# Chapter 12: Rigid Transformations

**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

- 12.1. Reflection Symmetry
- 12.2. Rotation Symmetry
- 12.3. Geometric Translations
- 12.4. Rotations
- 12.5. Reflections
- 12.6. Composition of Transformations
- 12.7. Tessellations

### Chapter Summary

## 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
y− axis - Reflection over the
x− axis - Reflection over
x=a - Reflection over
y=b - Reflection over
y=x - Reflection over
y=−x - Rotation
- Center of Rotation
- Rotation of
180∘ - Rotation of
90∘ - Rotation of
270∘ - 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.

- Reflection over the
y− axis - A.(2a−x,y) - Reflection over the
x− axis - B.(−y,−x) - Reflection over
x=a - C.(−x,y) - Reflection over
y=b - D.(−y,x) - Reflection over
y=x - E.(x,−y) - Reflection over
y=−x - F.(x,2b−y) - Rotation of
180∘ - G.(x,y) - Rotation of
90∘ - H.(−x,−y) - Rotation of
270∘ - I.(y,−x) - Rotation of
360∘ - J.(y,x)

### 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.*