Finding the scale factor of a dilation involves comparing corresponding measurements of two similar figures (the pre-image and the image). Here’s a step-by-step guide:
Step 1: Identify Corresponding Points
Locate a pair of corresponding points on the pre-image and its dilated image. For example, if you’re working with triangles, pick matching vertices.
Step 2: Measure the Distances
Measure the distance between:
- The center of dilation and a point on the pre-image.
- The center of dilation and the corresponding point on the image.
If the center of dilation is not explicitly given, you may instead compare the lengths of corresponding sides.
Step 3: Calculate the Scale Factor
Use the formula for the scale factor (kk):
k=Distance from the center to a point on the Image distance from the center to the corresponding point on the pre-imagek = \frac{\text{Distance from the center to a point on the image}}{\text{Distance from the center to the corresponding point on the pre-image}}
or
k=Length of a side on the Image length of the corresponding side on the pre-imagek = \frac{\text{Length of a side on the image}}{\text{Length of the corresponding side on the pre-image}}
Step 4: Interpret the Result
- If k>1k > 1, the dilation is an enlargement (the image is larger than the pre-image).
- If 0<k<10 < k < 1, the dilation is a reduction (the image is smaller than the pre-image).
- If k=1k = 1, there is no dilation (the image and pre-image are the same size).
Example
Suppose the distance from the center of dilation to a point on the pre-image is 3 units, and the distance to the corresponding point on the image is 9 units. The scale factor is:
k=93=3k = \frac{9}{3} = 3
This means the dilation is an enlargement with a scale factor of 3.
By following these steps, you can easily find the scale factor for any dilation!