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Chapter 12: Rigid Transformations

Difficulty Level: At Grade Created by: CK-12

Introduction

The final chapter of Geometry explores transformations. A transformation is a move, flip, or rotation of an image. First, we will look at different types of symmetry and then discuss the different types of transformations. Finally, we will compose transformations and look at tessellations.

Chapter Outline

Summary

This chapter discusses transformations of figures in the two-dimensional space. It begins with an explanation of reflection and rotation symmetry. The chapter then branches out to discuss the different types of rigid transformations: translation (sliding a figure to a new position), rotation (rotating a figure with respect to an axis), and reflection (flipping a figure along a line of symmetry). Once the different types of basic transformations are discussed, the composition of these actions to create a new type of transformation is explored. The chapter wraps up with a detailed presentation of tessellations.

Chapter Keywords

• Line of Symmetry
• Line Symmetry
• Rotational Symmetry
• Center of Rotation
• angle of rotation
• Transformation
• Rigid Transformation
• Translation
• Vector
• Reflection
• Line of Reflection
• Reflection over the \begin{align*}y-\end{align*}axis
• Reflection over the \begin{align*}x-\end{align*}axis
• Reflection over \begin{align*}x = a\end{align*}
• Reflection over \begin{align*}y = b\end{align*}
• Reflection over \begin{align*}y = x\end{align*}
• Reflection over \begin{align*}y = -x\end{align*}
• Rotation
• Center of Rotation
• Rotation of \begin{align*}180^\circ\end{align*}
• Rotation of \begin{align*}90^\circ\end{align*}
• Rotation of \begin{align*}270^\circ\end{align*}
• Composition (of transformations)
• Glide Reflection
• Reflections over Parallel Lines Theorem
• Reflection over the Axes Theorem
• Reflection over Intersecting Lines Theorem
• Tessellation

Chapter Review

Match the description with its rule.

1. Reflection over the \begin{align*}y-\end{align*}axis - A. \begin{align*}(2a - x, y)\end{align*}
2. Reflection over the \begin{align*}x-\end{align*}axis - B. \begin{align*}(-y, -x)\end{align*}
3. Reflection over \begin{align*}x = a\end{align*} - C. \begin{align*}(-x, y)\end{align*}
4. Reflection over \begin{align*}y = b\end{align*} - D. \begin{align*}(-y, x)\end{align*}
5. Reflection over \begin{align*}y = x\end{align*} - E. \begin{align*}(x, -y)\end{align*}
6. Reflection over \begin{align*}y = -x\end{align*} - F. \begin{align*}(x, 2b - y)\end{align*}
7. Rotation of \begin{align*}180^\circ\end{align*} - G. \begin{align*}(x, y)\end{align*}
8. Rotation of \begin{align*}90^\circ\end{align*} - H. \begin{align*}(-x, -y)\end{align*}
9. Rotation of \begin{align*}270^\circ\end{align*} - I. \begin{align*}(y, -x)\end{align*}
10. Rotation of \begin{align*}360^\circ\end{align*} - J. \begin{align*}(y, x)\end{align*}

Texas Instruments Resources

In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9697.

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