# 12.4: Rotations

## Learning Objectives

- Find the image of a figure in a rotation in a coordinate plane.

## Review Queue

- Reflect with vertices and over the axis. What are the vertices of ?
- Reflect over the axis. What are the vertices of ?
- How do the coordinates of relate to ?

**Know What?** The international symbol for recycling is to the right. It is three arrows rotated around a point. Let’s assume that the arrow on the top is the preimage and the other two are its images. Find the center of rotation and the angle of rotation for each image.

## Defining Rotations

**Rotation:** A transformation where a figure is turned around a fixed point to create an image.

The lines drawn from the preimage to the ** center of rotation**, and from the center of rotation to the image form the

**. In this section, we will only do**

*angle of rotation***.**

*counterclockwise rotations*

**Example 1:** A rotation of clockwise is the same as what counterclockwise rotation?

**Solution:** There are around a point. So, an rotation clockwise is the same as a rotation counterclockwise.

**Example 2:** A rotation of counterclockwise is the same as what clockwise rotation?

**Solution:** clockwise rotation.

**Investigation 12-1: Drawing a Rotation of **

Tools Needed: pencil, paper, protractor, ruler

- Draw and a point .
- Draw .
- Place the center of a protractor on and the line on . Mark a angle.
- Mark on the line so .
- Repeat steps 2-4 with and .
- Make .

Use this process to rotate any figure.

**Example 3:** Rotate rectangle counterclockwise around .

**Solution:** Use Investigation 12-1. In step 3, change the angle to . Each angle of rotation is .

** Rotation**

To rotate a figure , in the plane, we ** use the origin as the center of the rotation**. A angle is called a straight angle. So, an image rotated over the origin will be on the same line and the same distance away from the origin as the preimage, but on the other side.

**Example 4:** Rotate , with vertices , and , . Find the coordinates of .

**Solution:** You can either use Investigation 12-1 or the hint given above to find . First, graph the triangle. If is (7, 4), that means it is 7 units to the ** right** of the origin and 4 units

**. would then be 7 units to the**

*up***of the origin and 4 units**

*left***.**

*down*
**Rotation of :**

Recall from the second section that ** a rotation is an isometry**. This means that . You can use the distance formula to show this.

** Rotation**

Similar to the rotation, the image of a will be the same distance away from the origin as its preimage, but rotated .

**Example 5:** Rotate .

**Solution:** When rotating something , use Investigation 12-1 to see if there is a pattern.

**Rotation of :**

**Rotation of **

A rotation of counterclockwise would be the same as a rotation of *plus* a rotation of . So, if the values of a rotation are , then a rotation would be the opposite sign of each, or .

**Rotation of :**

**Example 6:** Find the coordinates of after a rotation.

**Solution:** Using the rule, we have:

While we can rotate any image any amount of degrees, only and have special rules. To rotate a figure by an angle measure other than these three, you must use Investigation 12-1.

**Example 7:** ** Algebra Connection** The rotation of a quadrilateral is shown below. What is the measure of and ?

**Solution:** Because a rotation is an isometry, we can set up two equations to solve for and .

**Know What? Revisited** The center of rotation is shown in the picture to the right. If we draw rays to the same place in each arrow, the two images are a rotation in either direction.

## Review Questions

- Questions 1-10 are similar to Examples 1 and 2.
- Questions 11-16 are similar to Investigation 12-1 and Example 3.
- Questions 17-25 are similar to Examples 4-6.
- Questions 26-28 are similar to Example 7.
- Questions 29-34 are similar to Examples 4-6.
- Questions 34-37 are a review.
- Question 38 is similar to Example 4.

In the questions below, every rotation is *counterclockwise*, unless otherwise stated.

- If you rotated the letter counterclockwise, what letter would you have?
- If you rotated the letter
*clockwise*, what letter would you have? - A clockwise rotation is the same as what counterclockwise rotation?
- A clockwise rotation is the same as what counterclockwise rotation?
- A counterclockwise rotation is the same as what clockwise rotation?
- A counterclockwise rotation is the same as what clockwise rotation?
- A counterclockwise rotation is the same as what clockwise rotation?
- Rotating a figure is the same as what other rotation?
- Does it matter if you rotate a figure clockwise or counterclockwise? Why or why not?
- When drawing a rotated figure and using your protractor, would it be easier to rotate the figure counterclockwise or clockwise? Explain your reasoning.

Using Investigation 12-1, rotate each figure around point the given angle measure.

Rotate each figure in the coordinate plane the given angle measure. The center of rotation is the origin.

** Algebra Connection** Find the measure of in the rotations below. The blue figure is the preimage.

Find the angle of rotation for the graphs below. The center of rotation is the origin and the blue figure is the preimage. Your answer will be , or .

** Two Reflections** The vertices of are , and . Use this information to answer questions 24-27.

- Plot on the coordinate plane.
- Reflect over the axis. Find the coordinates of .
- Reflect over the axis. Find the coordinates of .
- What
transformation would be the same as this double reflection?*one*

## Review Queue Answers

- is the double negative of