Title: READ: Rotations and Dilations Reading
Course: Geometry (B)

Rotations

Learning Objectives

- Find the image of a point in a rotation in a coordinate plane.
- Recognize that a rotation is an isometry.

Sample Rotations

In this lesson we limit our study to rotations centered at the origin of a coordinate plane. We begin with some specific examples of rotations. Later we’ll see how these rotations fit into a general formula.

Remember how a rotation is defined. In a rotation centered at the origin with an angle of rotation of a point moves counterclockwise along an arc of a circle. The central angle of the circle measures The original point is one endpoint of the arc, and the image of the original point is the other endpoint of the arc.

180° Rotation

Our first example is rotation through an angle of .

In a rotation, the image of is the point .

Notice:

- and are the endpoints of a diameter.
- The rotation is the same as a “reflection in the origin.”

A rotation is an isometry. The image of a segment is a congruent segment.

If is a polygon matrix, then the matrix for the image of the polygon in a rotation is the product . The Lesson Exercises include exploration of this matrix for a rotation.

90° Rotation

The next example is a rotation through an angle of .

In a rotation, the image of is the point .

Notice:

- and are radii of the same circle, so .
- is a right angle.
- The acute angle formed by and the axis and the acute angle formed by and the axis are complementary angles.

A rotation is an isometry. The image of a segment is a congruent segment.

Dilations

Learning Objectives

- Use the language of dilations.
- Calculate and apply scalar products.
- Use scalar products to represent dilations.

Introduction

We begin the lesson with a review of dilations, which were introduced in an earlier chapter. Like the other transformations, dilations can be expressed using matrices. Before we can do that, though, you will learn about a second kind of multiplication with matrices called scalar multiplication.

Dilation Refresher

The image of point in a dilation centered at the origin, with a scale factor , is the point .

For , the dilation is an enlargement.

For , the dilation is a reduction.

Any linear feature of an image is times as long as the length in the original figure.

Areas in the image are times the corresponding area in the original figure.

Scalar Products for Dilations

Recall from the Dilation Refresher above:

The image of point in a dilation centered at the origin, with a scale factor , is the point .

This is exactly the tool we need in order to use matrices for dilations.

Example 3

The following rectangle is dilated with a scale factor of .

Compositions with Dilations

Dilations can be one of the transformations in a composition, just as translations, reflections, and rotations can.

Example 4

We will use two transformations to move the circle below.

First we will dilate the circle with scale factor . Then, we will translate the new image right and up.

We can call this a translation-dilation.

a) What are the coordinates of the center of the final image circle?

b) What is the radius of the final image?

c) What is the circumference of the final image?

d) What is the area of the original circle?

e) What is the area of the final image circle?