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Dilations &
Scale Factor

Learn how dilations transform figures by a scale factor from a center point: preserving shape while changing size.

Geometry

What Is a Dilation?

A dilation is a transformation that produces an image that is the same shape as the original, but a different size. Every dilation has two defining components:

Center of Dilation
A fixed point from which all distances are measured. The image expands or contracts from this point.
Scale Factor \( k \)
The ratio that determines how much larger or smaller the image becomes relative to the original.

Visualizing a Dilation

O A B C A' B' C'
\( k = 2 \)
 Original Image

Why Dilations Matter

Dilations are the foundation of similarity in geometry. Two figures are similar if and only if one can be obtained from the other by a sequence of dilations and rigid motions.

Architecture
Scale models preserve proportions of a building at a fraction of its size.
Maps
Cartography uses scale factors so distances on paper correspond to real-world distances.
Photography
Zoom lenses create dilated images on the sensor while preserving shape.

The Dilation Formula

The formula subtracts the center coordinates, scales by \( k \), then adds the center back. When the center is at the origin, it simplifies to multiplying each coordinate by \( k \).

Properties Preserved

1
Angle measures: corresponding angles are congruent.
2
Parallelism: a line and its image are parallel (or the same line).
3
Collinearity: points that are collinear remain collinear.
4
Betweenness: if \( B \) is between \( A \) and \( C \), then \( B' \) is between \( A' \) and \( C' \).
5
Proportional side lengths: all side lengths are multiplied by \( |k| \).

Enlargement vs Reduction

Enlargement: \( k = 2 \)
 O
Reduction: \( k = 0.5 \)
 O
\( k > 1 \)
Image is larger than the original: an enlargement.
\( 0 < k < 1 \)
Image is smaller than the original: a reduction.
\( k = 1 \)
Image is congruent to the original: an identity.

Negative Scale Factor

O A B C A' B' C'
\( k = -1 \)

How to Apply a Dilation

1
Identify the center of dilation \( O = (a, b) \) and the scale factor \( k \).
2
Subtract the center from each vertex: compute \( (x - a, \; y - b) \).
3
Multiply each result by \( k \): get \( (k(x-a), \; k(y-b)) \).
4
Add the center back: the image point is \( (a + k(x-a), \; b + k(y-b)) \).
5
Plot and connect the image points. Verify the shape looks proportional.

Example 1: Center at Origin

Dilate triangle with vertices \( A(1,2) \), \( B(4,2) \), \( C(2,5) \) by scale factor \( k = 2 \), center at the origin.

1 2 4 8 2 4 5 6 10 O A B C A' B' C'
\( k = 2 \)
 Original Image

Example 2: Center at \( (2, 1) \)

Dilate point \( P(5, 3) \) with center \( O(2, 1) \) and scale factor \( k = 3 \).

1
Subtract center: \( (5 - 2, \; 3 - 1) = (3, \; 2) \)
2
Multiply by \( k \): \( (3 \cdot 3, \; 3 \cdot 2) = (9, \; 6) \)
3
Add center back: \( (2 + 9, \; 1 + 6) = (11, \; 7) \)

Example 3: Finding \( k \)

A triangle has sides of length 6, 8, and 10. After a dilation, the corresponding sides are 15, 20, and 25. Find the scale factor.

6 8 10 ~ 15 20 25

Effect on Perimeter and Area

Perimeter
Scales by \( |k| \). If the original perimeter is \( P \), the image perimeter is \( |k| \cdot P \).
Area
Scales by \( k^2 \). If the original area is \( A \), the image area is \( k^2 \cdot A \).

Dilation vs Rigid Motion

Property Dilation Rigid Motion
Preserves shape ✓ ✓
Preserves angles ✓ ✓
Preserves side lengths ✕ (unless \( |k| = 1 \)) ✓
Preserves area ✕ (scales by \( k^2 \)) ✓
Produces congruent figures ✕ ✓
Produces similar figures ✓ ✓

Common Mistakes

✕ Forgetting the Center
Multiplying coordinates directly by \( k \) without adjusting for a non-origin center. Always subtract first, scale, then add back.
✕ Scaling Area by \( k \)
Area scales by \( k^2 \), not \( k \). A scale factor of 3 makes area 9 times larger, not 3.
✕ Confusing \( k \) Direction
Thinking \( k < 1 \) means enlargement. Remember: \( 0 < k < 1 \) is a reduction, \( k > 1 \) is an enlargement.

Question 1 of 5

A triangle is dilated by a scale factor of \( k = 4 \) from the origin. If vertex \( A = (2, 3) \), what is \( A' \)?

Question 2 of 5

A figure has an area of 20 square units. After a dilation with scale factor \( k = 3 \), what is the area of the image?

Question 3 of 5

Point \( P(6, 4) \) is dilated with center \( O(2, 0) \) and scale factor \( k = \frac{1}{2} \). What is \( P' \)?

Question 4 of 5

Which property is not preserved by a dilation with \( k \neq 1 \)?

Question 5 of 5

A segment has length 12 cm. After a dilation, the image segment has length 9 cm. What is the scale factor?