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: Dilation changes size but preserves shape and angle measures.

Common stuck point: Scale factor > 1 enlarges, 0 < \text{scale} < 1 shrinks, negative reflects.

Sense of Study hint: Draw a line from the center of dilation through each vertex. Multiply each distance by the scale factor to find the new point.

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'.

Solution

  1. 1
    Step 1: Recall the dilation rule from the origin: (x, y) \to (kx, ky) where k is the scale factor.
  2. 2
    Step 2: Apply to A(2, 4): A' = (3 \cdot 2,\, 3 \cdot 4) = (6, 12).
  3. 3
    Step 3: Apply to B(6, 0): B' = (3 \cdot 6,\, 3 \cdot 0) = (18, 0).
  4. 4
    Step 4: Apply to C(4, 8): C' = (3 \cdot 4,\, 3 \cdot 8) = (12, 24).

Answer

A'(6, 12), B'(18, 0), C'(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.

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.
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Background Knowledge

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

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