Math Core

Translations, reflections and rotations move a figure without changing its size. A dilation is different: it enlarges or shrinks a figure while keeping its shape. Dilations are the foundation of similarity, which you'll study later in this course, and they explain how maps, scale models and zoomed photos work.

Center and scale factor

Definition

Dilation

A dilation with center $C$ and scale factor $k$ (with $k > 0$ ) maps each point $P$ to a point $P'$ on ray $\overrightarrow{CP}$ so that

CP' = k \cdot CP.

The center maps to itself.

If $k > 1$ , the dilation is an enlargement: points move away from the center.
If $0 < k < 1$ , it's a reduction: points move toward the center.
If $k = 1$ , nothing moves.

Dilating on the coordinate plane

When the center is the origin, multiply both coordinates by $k$ :

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

For any other center, use the same shift-and-return idea as for rotations. Find the point's position relative to the center, scale that by $k$ , and add the center back.

Dilation rules

Center at the origin: $(x, y) \rightarrow (kx,\ ky)$ .
Center at $(a, b)$ : $(x, y) \rightarrow \big(a + k(x - a),\ b + k(y - b)\big)$ .

Worked example: A center that isn't the origin

Dilate triangle $ABC$ with $A(2, 2)$ , $B(4, 2)$ and $C(2, 3)$ by a scale factor of $2$ about the center $P(1, 1)$ .

Find each vertex relative to $P$ , double it, and add $(1, 1)$ back:

vertex	relative to $P$	times $2$	image
$A(2, 2)$	$(1, 1)$	$(2, 2)$	$A'(3, 3)$
$B(4, 2)$	$(3, 1)$	$(6, 2)$	$B'(7, 3)$
$C(2, 3)$	$(1, 2)$	$(2, 4)$	$C'(3, 5)$

Triangle ABC dilated by a scale factor of 2 about P(1, 1).Open in grapher →

Each side doubled: $AB = 2$ and $A'B' = 4$ ; $AC = 1$ and $A'C' = 2$ . The points $P$ , $A$ and $A'$ lie on one line, as the definition requires.

Common mistake

Multiplying the coordinates by $k$ only works when the center is the origin. With the center $P(1, 1)$ above, doubling $A(2, 2)$ to $(4, 4)$ would be wrong. Always measure from the center.

What a dilation preserves

A dilation is not a rigid motion, because it changes lengths. But it keeps the shape.

Properties of dilations

Every length is multiplied by $k$ , so $A'B' = k \cdot AB$ .
Angle measures are preserved, so the image is similar to the pre-image.
A line through the center maps to itself. A line not through the center maps to a parallel line.
Areas are multiplied by $k^2$ (perimeters by $k$ ).

The area fact follows from the length fact. If a rectangle is $\ell$ by $w$ , its image is $k\ell$ by $kw$ , with area $k^2 \ell w$ . Any figure can be approximated by tiny rectangles, so the same factor $k^2$ applies to every area.

Worked example: Dilating a line

Find the image of the line $y = 2x + 3$ under a dilation with center at the origin and scale factor $3$ .

The line doesn't pass through the origin, so its image is a parallel line with slope $2$ . To find it, dilate one point: $(0, 3) \rightarrow (0, 9)$ . The image is $y = 2x + 9$ .

Check with another point: $(1, 5)$ is on the original line and maps to $(3, 15)$ , and $2(3) + 9 = 15$ . ✓

Finding the scale factor and center

Compare any length in the image with the matching length in the pre-image:

k = \frac{\text{image length}}{\text{pre-image length}}.

To find the center, draw lines through matching points, like $P$ and $P'$ , $Q$ and $Q'$ . The center is where they meet. With coordinates, you can solve $P' = C + k(P - C)$ for $C$ .

Worked example: Scale factor and center from coordinates

A dilation maps $P(2, 3)$ to $P'(6, 11)$ and $Q(4, 1)$ to $Q'(10, 7)$ . Find the scale factor and the center.

The segment $\overline{PQ}$ goes $\langle 2, -2 \rangle$ , and $\overline{P'Q'}$ goes $\langle 4, -4 \rangle$ . The image segment is twice as long, so $k = 2$ .

With $k = 2$ , $P' = C + 2(P - C) = 2P - C$ , so $C = 2P - P' = (4 - 6,\ 6 - 11) = (-2, -5)$ .

Check with $Q$ : $2Q - Q' = (8 - 10,\ 2 - 7) = (-2, -5)$ . ✓ The center is $(-2, -5)$ .

Tip

A quick sanity check: for an enlargement the image is farther from the center than the pre-image, and for a reduction it's closer. If a scale factor of $\dfrac{1}{2}$ pushed a point away from the center, something went wrong.

Practice

Practice 1

Dilate the point $(-4, 6)$ by a scale factor of $\dfrac{1}{2}$ with center at the origin. What is the image?

Enter a point like (2, -3)

Practice 2

Dilate the point $(5, 3)$ by a scale factor of $3$ with center $(2, 1)$ . What is the image?

Enter a point like (2, -3)

Practice 3

Dilate the point $(7, -2)$ by a scale factor of $\dfrac{1}{3}$ with center $(1, 4)$ . What is the image?

Enter a point like (2, -3)

Practice 4

Under a dilation, segment $\overline{AB}$ with length $6$ maps to $\overline{A'B'}$ with length $15$ . What is the scale factor?

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

Practice 5

A triangle with area $12$ square units is dilated by a scale factor of $3$ . What is the area of the image, in square units?

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

Practice 6

A dilation with center at the origin maps $(6, -9)$ to $(4, -6)$ . What is the scale factor?

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

Practice 7

The line $y = -x + 4$ is dilated with center at the origin and scale factor $\dfrac{1}{2}$ . Which is an equation of the image?

Practice 8

A dilation maps $P(1, 3)$ to $P'(5, 5)$ and $Q(3, 1)$ to $Q'(9, 1)$ . What is the center of the dilation?

Enter a point like (2, -3)