Skill: Determine scale factors of dilations

A scale factor tells how much every length changes in a dilation.

Use the formula

$$\text{scale factor}=\frac{\text{image length}}{\text{preimage length}}.$$

The preimage is the original figure, and the image is the dilated figure.

• If the scale factor is greater than $1$, the figure is enlarged.
• If the scale factor is between $0$ and $1$, the figure is reduced.

Example: A side of a triangle is $6$ centimeters in the preimage and $9$ centimeters in the image. Find the scale factor.

1. Identify the corresponding lengths:

   • Preimage: $6$ cm
   • Image: $9$ cm

2. Divide the image length by the preimage length:

$$k=\frac{9}{6}$$

3. Simplify:

$$k=\frac{3}{2}=1.5$$

The scale factor is

$$\boxed{1.5}$$

This means every side length in the image is $1.5$ times the corresponding side length in the preimage, so the dilation is an enlargement. For example, a preimage side of $8$ cm would become

$$1.5(8)=12\text{ cm}.$$