Dilations are transformations that change figures in specific ways. Learn about these changes and how to complete dilations here. Then test your understanding with a quiz.

## What Is Dilation?

Have you ever gone to the movies and wondered where the movie was being projected from? How did the person in the back office get the film to fit perfectly on the huge screen? Well, if you didn’t already know, the process used to get films to fit a movie screen is a perfect example of how dilations are used in real life.

A **dilation** is a transformation that changes the size of a figure. It can become larger or smaller, but the shape of the figure does not change. To complete a dilation, two things are needed. The first is a center point (or fixed point), which is usually only mentioned when the dilation must be drawn.

The second is a **scale factor** or ratio, which is often represented by the variable ** r**. Here, you can see how this relates to the everyday movie experience. The light beam would be the center point, the film strip would be the pre-image, and the movie displayed on the big screen would be the image.

When completing dilations, we often use the terminology from the above example. The original figure is referred to as the pre-image and the newly dilated figure, denoted with prime marks, is called the image. In this example, our pre-image is triangle ABC, and it is dilated to produce the image of triangle A’B’C’.

Knowing the scale factor allows you to predict what the image will look like after the dilation. If the absolute value of the scale factor is less than 1, then the image will be smaller than the pre-image. If the absolute value of the scale factor is greater than 1, then the image will be larger than the pre-image. Additionally, a negative scale factor causes the dilation to rotate 180 degrees.

Dilations can occur both on a coordinate plane and not on a coordinate plane. Let’s take a look at each.

## Dilations Not on a Coordinate Plane

Without a coordinate plane, your primary goal will be to calculate the segment length of a dilated image. A general formula to use is ** Image = (Pre-Image)*|Scale Factor|** . In other words, multiply the pre-image by the absolute value of the scale factor. We must use absolute value because lengths should always be positive (you cannot have a negative length or distance).

For our first example, let AB = 8 and let’s dilate it by a scale factor of -2. The absolute value of this scale factor is positive 2, which is larger than 1. Therefore, we can predict that segment A’B’ will be larger than segment AB. Using the formula above, we see that A’B’ = (8)*(2) = 16.

In our second example, we will use the same pre-image, but this time, we will dilate it by a scale factor of .25. The absolute value of this scale factor is .25, which is less than 1. With this, we can predict that segment A’B’ will be smaller than segment AB. Once again, to find the length of A’B’, we will multiply by the absolute value of the scale factor. Therefore, A’B’ = (8)*(.25) = 2.

## Dilations on a Coordinate Plane

Now, when completing dilations on a coordinate plane, your primary goal will be to find the coordinates of the image. Since we are not finding the length of segments, we do not need to multiply by the absolute value of the scale factor. Instead, just multiply each coordinate by the scale factor. However, we must still use the absolute value to predict whether the image will increase or decrease in size. For the examples in this lesson, we will use the origin (0, 0) as our center point. Let’s do an example.

Let D (2, -2) and E (6, -4) be the coordinates for our pre-image and let’s dilate segment DE with a scale factor of .5. To complete this, we will multiply each coordinate by .5. In doing so, we see that the coordinates for our image are D’ (1, -1) and E’ (3, -2). We can also see that our image is smaller than our pre-image, since the absolute value of our scale factor is less than 1.

For our second example, let’s use the same pre-image and dilate it with a scale factor of -2. Since this scale factor is negative, our image will be a 180-degree rotation of the pre-image. Because absolute values are always positive, the absolute value of this scale factor is positive 2, causing our image to be twice as large as the pre-image. To determine the new coordinates, we will multiply points D and E by -2 to get D’ (-4, 4) and E’ (-12, 8), like you see here.

## Lesson Summary

**Dilations** are transformations that change the size of the figure. The scale factor will help you determine whether the image will be smaller or larger than the pre-image. To complete dilations, multiply the coordinates by the scale factor or multiply the side lengths by the absolute value of the scale factor.

## Points of Emphasis

- Dilation: a transformation that changes the size of a figure
- Scale factor: a ratio
- General formula to use to calculate the segment length of a dilated image: Image = (Pre-Image)*|Scale Factor|

## Learning Outcomes

Watch and read through the entire lesson so that you can easily:

- Describe the characteristics of a dilation
- Name the two things necessary to complete a dilation
- Understand what your primary goals would be when completing dilations on a coordinate plane or without a coordinate plane