Math Core

When you zoom in on a photo, everything gets bigger, but nothing gets distorted: faces keep their shape and straight lines stay straight. Translations, reflections and rotations never change size. A dilation is the transformation that does.

Scale factor and center

Definition

Dilation

A dilation enlarges or shrinks a figure by a scale factor $k$ from a fixed point called the center of dilation. Every point moves along a line through the center so that its distance from the center is multiplied by $k$ .

The scale factor tells you what happens to the size:

If $k > 1$ , the image is an enlargement (bigger).
If $0 < k < 1$ , the image is a reduction (smaller).
If $k = 1$ , the image is exactly the same as the pre-image.

When the center is the origin, the rule is short:

(x, y) \rightarrow (kx,\ ky)

Multiply both coordinates by the scale factor.

Worked example: An enlargement

Dilate triangle $ABC$ with $A(1, 1)$ , $B(3, 1)$ and $C(1, 2)$ by a scale factor of $2$ , centered at the origin.

\begin{aligned} A(1, 1) &\rightarrow A'(2, 2) \\ B(3, 1) &\rightarrow B'(6, 2) \\ C(1, 2) &\rightarrow C'(2, 4) \end{aligned}

Triangle ABC dilated by a scale factor of 2 about the origin.Open in grapher →

Side $AB$ is 2 units long, and $A'B'$ is 4 units long, twice as long. Side $AC$ is 1 unit and $A'C'$ is 2 units. Every length doubled, but the triangle has the same shape.

Worked example: A reduction

Dilate triangle $DEF$ with $D(-4, 4)$ , $E(4, 4)$ and $F(0, -2)$ by a scale factor of $\dfrac{1}{2}$ , centered at the origin.

Multiply each coordinate by $\dfrac{1}{2}$ (that is, divide by 2): $D'(-2, 2)$ , $E'(2, 2)$ and $F'(0, -1)$ .

Triangle DEF dilated by a scale factor of 1/2 about the origin.Open in grapher →

The image is half as wide and half as tall, and it sits closer to the origin.

Common mistake

A dilation multiplies; it doesn't add. Dilating $(3, 4)$ by a scale factor of 2 gives $(6, 8)$ , not $(5, 6)$ . Adding 2 to each coordinate would be a translation.

What changes and what doesn't

A dilation is not a rigid motion, because lengths change. But the shape is kept.

Dilations

Angle measures stay the same.
Every length is multiplied by $k$ . So $\dfrac{\text{image length}}{\text{pre-image length}} = k$ for every pair of matching sides.
Each side of the image is parallel to the matching side of the pre-image (or on the same line).

Finding the scale factor

Because every length is multiplied by $k$ , you can find $k$ by dividing an image measurement by the matching pre-image measurement.

Worked example: Working out k

A dilation centered at the origin sends $A(2, -3)$ to $A'(5, -7.5)$ . What is the scale factor? If a side of the pre-image is 8 units long, how long is the matching side of the image?

Divide matching coordinates: $5 \div 2 = 2.5$ and $-7.5 \div (-3) = 2.5$ . The scale factor is $k = 2.5$ . Since $k > 1$ , it's an enlargement.

The side of length 8 becomes $8 \times 2.5 = 20$ units long.

Tip

Check whether you expect the image to be bigger or smaller before you calculate. If $k$ is less than 1 but your image came out bigger, something went wrong.

Practice

Practice 1

Dilate the point $(3, -5)$ by a scale factor of $4$ , centered at the origin. What is the image?

Enter a point like (2, -3)

Practice 2

Dilate the point $(-8, 6)$ by a scale factor of $\dfrac{1}{2}$ , centered at the origin. What is the image?

Enter a point like (2, -3)

Practice 3

A dilation centered at the origin sends $A(4, 10)$ to $A'(6, 15)$ . What is the scale factor?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

A figure is dilated by a scale factor of $0.6$ . Which describes the image?

Practice 5

A triangle has a side that is 9 cm long. The triangle is dilated by a scale factor of $\dfrac{2}{3}$ . How long is the matching side of the image, in centimeters?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

After a dilation with scale factor $3.5$ , one side of the image is 21 inches long. How long was the matching side of the pre-image, in inches?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

A quadrilateral is dilated by a scale factor of 3. Which of these is the same in the image as in the pre-image?

Practice 8

The point $(6, -9)$ is dilated by a scale factor of $\dfrac{1}{3}$ centered at the origin, and then the image is translated 2 units right. Where does the point end up?

Enter a point like (2, -3)