Dilation Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Dilation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A transformation that enlarges or shrinks a figure by a scale factor from a center point.

Like zooming in or out on a photo—everything gets bigger or smaller proportionally.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A dilation pushes every point toward or away from a fixed center, multiplying its distance by the scale factor $k$ .

Common stuck point: The procedure for dilation is the easy part; the trap is adding $k$ to coordinates instead of multiplying. Asking "Is the image the same shape but a scaled size, made by multiplying distances from a center by $k$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the image the same shape but a scaled size, made by multiplying distances from a center by $k$ ?

Worked Examples

Example 1

easy

Triangle

ABC

has vertices

A(2, 4)

B(6, 0)

C(4, 8)

. Apply a dilation from the origin with scale factor

k = 3

. Find the image vertices

A'

B'

C'

Answer

A'(6, 12)

B'(18, 0)

C'(12, 24)

First step

Step 1: Recall the dilation rule from the origin:

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

where

k

is the scale factor.

Full solution

2
Step 2: Apply to $A(2, 4)$ : $A' = (3 \cdot 2,\, 3 \cdot 4) = (6, 12)$ .
3
Step 3: Apply to $B(6, 0)$ : $B' = (3 \cdot 6,\, 3 \cdot 0) = (18, 0)$ .
4
Step 4: Apply to $C(4, 8)$ : $C' = (3 \cdot 4,\, 3 \cdot 8) = (12, 24)$ .

Dilation from the origin multiplies every coordinate by the scale factor. With

k=3

the triangle is enlarged to three times its original size, keeping the same shape and orientation relative to the origin.

Example 2

medium

Point

P(8, 12)

is dilated from the origin with scale factor

k = \dfrac{1}{4}

. Find the image

P'

and compare the distance from the origin to

P'

vs. to

P

Example 3

easy

A triangle with side lengths

3,\ 4,\ 5

is dilated by factor

4

. Find its new side lengths.

Example 4

hard

Triangle

ABC

with

A(1,1),\ B(4,1),\ C(1,5)

is dilated by factor

-1

from the origin. Find the image vertices.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Square

ABCD

has vertices

A(1,1)

B(3,1)

C(3,3)

D(1,3)

. After a dilation from the origin with scale factor

k = 2

, find the new vertices and the new side length.

Example 2

hard

After a dilation from the origin, point

Q(3, 5)

maps to

Q'(7.5, 12.5)

. Find the scale factor

k

. Then determine by what factor the area of any figure changes under this dilation.

Example 3

easy

Dilate the point

(3, 2)

by scale factor 2 from the origin.

Example 4

easy

Dilate

(4, 6)

by scale factor

\tfrac{1}{2}

from the origin.

Example 5

easy

Is a dilated image congruent or similar to the original (factor not 1)?

Example 6

easy

What stays fixed during a dilation?

Example 7

easy

A dilation has scale factor 3. By what factor do lengths change?

Example 8

easy

A dilation has scale factor 3. By what factor does area change?

Example 9

easy

Does dilation preserve angle measures?

Example 10

easy

Zooming in on a photo is an example of which transformation?

Example 11

medium

Triangle vertices

(1,1)

(2,3)

(4,1)

are dilated by factor 2 from the origin. Find the images.

Example 12

medium

A segment of length 5 is dilated by factor 4. Find the image's length.

Example 13

medium

A figure with area 12 is dilated by factor 3. Find the image's area.

Example 14

medium

A dilation maps

(2, 4)

(6, 12)

from the origin. Find the scale factor.

Example 15

medium

Why must you always state the center of dilation, not just the scale factor?

Example 16

medium

A dilation has scale factor 2 from the origin. How does the perimeter of a figure change?

Example 17

medium

Dilate

(5, 3)

by factor 2 from the center

(1, 1)

Example 18

medium

Does a dilation preserve orientation (for a positive scale factor)?

Example 19

challenge

A dilation by factor

-2

from the origin is applied to

(3, 1)

. Find the image and describe the effect of the negative sign.

Example 20

challenge

Two similar figures are related by a dilation. The smaller has area 20; the scale factor to the larger is 2.5. Find the larger's area.

Example 21

challenge

A dilation from the origin maps

(2, 5)

(2, 5)

— itself. What scale factor and what does this imply about the point?

Example 22

challenge

Explain why a dilation (factor not 1) produces a similar but never congruent figure, and what single property makes it 'not rigid'.

Example 23

easy

Dilate the point

(2, 7)

by scale factor

3

from the origin.

Example 24

easy

Dilate the point

(10, -6)

by scale factor

\tfrac{1}{2}

from the origin.

Example 25

easy

A dilation centered at the origin maps

(4, 8)

(1, 2)

. What is the scale factor?

Example 26

easy

What is the image of the origin under any dilation centered at the origin?

Example 27

medium

Dilate

(3, 4)

by factor

2

from center

(1, 1)

Example 28

medium

A rectangle has perimeter

30

. After a dilation by factor

\tfrac{2}{3}

, what is the new perimeter?

Example 29

medium

A figure with area

50

is dilated. The image has area

200

. What was the scale factor?

Example 30

medium

A solid has volume

24\,\text{cm}^3

and is dilated by factor

3

. Find the new volume.

Example 31

medium

Are dilation images always similar to the original?

Example 32

medium

A triangle with vertices

(0,0),\ (2,0),\ (0,3)

is dilated by factor

4

from the origin. Find the image area.

Example 33

medium

After a dilation centered at

(2, 2)

, point

(4, 6)

maps to

(8, 14)

. Find the scale factor.

Example 34

medium

A rectangle has dimensions

4 \times 7

. After a dilation by factor

\tfrac{3}{2}

, find its new dimensions and area.

Example 35

hard

If a dilation centered at

(3, 1)

has scale factor

\tfrac{1}{2}

, find the image of

(7, 9)

Example 36

hard

Two similar triangles have side ratio

2:5

. What is the ratio of their areas?

Example 37

hard

After dilation from the origin, segment

\overline{AB}

with

A(2,1)

and

B(6,4)

has image of length

25

. Find the scale factor.

Example 38

hard

A circle has radius

r

and is dilated by factor

k

. Find the area ratio of image to original.

Example 39

hard

A scale model of a building uses scale factor

\tfrac{1}{100}

. If the building has volume

V

, what is the model's volume?

Example 40

hard

Why is the center of dilation the only fixed point (assuming

k \ne 1

Example 41

hard

Apply two dilations centered at the origin with factors

2

and

3

. What single dilation is equivalent?

Example 42

challenge

A photograph

10\text{ cm} \times 15\text{ cm}

is enlarged so its area becomes

600\,\text{cm}^2

. Find the scale factor.

Example 43

challenge

Does a dilation centered at a non-origin point preserve the origin?

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Related Concepts

Geometric Transformation Similarity Scaling in Space Proportional Geometry

Background Knowledge

These ideas may be useful before you work through the harder examples.

transformation geo