A dilation is a transformation that enlarges or shrinks a figure by a scale factor k from a fixed center point. Each point moves along the ray from the center to k times its distance. With center at the origin, (x, y) → (kx, ky). The image keeps its shape, so it is similar, not congruent.
Each image point is on the same ray from O, twice as far away. The shape stays the same; only the size changes.
Formal definition and notation
A dilation with center O and scale factor k > 0 maps O to itself and maps every other point P to the point P′ on ray OP with OP′ = k · OP. It multiplies every length by k, keeps every angle measure, and sends each line not through O to a parallel line. If k > 1 the image is an enlargement, if 0 < k < 1 it is a reduction, and if k = 1 nothing moves. Dilations produce similar figures.
(x, y) → (kx, ky) and k = image lengthoriginal length
k
the scale factor
perimeter
multiplied by k
area
multiplied by k2
For a center other than the origin, subtract the center’s coordinates, multiply by k, then add them back. Center (1, 1), k = 3: the point (2, 3) goes to (1 + 3 · 1, 1 + 3 · 2) = (4, 7).
Try it: resize from the origin
Interactive
Dilate is selected. Change the scale factor and watch each image vertex move along its ray from the origin. Try a factor less than 1 to shrink the triangle.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Dilation examples, step by step
Example 1: Dilate one point
Warm-up
Dilate the point (3, −2) by a scale factor of 4 with center at the origin.
Rule: (x, y) → (4x, 4y).
4 × 3 = 12 and 4 × (−2) = −8.
The image is (12, −8).
Answer: The image is (12, −8).
Example 2: Shrink a triangle
Core
Triangle ABC has A(4, 8), B(10, 2) and C(6, −4). Dilate it by k = 12 about the origin.
Multiply every coordinate by 12.
A′(2, 4) and B′(5, 1).
C′(3, −2).
Answer:A′(2, 4), B′(5, 1), C′(3, −2).
Example 3: Perimeter and area after a dilation
Test-style
A 4 cm by 6 cm rectangle is dilated by a scale factor of 1.5. Find the perimeter and the area of the image.
New sides: 4 × 1.5 = 6 cm and 6 × 1.5 = 9 cm.
Perimeter: 2(6 + 9) = 30 cm (the original 20 cm times 1.5).
Area: 6 × 9 = 54square cm (the original 24 times 1.52 = 2.25).
Answer: The image has perimeter 30 cm and area 54 square cm.
Common mistakes
✗ MistakeAdding the scale factor: dilating (3, 5) by 2 gives (5, 7).
✓ InsteadMultiply: (6, 10).
A dilation multiplies distances from the center. Adding a number to each coordinate is a translation.
✗ MistakeThinking the area doubles when a figure is dilated by 2.
✓ InsteadArea is multiplied by 2² = 4: a 3 by 3 square (area 9) becomes 6 by 6 (area 36).
Both the length and the width double, so the area grows by 2 × 2.
✗ MistakeCalling a dilated figure congruent to the original.
✓ InsteadIt is similar: same shape, different size (unless k = 1).
Congruent requires the same size, and a dilation changes size.
✗ MistakeFinding k as original over image: AB = 5, A′B′ = 20, so k = 14.
✓ Insteadk = 205 = 4.
The scale factor is image length divided by original length.
A translation slides a figure without changing size; a dilation changes size from a center point.
Quick check: 3 questions
Question 1 of 3
What is the image of (6, −9) after a dilation by 13 about the origin?
Show answer and explanation
Answer: A. Multiply both coordinates by 13: (2, −3). (18, −27) multiplies by 3, (3, −6) subtracts 3, and (6, −3) changes only one coordinate.
Question 2 of 3
Segment AB has length 5. After a dilation, A′B′ has length 20. What is the scale factor?
Show answer and explanation
Answer: C.k = 205 = 4. 15 is the difference, 14 divides the wrong way, and 25 adds.
Question 3 of 3
Which of these stays the same after a dilation with scale factor 3?
Show answer and explanation
Answer: B. Angles do not change, so the image is similar. Side lengths and perimeter are multiplied by 3, and area by 9.
Dilation: frequently asked questions
What is the rule for a dilation?
With the center at the origin, multiply both coordinates by the scale factor k: (x, y) → (kx, ky). With k = 3, the point (2, −1) goes to (6, −3). For any center, every image point lies on the ray from the center through the original point, k times as far away.
Is a dilation a rigid transformation?
No. Rigid transformations (translations, reflections and rotations) keep lengths the same. A dilation multiplies every length by the scale factor, so unless k = 1 the size changes. It still keeps angles and shape, so it is a similarity transformation rather than a rigid one.
What happens when the scale factor is less than 1?
The figure shrinks, which is called a reduction. Every point moves toward the center: with k = 12, each image point is halfway between the center and the original point, every side is half as long, and the area is 14 as large.
TransformationA transformation moves or changes a figure on the coordinate plane. The four common transformations are translations (slides), reflections (flips), rotations (turns) and dilations (resizing). The original figure is the preimage and the result is the image, labeled with primes such as A′. Only a dilation changes the size.Geometry · Grade 7, Grade 8, Algebra 1Scale FactorA scale factor is the number used to multiply side lengths when making a similar figure larger or smaller. A scale factor of 2 doubles every length, and a scale factor of 12 makes lengths half as large. Find it by dividing a new length by the matching original length. Areas change by the square of the scale factor.Ratios and Proportions · Grade 7, Grade 8, Algebra 1LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3Similar FiguresSimilar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.Geometry · Grade 7, Grade 8, Algebra 1OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2TranslationA translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).Geometry · Grade 8, GeometryReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, GeometryRotationA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).Geometry · Grade 8, GeometryTriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5PerimeterPerimeter is the total distance around a two-dimensional shape. To find it, add the lengths of all the sides. A rectangle with sides 5 and 3 has perimeter 5 + 3 + 5 + 3 = 16. Perimeter is a length, so it uses length units such as inches or meters, never square units.Geometry · Grade 3, Grade 4, Grade 5SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2CongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB ≅ CD reads “segment AB is congruent to segment CD.”Geometry · Grade 8, GeometryCoordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7ProportionA proportion is an equation that says two ratios are equal, such as 12 = 36. It is read 1 is to 2 as 3 is to 6. In a true proportion the cross products are equal, which lets you find a missing value. Proportions are often used to solve scale, percent, and rate problems.Ratios and Proportions · Grade 6, Grade 7, Grade 8