# 12.4: Rotations

## Learning Objectives

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

## 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 appears below. 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 by which a figure is turned around a fixed point to create an image.

**Center of Rotation:** The fixed point that a figure is rotated around.

Lines can be drawn from the preimage to the center of rotation, and from the center of rotation to the image. The angle formed by these lines is the angle of *rotation.*

In this section, our center of rotation will always be the ** origin.** Rotations can also be clockwise or counterclockwise. We will only do

**rotations, to go along with the way the quadrants are numbered.**

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

Tools Needed: pencil, paper, protractor, ruler

- Draw and a point outside the circle.
- Draw the line segment .
- Take your protractor, place the center on and the initial side on . Mark a angle.
- Find such that .
- Repeat steps 2-4 with points and .
- Connect and to form .

This is the process you would follow to rotate any figure counterclockwise. If it was a different angle measure, then in Step 3, you would mark a different angle. You will need to repeat steps 2-4 for every vertex of the shape.

** Rotation**

To rotate a figure in the coordinate plane, we use the origin as the center of the rotation. Recall, that a angle is the same as a straight line. So, a rotation of a point over the origin of will be on the same line and the same distance away from the origin.

**Example 1:** Rotate , with vertices and . Find the coordinates of .

**Solution:** You can either use Investigation 12-1 or the hint given above to find . It is very helpful to graph the triangle. Using the hint, if is , that means it is 7 units to the right of the origin and 4 units up. would then be 7 units to the ** left** of the origin and 4 units

**The vertices are:**

*down.*

The image has vertices that are the negative of the preimage. This will happen every time a figure is rotated .

**Rotation of :** If is rotated around the origin, then the image will be .

From this example, we can also see that ** a rotation is an isometry.** This means that . You can use the distance formula to verify that our assertion holds true.

** Rotation**

Similar to the rotation, a rotation (counterclockwise) is an isometry. Each image will be the same distance away from the origin as its preimage, but rotated .

**Example 2:** Rotate .

**Solution:** When we rotate something , you can use Investigation 12-1. Draw lines from the origin to and . The line from each point to the origin is going to be ** perpendicular** to the line from the origin to its image. Therefore, if is 6 units to the

**of the origin and 1 unit**

*right***will be 6 units**

*down,***and 1 to the**

*up*

*right.*Using this pattern, is (8, 2).

If you were to write the slope of each point to the origin, would be , and must be . Again, they are perpendicular slopes, following along with the rotation. Therefore, the and the values switch and the new value is the opposite sign of the original value.

**Rotation of :** If is rotated around the origin, then the image will be .

**Rotation of **

A rotation of counterclockwise would be the same as a clockwise rotation of . We also know that a rotation and a rotation are apart. We know that for every rotation, the and values are negated. So, if the values of a rotation are , then a rotation would be the opposite sign of each, or .

**Rotation of :** If is rotated around the origin, then the image will be .

**Example 3:** 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 4:** ** 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 below. If we draw rays to the same point in each arrow, we see that the two images are a rotation in either direction.

## Review Questions

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

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

- If you rotated the letter counterclockwise, what letter would you have?
- If you rotated the letter
*clockwise,*what letter would you have? Why do you think that is? - A clockwise rotation is the same as what counterclockwise rotation?
- A clockwise rotation is the same as what counterclockwise rotation?
- Rotating a figure is the same as what other rotation?

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.

** 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*

*Multistep Construction Problem*

- Draw two lines that intersect, and , and . Reflect over line to make . Reflect over line to get . Make sure does not intersect either line.
- Draw segments from the intersection point of lines and to and . Measure the angle between these segments. This is the angle of rotation between and .
- Measure the angle between lines and . Make sure it is the angle which contains in the interior of the angle.
- What is the relationship between the angle of rotation and the angle between the two lines of reflection?

## Review Queue Answers

- is the double negative of