Dilation of a Geometric Figure — Definition & Examples

Q: What is the difference between dilation and translation?

A dilation changes the size of a figure while keeping its shape, using a scale factor and a center point. A translation slides every point of a figure the same distance in the same direction without changing its size or shape at all. Dilation produces a similar figure; translation produces a congruent figure.

Q: What happens when the scale factor of a dilation is less than 1?

When the scale factor k is between 0 and 1, the image is smaller than the original — this is called a reduction or compression. Every point moves closer to the center of dilation. For example, a scale factor of 1/3 shrinks all distances from the center to one-third of their original length.

Q: Does dilation preserve angle measures and shape?

Yes. Dilation preserves all angle measures and the overall shape of a figure. Corresponding sides of the pre-image and image are proportional, so the two figures are similar. Dilation does not generally preserve side lengths or area unless the scale factor is 1.

Dilation of a Geometric Figure

A transformation in which all distances are lengthened by a common factor. This is done by stretching all points away from some fixed point P.

Point P, a small pre-image arrow shape, and a larger image arrow shape showing dilation stretching from fixed point P.

Key Formula

(x', y') = (k(x - a) + a, \; k(y - b) + b)

Where:

$(x, y)$ = Coordinates of any point on the original figure (pre-image)
$(x', y')$ = Coordinates of the corresponding point on the dilated figure (image)
$(a, b)$ = Coordinates of the center of dilation (the fixed point P)
$k$ = Scale factor — the ratio by which all distances from the center are multiplied

Worked Example

Problem: Triangle ABC has vertices A(1, 2), B(5, 2), and C(3, 6). Dilate the triangle by a scale factor of 2 with the center of dilation at the origin (0, 0).

Step 1: Write the dilation formula with center (0, 0) and scale factor k = 2. Since the center is the origin, the formula simplifies.

(x', y') = (2x, \; 2y)

Step 2: Apply the formula to vertex A(1, 2).

A' = (2 \cdot 1, \; 2 \cdot 2) = (2, 4)

Step 3: Apply the formula to vertex B(5, 2).

B' = (2 \cdot 5, \; 2 \cdot 2) = (10, 4)

Step 4: Apply the formula to vertex C(3, 6).

C' = (2 \cdot 3, \; 2 \cdot 6) = (6, 12)

Step 5: Verify that the side lengths doubled. Original AB = 4 units; new A'B' = 8 units. The shape is preserved but the figure is twice as large.

AB = 5 - 1 = 4, \quad A'B' = 10 - 2 = 8 = 2 \times 4 \; \checkmark

Answer: The dilated triangle has vertices A'(2, 4), B'(10, 4), and C'(6, 12).

Another Example

This example uses a non-origin center of dilation and a fractional scale factor (k < 1), which produces a reduction rather than an enlargement.

Problem: A square has vertices P(2, 3), Q(6, 3), R(6, 7), and S(2, 7). Dilate the square by a scale factor of 1/2 with the center of dilation at (4, 5).

Step 1: Write the dilation formula using center (a, b) = (4, 5) and k = 1/2.

(x', y') = \left(\tfrac{1}{2}(x - 4) + 4, \; \tfrac{1}{2}(y - 5) + 5\right)

Step 2: Apply to vertex P(2, 3).

P' = \left(\tfrac{1}{2}(2 - 4) + 4, \; \tfrac{1}{2}(3 - 5) + 5\right) = (3, 4)

Step 3: Apply to vertex Q(6, 3).

Q' = \left(\tfrac{1}{2}(6 - 4) + 4, \; \tfrac{1}{2}(3 - 5) + 5\right) = (5, 4)

Step 4: Apply to vertices R(6, 7) and S(2, 7).

R' = (5, 6), \quad S' = (3, 6)

Step 5: Check the result. The original side length was 4 units; the new side length is 2 units. The center (4, 5) stays fixed, and the figure shrinks toward it.

P'Q' = 5 - 3 = 2 = \tfrac{1}{2} \times 4 \; \checkmark

Answer: The dilated square has vertices P'(3, 4), Q'(5, 4), R'(5, 6), and S'(3, 6).

Frequently Asked Questions

What is the difference between dilation and translation?

A dilation changes the size of a figure while keeping its shape, using a scale factor and a center point. A translation slides every point of a figure the same distance in the same direction without changing its size or shape at all. Dilation produces a similar figure; translation produces a congruent figure.

What happens when the scale factor of a dilation is less than 1?

When the scale factor k is between 0 and 1, the image is smaller than the original — this is called a reduction or compression. Every point moves closer to the center of dilation. For example, a scale factor of 1/3 shrinks all distances from the center to one-third of their original length.

Does dilation preserve angle measures and shape?

Yes. Dilation preserves all angle measures and the overall shape of a figure. Corresponding sides of the pre-image and image are proportional, so the two figures are similar. Dilation does not generally preserve side lengths or area unless the scale factor is 1.

Dilation of a Geometric Figure vs. Compression of a Geometric Figure

	Dilation of a Geometric Figure	Compression of a Geometric Figure
Definition	Transformation that stretches a figure away from a center point	Transformation that shrinks a figure toward a center point
Scale factor	k > 1 (figure gets larger)	0 < k < 1 (figure gets smaller)
Effect on distances	All distances from the center are multiplied by a factor greater than 1	All distances from the center are multiplied by a factor less than 1
Shape preserved?	Yes — produces a similar figure	Yes — produces a similar figure
Formula	(x', y') = (k(x−a)+a, k(y−b)+b) with k > 1	Same formula but with 0 < k < 1

Why It Matters

Dilation appears throughout geometry courses when you study similarity, scale drawings, and coordinate transformations. Standardized tests frequently ask you to find the image of a figure after dilation or to determine the scale factor between two similar shapes. In real life, dilation describes how maps, blueprints, and photo enlargements work — all of which scale distances by a constant factor from a reference point.

Common Mistakes

Mistake: Multiplying the original coordinates directly by the scale factor when the center of dilation is not the origin.

Correction: You must first subtract the center coordinates, then multiply by k, then add the center coordinates back. The formula is (x', y') = (k(x − a) + a, k(y − b) + b). Only when the center is (0, 0) does this simplify to (kx, ky).

Mistake: Thinking that a scale factor less than 1 makes the figure negative or flipped.

Correction: A scale factor between 0 and 1 simply produces a smaller figure (a compression). The figure only reflects through the center when the scale factor is negative. A positive fractional k just reduces the size.

Related Terms

Transformations — Dilation is one of the four main transformations
Dilation — General concept covering all dilation types
Dilation of a Graph — Applies dilation to function graphs instead of figures
Compression of a Geometric Figure — A dilation with scale factor between 0 and 1
Pre-Image of a Transformation — The original figure before dilation is applied
Image of a Transformation — The resulting figure after dilation is applied
Point — The center of dilation is a fixed point
Fixed — The center of dilation remains fixed (unmoved)