Dilation and Similarity Transformations

A dilation is a transformation that enlarges or shrinks a figure by a scale factor k from a fixed center, and it produces a similar figure with the same shape but a different size. In brief: a dilation with center O and factor k maps each point P to P′ on the ray OP with OP′ = k · OP. A triangle dilated by k = 2 has sides twice as long, angles unchanged and area four times as large.

Dilations, combined with rigid motions, define similarity. Two figures are similar if one can be mapped onto the other by a sequence of dilations, reflections, rotations and translations. This guide explains the rules for dilations in the coordinate plane, the effect on lengths, areas and volumes, and how dilations lead to similarity.

What is a dilation?

A dilation is determined by a center O and a scale factor k ≠ 0. Each point P goes to P′ on the line OP so that OP′ = |k| · OP. If k > 0, P′ is on the same side of O as P. If k < 0, P′ is on the opposite side, so the figure is also rotated by 180°. The center is the only fixed point.

• If k > 1: enlargement.
• If 0 < k < 1: reduction.
• If k = 1: the identity, nothing changes.
• If k = −1: a half-turn about O.

Dilations preserve angles and make every segment parallel to its image. They do not preserve lengths, unless |k| = 1, so they are not rigid motions.

How do you dilate a figure in the coordinate plane?

For a center at the origin, multiply both coordinates by k: (x, y) → (kx, ky). For a center C(a, b), subtract the center, multiply and add the center back: (x, y) → (a + k(x − a), b + k(y − b)).

Example 1: center at the origin

Triangle A(1, 2), B(3, 2), C(2, 4) with k = 3: A′(3, 6), B′(9, 6), C′(6, 12). Side AB = 2 becomes A′B′ = 6, three times as long.

Example 2: center at another point

Dilate P(5, 4) with center C(1, 2) and k = 1/2. The vector from C to P is (4, 2). Half of it is (2, 1), so P′ = (3, 3). Check: P′ is the midpoint of CP ✓.

Example 3: negative factor

Dilate P(2, −3) with center the origin and k = −2: P′ = (−4, 6). The image is twice as far from the origin and on the opposite side.

How does a dilation change length, area and volume?

With scale factor k:

• Lengths are multiplied by |k|.
• Angles stay the same.
• Areas are multiplied by k².
• Volumes are multiplied by |k|³.

Example 4

A photo of 10 cm by 15 cm is enlarged with k = 1.5. The new size is 15 cm by 22.5 cm. The area changes from 150 cm² to 337.5 cm², which is 2.25 times, and 1.5² = 2.25 ✓.

Example 5

A model of a building is made with scale 1 : 50. Lengths shrink by 1/50, the floor area by 1/2500 and the volume by 1/125,000. A room of 5 m height appears as 10 cm on the model.

What is a similarity transformation?

A similarity transformation is a composition of a dilation and zero or more rigid motions. Two figures are similar if a similarity transformation maps one onto the other. Similar figures have equal corresponding angles and proportional corresponding sides, with the ratio equal to the scale factor.

For example, a 3-4-5 triangle and a 6-8-10 triangle are similar with k = 2. A 3-4-5 triangle moved to another place and enlarged by 2 is also similar to the first one. This is the basis for the similarity criteria for triangles. All circles are similar to each other, and so are all squares, but not all rectangles.

Effects of different scale factors

Where do these ideas come from?

Similar figures were used by Greek mathematicians, and Thales is traditionally said to have measured the height of a pyramid with the help of its shadow, although the exact form of the story varies among sources. Euclid’s Elements develops the theory of similar figures in Book VI. The transformation language, with dilation as a mapping of the plane, belongs to the modern treatment of geometry.

Where are dilations used today?

Maps, plans, scale models and photographs are all dilations. Computer graphics zoom and resize images by scale factors. Fractals such as the Sierpinski triangle are built from dilated copies of themselves. In science, scaling laws say how area and volume change with size: an animal twice as long has four times the cross-section of its bones but eight times the weight.

Which mistakes are common?

Scaling the area by k

Area scales by k², not k. A scale factor of 3 makes the area 9 times larger.

Forgetting the center

With a center other than the origin, subtract the center first. Otherwise the image is in the wrong place.

Mixing up enlargement and reduction

A scale factor between 0 and 1 gives a smaller figure, even if the number looks large, as in 0.9.

Assuming all shapes with equal angles are similar

For triangles equal angles are enough, but for rectangles a 2 × 4 and a 3 × 4 have equal angles and are not similar. The sides must also be proportional.