Properties of Dilations: Parallel Lines and Scaled Lengths

Quick answer

A dilation with center C and scale factor k sends each point P to the point on ray CP that is k times as far from C. Two properties follow and can be checked with coordinates: a line not through the center goes to a parallel line, while a line through the center stays where it is, and every segment's length is multiplied by k. Both come from one fact about the step between two points.

What you'll learn

Dilate a point from any center with a given scale factor
Verify that a dilation takes a line to a parallel line or to itself
Verify that a dilation multiplies every length by the scale factor

A dilation from any center

A dilation needs two things: a center $C$ and a scale factor $k$ . It sends each point $P$ to the point $P'$ on ray $CP$ that is $k$ times as far from $C$ .

About the origin that is the familiar rule $(x, y) \to (kx, ky)$ . About any other center, do it in three moves: find the step from $C$ to $P$ , multiply the step by $k$ , and add it back to $C$ .

C = (a, b) \qquad (x, y) \;\longrightarrow\; \bigl(a + k(x - a),\; b + k(y - b)\bigr)

Checking it with a triangle

Dilate triangle $A(2, 2)$ , $B(4, 2)$ , $C(2, 4)$ from center $(1, 1)$ with scale factor $2$ .

A' = (3, 3) \qquad B' = (7, 3) \qquad C' = (3, 7)

Triangle ABC dilated by 2 from center (1, 1)

Measure each side before and after.

Side	Slope	Length	Image side	Slope	Length
$AB$	$0$	$2$	$A'B'$	$0$	$4$
$AC$	undefined	$2$	$A'C'$	undefined	$4$
$BC$	$-1$	$2\sqrt{2}$	$B'C'$	$-1$	$4\sqrt{2}$

Two patterns stand out. Every slope is unchanged, so each side is parallel to its image. Every length is doubled, matching the scale factor.

Why the image of a line is parallel

Take any two points $A$ and $B$ . The dilation multiplies each one’s step from the center by $k$ :

A' - C = k(A - C) \qquad B' - C = k(B - C)

Subtract the first from the second. The center cancels:

B' - A' = k(B - A)

So the step from $A'$ to $B'$ is $k$ times the step from $A$ to $B$ . Both the run and the rise are multiplied by $k$ .

That one fact gives both properties.

The slope is kept. A slope is rise over run, and multiplying both by $k$ leaves the fraction unchanged:

\frac{k \cdot \text{rise}}{k \cdot \text{run}} = \frac{\text{rise}}{\text{run}}

The image line has the same slope, so it is parallel to the original — or it is the same line.

The length is scaled. Multiplying the run and rise by $k$ multiplies the distance $\sqrt{\text{run}^2 + \text{rise}^2}$ by $k$ . Every segment’s image is exactly $k$ times as long.

Lines through the center stay put

A line through the center is the exception to “parallel”: it lands on itself.

Each point $P$ on such a line moves along ray $CP$ , and that ray lies on the line. The point slides along the line it started on, never off it.

Line	Image
does not pass through the center	a different line, parallel to it
passes through the center	the same line

Scale factors below 1

A scale factor between $0$ and $1$ shrinks. With $k = \tfrac{1}{2}$ , every point moves halfway toward the center and every length halves. The parallel property holds for every $k$ other than $0$ , because the argument never used the size of $k$ .

Worked examples

Common mistakes

Practice problems

Dilate $(4, 6)$ about the origin with $k = \tfrac{1}{2}$ .

Answer

$(2, 3)$

Full solution

About the origin, multiply each coordinate by $\tfrac{1}{2}$ .
Dilate $(5, 3)$ from center $(1, 1)$ with $k = 3$ .

Answer

$(13, 7)$

Full solution

The step is $4$ right and $2$ up. Tripled: $12$ right and $6$ up. Adding to $(1, 1)$ gives $(13, 7)$ .
A segment of length $6$ is dilated with $k = 2.5$ . How long is the image?

Answer

$15$

Full solution

Every length is multiplied by $k$ : $6 \times 2.5 = 15$ .
A segment not through the center has slope $\tfrac{3}{4}$ . After a dilation with $k = 5$ , what is the slope of its image?

