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E1.3 · Perform dilations and describe similar shapes

Learn to perform dilations and describe similar shapes through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Spatial Sense

How a shape grows or shrinks while keeping its form

Imagine tracing a triangle and making the tracing larger without stretching it in just one direction. The new triangle has the same form, but its size has changed. This kind of change is called a dilation. In this lesson, you will use a scale factor to enlarge or reduce shapes, find new vertex coordinates, and describe the relationship between the original shape and its image.

What you will learn

  • Explain what a dilation does to a shape.
  • Use a scale factor to find image coordinates when the centre is the origin.
  • Describe how a dilated shape is similar to the original.

What a dilation does

A shape is a flat figure, such as a triangle or rectangle. A dilation changes a shape's size from a fixed point called the centre of dilation. The original shape is the shape before the change. The image is the new shape after the change.
A scale factor tells how much each distance from the centre changes. For example, a scale factor of 2 doubles each distance. A scale factor of one-half makes each distance half as long. A factor greater than 1 makes an enlargement. A factor between 0 and 1 makes a reduction. A factor of 1 leaves the size unchanged.
Picture a point on a transparent sheet and a pin at the centre. If you pull the point so it moves twice as far from the pin, it has been moved using a scale factor of 2. In this lesson, the centre will be the origin: the point where the two grid axes meet.
  • A dilation changes size from a fixed centre.
  • One scale factor applies to the whole shape.
  • A factor above 1 enlarges; a factor between 0 and 1 reduces.

Dilate points on a grid

A coordinate is a pair of numbers that locates a point on a grid. The first number tells its horizontal position. The second tells its vertical position. For example, (3, 2) is 3 units right and 2 units up from the origin.
When the centre is the origin, multiply both coordinates of every vertex by the scale factor. A vertex is a corner of a shape. This rule changes both the horizontal and vertical distances by the same amount. Then join the new vertices in the same order as the original vertices.
For a scale factor of 2, the point (3, 2) becomes (6, 4). For a factor of one-half, it becomes (1.5, 1). The decimal 1.5 is halfway between 1 and 2 on the grid. It is fine for an image point to fall between grid lines.
(x,y)→(kx,ky)(x,y)\rightarrow(kx,ky)
  • For a centre at the origin, multiply both coordinates by the scale factor.
  • Use the same factor for every vertex.
  • Connect image vertices in the same order as the original vertices.

Describe similar shapes

Similar shapes have the same form, though they may be different sizes. Their matching angles are equal. Their matching side lengths all change by the same scale factor. Matching parts in the same positions are called corresponding parts. For example, the bottom side of one shape may correspond to the bottom side of its image.
A dilation makes an image similar to its original shape. To describe the result, name the scale factor and say whether the image is an enlargement or a reduction. For example, a scale factor of 3 means every distance from the centre is tripled.
To check a dilation, compare the matching distances from the centre. Each must use the same factor. If one part is doubled while another is tripled, those changes do not describe one dilation.
  • Similar shapes have equal corresponding angles.
  • Corresponding side lengths use one common scale factor.
  • A dilation produces an image similar to its original.

A reliable check

Start by identifying the centre and the scale factor. Find the image coordinates for each vertex. Connect them in order. Then describe the change as an enlargement, a reduction, or no size change.
Check more than one vertex. With a centre at the origin, each coordinate pair must use the same factor. Also check that a factor greater than 1 puts image points farther from the origin, while a factor between 0 and 1 puts them closer. These checks help you catch a calculation or drawing error.
  • Identify the centre and factor before calculating.
  • Check every vertex using the same rule.
  • Use the factor to describe whether the image grows or shrinks.

Worked example

Double a triangle

Triangle ABC has vertices A(1, 1), B(3, 1), and C(1, 2). Dilate it from the origin by a scale factor of 2. Give the new vertices and describe the result.
Image triangle after dilation
Image triangle after dilationThe ordered image vertices are A′, B′, and C′. The original vertices are given in the problem.A′B′C′

The ordered image vertices are A′, B′, and C′. The original vertices are given in the problem.

  1. Apply the same change to each vertex
    The centre is the origin, so multiply both coordinates of each vertex by 2. This doubles each coordinate and each distance from the centre.
    (x,y)→(2x,2y)(x,y)\rightarrow(2x,2y)
  2. Find the image vertices
    For example, A(1, 1) becomes A′(2, 2). Use the same calculation for B and C.
    A′(2,2), B′(6,2), C′(2,4)A′(2,2),\ B′(6,2),\ C′(2,4)
Answer: The image vertices are A′(2, 2), B′(6, 2), and C′(2, 4). This is an enlargement with scale factor 2. The image is similar to triangle ABC.
Check: Each image coordinate is twice its original coordinate. For example, C(1, 2) becomes C′(2, 4).

