coordinates of each point of If is the image of after a translation, then which point is polygon ? the image of after the same translation? 87 Section 3: Rigid Transformations and Symmetry Section 3 – Topic 4 Make generalizations about reflections to complete the Examining and Using Reflections following table. A reflection is a mirrored version of an object. The image does Reflection over Notation New coordinates not change __________, but the figure itself reverses. -axis -axis The function reflects the point over the given line. For instance, reflects the point over the -axis. Let’s examine the line reflections of the point over the -axis, -axis, , and . etsractice 1. Suppose the