Answer

$\tfrac{3}{4}$

Full solution

Rise and run are both multiplied by $5$ , which leaves their ratio at $\tfrac{3}{4}$ . The image is parallel to the original.
A line passes through the center of a dilation. Where is its image?

Answer

On the same line.

Full solution

Each point on the line moves along its ray from the center, and that ray lies on the line.
Dilate $A(0, 4)$ and $B(4, 0)$ about the origin with $k = 3$ . Check that $A'B'$ is parallel to $AB$ and three times as long.

Answer

$A'(0, 12)$ and $B'(12, 0)$ . Both slopes are $-1$ , and the lengths are $4\sqrt{2}$ and $12\sqrt{2}$ .

Full solution

Slopes: $\tfrac{0 - 4}{4 - 0} = -1$ and $\tfrac{0 - 12}{12 - 0} = -1$ . Parallel ✓

Lengths: $\sqrt{16 + 16} = 4\sqrt{2}$ and $\sqrt{144 + 144} = 12\sqrt{2}$ , three times as long ✓
A segment $12$ units long is dilated and its image is $4$ units long. Find $k$ . Did the figure grow or shrink?

Answer

$k = \tfrac{1}{3}$ ; it shrank.

Full solution

$k = \tfrac{4}{12} = \tfrac{1}{3}$ . A scale factor between $0$ and $1$ shrinks a figure.
A square of area $9$ is dilated by $k = 2$ . Find the area of the image.

Hint

Find the side length first.

Answer

$36$

Full solution

The side is $3$ , and the image side is $6$ . The area is $6^2 = 36$ , which is $2^2 = 4$ times the original.
Explain why a dilation with $k = 1$ changes nothing.

Answer

Every step from the center is multiplied by $1$ , so every point stays where it is.

Full solution

$P' = C + 1 \cdot (P - C) = C + P - C = P$ . Each point is its own image.
Asked to dilate $(5, 3)$ from center $(1, 1)$ by $2$ , Leo doubles both coordinates and gets $(10, 6)$ . Find his error.

Hint

Where is the center of the dilation he performed?

Answer

Doubling the coordinates dilates about the origin. From $(1, 1)$ the image is $(9, 5)$ .

Full solution

Doubling coordinates doubles each point’s step from $(0, 0)$ , so Leo used the origin as the center.

The center here is $(1, 1)$ . The step from it to $(5, 3)$ is $4$ right and $2$ up; doubled, $8$ right and $4$ up. Adding to $(1, 1)$ gives $(9, 5)$ .

A check with the ray: $(9, 5)$ is on the line from $(1, 1)$ through $(5, 3)$ , since both steps have slope $\tfrac{1}{2}$ , and it is twice as far out. Leo’s $(10, 6)$ gives a step of slope $\tfrac{5}{9}$ , so it is off that ray.

Frequently asked questions

How do I dilate a point from a center that is not the origin?

Find the step from the center to the point, multiply it by k, and add it back to the center. From center (a, b), the point (x, y) goes to (a + k(x − a), b + k(y − b)).

Why is the image of a line parallel to the original?

The step between any two points on the line is multiplied by k. Rise and run are both multiplied by k, so the slope is unchanged.

What happens to a line through the center?

It maps onto itself. Every point on it moves along its own ray from the center, and that ray lies on the same line.

What happens to lengths under a dilation?

Every length is multiplied by the scale factor k. A segment of length 5 dilated by k = 3 has length 15.

Do angles change under a dilation?

No. Parallel lines go to parallel lines and slopes are kept, so every angle keeps its size.

What to learn next

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.HSG.SRT.A.1Similarity, Right Triangles, and TrigonometryVerify experimentally the properties of dilations given by a center and a scale factor:
CCSS.MATH.CONTENT.HSG.SRT.A.1aSimilarity, Right Triangles, and TrigonometryA dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
CCSS.MATH.CONTENT.HSG.SRT.A.1bSimilarity, Right Triangles, and TrigonometryThe dilation of a line segment is longer or shorter in the ratio given by the scale factor.

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