Worked example

Reduce a rectangle

Rectangle PQRS has vertices P(2, 2), Q(6, 2), R(6, 4), and S(2, 4). Dilate it from the origin by a scale factor of one-half. Give the image vertices and state whether the image is larger or smaller.
Image rectangle after reduction
Image rectangle after reductionThe ordered image vertices are P′, Q′, R′, and S′. The original vertices are given in the problem.P′Q′R′S′

The ordered image vertices are P′, Q′, R′, and S′. The original vertices are given in the problem.

  1. Use one-half for both coordinates
    A factor between 0 and 1 reduces distances from the centre. Multiply each coordinate by one-half.
    (x,y)→(x2,y2)(x,y)\rightarrow\left(\frac{x}{2},\frac{y}{2}\right)
  2. Find the image vertices
    For example, Q(6, 2) becomes Q′(3, 1). Repeat the calculation for all four vertices.
    P′(1,1), Q′(3,1), R′(3,2), S′(1,2)P′(1,1),\ Q′(3,1),\ R′(3,2),\ S′(1,2)
Answer: The image vertices are P′(1, 1), Q′(3, 1), R′(3, 2), and S′(1, 2). The image is smaller than the original, with scale factor one-half. The two rectangles are similar.
Check: The original width is 4 units and the image width is 2 units. The original height is 2 units and the image height is 1 unit. Both lengths are halved.

Worked example

Find a scale factor

Point M(2, 3) moves to M′(6, 9) in a dilation from the origin. Find the scale factor. Then describe what happens to distances from the origin.
  1. Compare the coordinates
    The first coordinate changes from 2 to 6, which is multiplication by 3. Check the second coordinate: 3 changes to 9, also multiplication by 3.
    6÷2=3,9÷3=36\div2=3,\quad 9\div3=3
  2. Describe the dilation
    Both coordinates use the same factor, so the scale factor is 3. Every distance from the origin is tripled. This is an enlargement.
    (x,y)→(3x,3y)(x,y)\rightarrow(3x,3y)
Answer: The scale factor is 3. Every distance from the origin is tripled, so a shape undergoing this dilation becomes an enlargement. Its image is similar to the original.
Check: Both coordinates of M are multiplied by 3 to make M′(6, 9).

Common mistakes and how to avoid them

Multiplying only one coordinate by the scale factor.
Correction: When the centre is the origin, multiply both coordinates of every vertex by the same factor.
Using a different factor for different vertices.
Correction: A dilation uses one scale factor for the whole shape.
Calling every changed shape similar.
Correction: Check that corresponding angles are equal and corresponding side lengths use one scale factor.
Thinking a factor between 0 and 1 makes the image larger.
Correction: A factor between 0 and 1 makes each distance from the centre smaller.

Lesson summary

  • A dilation enlarges or reduces a shape from a fixed centre.
  • The scale factor tells how each distance from the centre changes.
  • With the origin as centre, multiply both coordinates of each vertex by the scale factor.
  • A dilated image is similar to its original: corresponding angles are equal and corresponding lengths use one scale factor.

Check your understanding

Question 1

Point (4, 2) is dilated from the origin by a scale factor of 2. What are its new coordinates?
  1. (8, 4)
  2. (6, 4)
  3. (8, 2)
  4. (2, 1)
Show answer and explanation
(8, 4)
Multiply both coordinates by 2: 4 becomes 8 and 2 becomes 4.

Question 2

A dilation uses a scale factor of one-half. What happens to distances from the centre?
  1. They are halved.
  2. They are doubled.
  3. They stay the same.
  4. Only horizontal distances are halved.
Show answer and explanation
They are halved.
Each distance from the centre is multiplied by one-half, so it is halved.

Question 3

Which statement best describes similar shapes?
  1. Their matching angles are equal, and their corresponding lengths use one scale factor.
  2. They must have the same side lengths.
  3. They must face the same direction on the page.
  4. Only one pair of corresponding side lengths can change.
Show answer and explanation
Their matching angles are equal, and their corresponding lengths use one scale factor.
Similar shapes may have different sizes, but their matching angles are equal and their corresponding lengths use one common factor.

Key terms

Dilation
A change that enlarges or reduces a shape from a fixed centre.
Centre of dilation
The fixed point from which distances are enlarged or reduced.
Scale factor
The number used to multiply distances from the centre.
Corresponding parts
Matching parts of two shapes, such as sides or angles in the same positions.
Similar shapes
Shapes with equal matching angles and corresponding side lengths that use the same scale factor.
Image
The new shape made by a change such as a dilation.

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