Dilations
G.SRT.1.a A dilation takes a line not passing through the center of the dilation to a parallel line.
Goals for this section:
Students will learn how to apply dilations and will know that a dilation is simply making an image smaller or larger. This lesson will also introduce ratios which will help prepare them for the subject of similarity in geometric figures.
What is a dilation?
Dilation in math is changing the size of an object by a scale factor.
Think of your pupils in the eyes. We say your pupils "dilate" when they get larger.
We use special notation for dilations
- D stands for dilation.
- n stands for the scale factor.
- C stands for the center of the dilation. Sometimes C is omitted which implies that the center is the origin.
What is a scale factor?
A scale factor is a number that is multiplied to a preimage.
- A scale factor larger than 1 will make the preimage larger.
- A scale factor that is a fraction will make the preimage smaller.
For this lesson, we will not consider negative scale factors.
External Resource
This video shows the basic concept of a dilation and how to recognize if a figure is a dilation of another.
Finding the scale factor
The following example will show you how to find the scale factor given two images. We will also introduce the concept of ratios.
Step 1
Identify your preimage and image.
Step 2
Identify the corresponding parts from the preimage to the image.
Step 3
Create a ratio with the image part in the numerator and the preimage part in the denominator.
Step 4
Simplify ratio if possible. This is your scale factor!
External Resource
This video shows how to find the scale factor given two objects that are similar.
Dilation about the origin
Dilation about the origin means to do the dilation with respect to the origin. Simply, this means to multiply the scale factor directly to the preimage. In this case we will have a preimage on a coordinate plane and apply a dilation to it.
Dilation about a given center
Dilations about a given center means that the preimage will be dilated in relation to the center. The strategy is to dilate the distances from the center.
Step 1
Pick one point and find the horizontal (x-axis) and vertical (y-axis) distances from the center. It will always be the point minus the center.
Step 2
Multiply the scale factor to the distances.
Step 3
Add the distances to the center point to get your new transformed coordinate.
