Dilation Examples in Math

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 kk.

Common stuck point: The procedure for dilation is the easy part; the trap is adding kk to coordinates instead of multiplying. Asking "Is the image the same shape but a scaled size, made by multiplying distances from a center by kk?" 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 kk?

Worked Examples

Example 1

easy
Triangle ABCABC has vertices A(2,4)A(2, 4), B(6,0)B(6, 0), C(4,8)C(4, 8). Apply a dilation from the origin with scale factor k=3k = 3. Find the image vertices A′A', B′B', C′C'.

Answer

A′(6,12)A'(6, 12), B′(18,0)B'(18, 0), C′(12,24)C'(12, 24)

First step

1
Step 1: Recall the dilation rule from the origin: (x,y)→(kx,ky)(x, y) \to (kx, ky) where kk is the scale factor.

Full solution

  1. 2
    Step 2: Apply to A(2,4)A(2, 4): A′=(3⋅2, 3⋅4)=(6,12)A' = (3 \cdot 2,\, 3 \cdot 4) = (6, 12).
  2. 3
    Step 3: Apply to B(6,0)B(6, 0): B′=(3⋅6, 3⋅0)=(18,0)B' = (3 \cdot 6,\, 3 \cdot 0) = (18, 0).
  3. 4
    Step 4: Apply to C(4,8)C(4, 8): C′=(3⋅4, 3⋅8)=(12,24)C' = (3 \cdot 4,\, 3 \cdot 8) = (12, 24).
Dilation from the origin multiplies every coordinate by the scale factor. With k=3k=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)P(8, 12) is dilated from the origin with scale factor k=14k = \dfrac{1}{4}. Find the image P′P' and compare the distance from the origin to P′P' vs. to PP.

Example 3

easy
A triangle with side lengths 3, 4, 53,\ 4,\ 5 is dilated by factor 44. Find its new side lengths.

Example 4

hard
Triangle ABCABC with A(1,1), B(4,1), C(1,5)A(1,1),\ B(4,1),\ C(1,5) is dilated by factor −1-1 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 ABCDABCD has vertices A(1,1)A(1,1), B(3,1)B(3,1), C(3,3)C(3,3), D(1,3)D(1,3). After a dilation from the origin with scale factor k=2k = 2, find the new vertices and the new side length.

Example 2

hard
After a dilation from the origin, point Q(3,5)Q(3, 5) maps to Q′(7.5,12.5)Q'(7.5, 12.5). Find the scale factor kk. Then determine by what factor the area of any figure changes under this dilation.

Example 3

easy
Dilate the point (3,2)(3, 2) by scale factor 2 from the origin.

Example 4

easy
Dilate (4,6)(4, 6) by scale factor 12\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)(1,1), (2,3)(2,3), (4,1)(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)(2, 4) to (6,12)(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)(5, 3) by factor 2 from the center (1,1)(1, 1).

Example 18

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

Example 19

challenge
A dilation by factor −2-2 from the origin is applied to (3,1)(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) to (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)(2, 7) by scale factor 33 from the origin.

Example 24

easy
Dilate the point (10,−6)(10, -6) by scale factor 12\tfrac{1}{2} from the origin.

Example 25

easy
A dilation centered at the origin maps (4,8)(4, 8) to (1,2)(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)(3, 4) by factor 22 from center (1,1)(1, 1).

Example 28

medium
A rectangle has perimeter 3030. After a dilation by factor 23\tfrac{2}{3}, what is the new perimeter?

Example 29

medium
A figure with area 5050 is dilated. The image has area 200200. What was the scale factor?

Example 30

medium
A solid has volume 24 cm324\,\text{cm}^3 and is dilated by factor 33. 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)(0,0),\ (2,0),\ (0,3) is dilated by factor 44 from the origin. Find the image area.

Example 33

medium
After a dilation centered at (2,2)(2, 2), point (4,6)(4, 6) maps to (8,14)(8, 14). Find the scale factor.

Example 34

medium
A rectangle has dimensions 4×74 \times 7. After a dilation by factor 32\tfrac{3}{2}, find its new dimensions and area.

Example 35

hard
If a dilation centered at (3,1)(3, 1) has scale factor 12\tfrac{1}{2}, find the image of (7,9)(7, 9).

Example 36

hard
Two similar triangles have side ratio 2:52:5. What is the ratio of their areas?

Example 37

hard
After dilation from the origin, segment AB‾\overline{AB} with A(2,1)A(2,1) and B(6,4)B(6,4) has image of length 2525. Find the scale factor.

Example 38

hard
A circle has radius rr and is dilated by factor kk. Find the area ratio of image to original.

Example 39

hard
A scale model of a building uses scale factor 1100\tfrac{1}{100}. If the building has volume VV, what is the model's volume?

Example 40

hard
Why is the center of dilation the only fixed point (assuming k≠1k \ne 1)?

Example 41

hard
Apply two dilations centered at the origin with factors 22 and 33. What single dilation is equivalent?

Example 42

challenge
A photograph 10 cm×15 cm10\text{ cm} \times 15\text{ cm} is enlarged so its area becomes 600 cm2600\,\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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Background Knowledge

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

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