Dilation Geometry: Scale Factor, Center, and Similarity

In dilation geometry, a figure resizes by a scale factor k about a center of dilation, changing every side length but never the shape, angles, or proportions. About the origin, (x, y) becomes (kx, ky); k > 1 enlarges the figure and 0 < k < 1 shrinks it. Every length multiplies by k, and every area multiplies by k², so the image stays similar to the pre-image. For a center elsewhere, use P′ = C + k(P − C).

What Is a Dilation in Geometry?

A dilation changes a figure's size, never its shape. The original figure is the pre-image, labeled ABC, and the resized copy is the image, labeled A′B′C′. Corresponding angles stay equal, and every distance scales by the same factor, which is exactly why the image and pre-image are similar. Cuemath's dilation geometry guide uses this image-and-pre-image wording throughout.

Dilations sit inside the transformation family mapped in the geometry concepts guide. This guide covers two-dimensional dilations only. Rotation and reflection are separate rigid transformations covered in companion guides; symmetry as a property is assumed familiar from the geometric shapes guide.

Center of Dilation and Scale Factor

The center of dilation is the fixed point the resize is measured from. Draw rays from it through each pre-image vertex; the image vertex sits on the same ray, k times farther out. The center itself never moves, and every other point slides outward or inward along its ray.

Enlargement vs Reduction

The scale factor k decides which way the size moves. k > 1 is an enlargement: the image grows larger than the pre-image. 0 < k < 1 is a reduction: the image shrinks. The identity k = 1 leaves the figure untouched, and dilations assume k ≠ 0. With a positive k, the image stays on the same side of the center as the pre-image, and orientation is preserved: the vertices run in the same order they did before.

Finding the Scale Factor

The scale factor is the new dimension divided by the original one: k = image length ÷ pre-image length. Pick any pair of corresponding sides and divide. When a side of length 2 grows to length 6, the factor is k = 6 ÷ 2 = 3. Keep the formula beside the other transformation formulas in the geometry formulas reference.

The Coordinate Rule: Dilation About the Origin

With the center at the origin, dilation reduces to coordinate multiplication on the plane set up in the plane geometry guide. Every coordinate multiplies by k, so (x, y) becomes (kx, ky).

Take P(2, 3) about the origin. k = 2 sends it to (4, 6); k = 1/2 sends it to (1, 1.5). Both images stay on the ray from the origin through P, just at different distances. Every vertex of a figure follows the same multiplication, so the whole triangle grows or shrinks about (0, 0) together. Cuemath's worked coordinate cases walk through the same origin examples.

Dilating About a Center That Is Not the Origin

The origin rule assumes C = (0, 0). For any other center C(h, k), one vector formula covers it: P′ = C + k(P − C). Subtract the center from the point, multiply the difference by the scale factor, then add the center back. The ray picture still works: plot the center, draw a ray through each vertex, and place the image vertex k times as far along it.

Worked example: dilate P(4, 5) by k = 3 about C(1, 2). Subtract: (4 − 1, 5 − 2) = (3, 3). Multiply: 3 × (3, 3) = (9, 9). Add the center back: (1 + 9, 2 + 9) = (10, 11), so P′(10, 11). Radius check: CP = √(3² + 3²) = 3√2, and CP′ = √(9² + 9²) = 9√2, exactly k times the original.

One boundary note: a negative scale factor is advanced material. With k < 0, the image lands on the opposite side of the center, on the extended ray rather than the ray itself, and distances scale by |k|. Standard Grades 9–12 dilation work uses positive factors, which keep the image on the same side.

Similarity, Length, and Area

After a dilation, corresponding angles match and corresponding sides stay parallel, so the image and pre-image are similar figures. They are not congruent: congruent means identical size, and dilation changes size on purpose. Similarity as a shape relationship is developed across the geometric shapes guide.

Length Scales by k

Every one-dimensional measurement multiplies by k: sides, perimeters, heights, and distances between any two points. A side of length 2 becomes 4 under k = 2, and the perimeter scales the same way. Corresponding sides of image and pre-image stay parallel, which is the visual signature of a dilation.

Area Scales by k Squared

Area is two-dimensional, so it multiplies by k², not by k. Worked example: a rectangle of width 2 and height 3 has area 6. Dilate by k = 2: the sides become 4 and 6, and the area is 24. Check: 6 × 2² = 24. Under k = 1/2, the area instead drops by (1/2)² = 1/4, from 6 to 1.5. The matching-corresponding-sides logic behind similar triangles is part of the geometry proofs guide.

Dilation vs Rotation and Reflection

Rotation and reflection are rigid transformations: every distance stays fixed, the image is congruent to the pre-image, and size never changes. Dilation is the non-rigid member of the family: distances stretch or shrink, which is why the result is similar rather than congruent. A positive dilation keeps orientation, unlike a reflection, which flips it. The rigid side of the comparison is covered in the rotation geometry guide and the reflection geometry guide.

Common Mistakes to Avoid

• Treating k as an addition: dilation multiplies coordinates by k; it does not add k to them.
• Inverting the ratio: divide image length by pre-image length, never the reverse.
• Ignoring the center: the origin rule only works about (0, 0); anywhere else you need P′ = C + k(P − C).
• Letting k = 0: that collapses the whole figure to the center point, a degenerate case, not a similarity dilation.
• Scaling area by k: area scales by k², while lengths scale by k.
• Calling the image congruent: dilation changes size, so similar is the right word, congruent is not.

How to Practice Dilations

Start on graph paper with the origin as center: dilate a triangle by k = 2 and by k = 1/2, plotting pre-image and image together. Next, use a center like C(1, 2) and run the P′ = C + k(P − C) steps on two points. Finish by checking one side and one area: side length times k, area times k².

The Gauth Geometry course unit Rigid Transformations and Dilations covers the Coordinate-Plane Transformations lesson, which matches this topic. To drill at your own level, open Gauth Math Courses and enter dilations and scale factors.

Sources and Review

• Cuemath — Dilation Geometry: Definition, Scale Factor, reviewed 2026-10-28. Definition, center, scale factor, origin coordinate rule.