Lesson 6: Dilations and Scale Factors

Property

A dilation is given by a point $C$, the center of the dilation, and a positive number $r$, the factor of the dilation. The dilation moves each point $P$ to a point $P'$ on the ray $CP$ so that the ratio of the length of the image to the length of the original is $r$: $|CP'|/|CP| = r$.

• If $r > 1$, the figure expands.
• If $r < 1$, the figure contracts.
• If $r = 1$, the figure is unchanged.
• Lengths are multiplied by the scale factor $r$, while areas are multiplied by $r^2$.

Examples

• Dilating the point $(4, 8)$ from the origin by a factor of $r=3$ results in the new point $(4 \times 3, 8 \times 3) = (12, 24)$.
• A rectangle with sides of length 6 and 10 is dilated by a factor of $r=0.5$. The new side lengths are 3 and 5, and the new area is $15$, which is $60 \times (0.5)^2$.
• A line segment from $(1, 2)$ to $(4, 2)$ has length 3. After a dilation with factor $r=4$ from the origin, the new segment is from $(4, 8)$ to $(16, 8)$ and has length 12, which is $3 \times 4$.

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).