Section 2.7: Dilations

Property

A dilation is a transformation that changes the size of a figure by a scale factor $k$ with respect to a fixed point called the center of dilation.

All points on the original figure are mapped to corresponding points on the image such that the distance from the center to each image point is $k$ times the distance from the center to the corresponding original point.

Property

If you are looking at a pre-image and its dilated image, you can work backwards to find the exact Center of Dilation. Because dilations expand outward in straight lines, drawing straight lines through corresponding vertices (connecting A to A', B to B', C to C', and extending them) will eventually make all the lines intersect at one single point. That intersection is the Center of Dilation.

Examples

• Finding the Center: You have a small square PQRS and a large square P'Q'R'S'. Place a ruler on point P and point P', draw a long line. Do the same for Q and Q'. The exact spot on the graph where those two lines cross each other is your center of dilation.

Explanation

Think of the Center of Dilation as a flashlight, and the shape as an object casting a shadow. The light rays travel in perfectly straight lines through the corners of the object to create the enlarged shadow. By tracing the lines backwards from the shadow (image) through the object (pre-image), you will always find the flashlight (center). If you draw the lines and they are perfectly parallel and never cross, then the shape wasn't dilated—it was translated!

Property

The scale factor is the ratio of the lengths in the new figure to the corresponding lengths in the original figure.

Examples

• A map has a scale factor of $\frac{1}{10000}$. A road that is 3 cm long on the map is $3 \times 10000 = 30000$ cm, or 300 meters, in real life.
• A triangle with a base of 8 inches is enlarged to a similar triangle with a base of 20 inches. The scale factor is $\frac{20}{8} = 2.5$.
• To reduce a 12-foot wall to fit on a blueprint with a scale factor of $\frac{1}{48}$, its length on the blueprint is $12 \times \frac{1}{48} = \frac{1}{4}$ foot, or 3 inches.

Explanation

The scale factor is the number you multiply by to change the size of a figure. A scale factor greater than 1 makes the figure bigger (an enlargement), while a factor between 0 and 1 makes it smaller (a reduction).