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Unit J1 Section 3
Enlargements

An enlargement is a transformation which enlarges (or reduces) the size of an image. Each enlargement is described in terms of a centre of enlargement and a scale factor.

The example shows how the original, A, was enlarged with scale factors 2 and 4. A line from the centre of enlargement passes through the corresponding vertex of each image.

Note

The distances, OA′ and OA′′, are related to OA:

OA′ = 2 × OA

OA′′ = 4 × OA

The same is true of all the other distances between O and corresponding points on the images.

Worked Examples

1

Enlarge the triangle shown using the centre of enlargement marked and scale factor 3.

The first step is to draw lines from the centre of enlargement through each vertex of the triangle as shown below.

As the scale factor is 3, then

OA′ = 3 × OA

OB′ = 3 × OB

OC′ = 3 × OC

The points A′, B′ and C′ have also been marked on the diagram. Once these points have been found they can be used to draw the enlarged triangle.
2

Enlarge the pentagon with scale factor 2 using the centre of enlargement marked on the diagram.

The first step is to draw lines from the centre of enlargement which pass through the five vertices of the pentagon.

As the scale factor is 2 the distances from the centre of enlargement to the vertices of the image will be

OA′ = 2 × OA

OB′ = 2 × OB

OC′ = 2 × OC

OD′ = 2 × OD

OE′ = 2 × OE.

These points can then be marked and joined to give the enlargement.
3

The diagram shows the square ABCD which has been enlarged to give the squares A′B′C′D′ and A′′B′′C′′D′′ .

(a)

Find the scale factor for each enlargement.

The sides of the square ABCD are each 1.5 cm. The sides of the square A′B′C′D′ are 3 cm. As these are twice as long as the original, the scale factor for this enlargement is 2.

The sides of the square A′′B′′C′′D′′ are 6 cm, which is 4 times longer than the original square. So the scale factor for this enlargement is 4.
(b)

Find the centre of enlargement.

To find the centre of enlargement draw lines through A, A′ and A′′ , then repeat for B, B′ and B′′ , C, C′ and C′′ and D, D′ and D′′.

These lines cross at the centre of enlargement as shown in the diagram.

Note

When the scale factor of an enlargement is a fraction, the size of the enlargement is reduced. The image of the original is then between the centre of enlargement and the original.

4

The diagram shows three triangles. ABC was enlarged with different scale factors to give A′B′C′ and A′′B′′C′′.

(a)

Find the centre of enlargement.

To find the centre of enlargement, lines should be drawn through the corresponding points on each figure.
(b)

Find the scale factor for each enlargement.

To find the scale factors, compare the lengths of sides in the different triangles. First consider triangles ABC and A′B′C′ :

AC = 6 cm   and   A′C′ = 3 cm,

so

A′C′ = × AC

which means that the scale factor is .

For triangles ABC and A′′B′′C′′,

AC = 6 cm   and   A′′C′′ = 1.5 cm,

so

A′′C′′ = × AC

which means that the scale factor is .
5

Enlarge the triangle shown with scale factor and centre of enlargement as shown.

The first stage is to draw lines from each corner of the triangle through the centre of enlargement.

These points can then be joined to give the image.

Then the corners of the image should be fixed so that

OA′ = × OA

OB′ = × OB

OC′ = × OC

Exercises

In each diagram below, the smaller shape has been enlarged to obtain the larger shape. For each example state the scale factor.

(a)
(b)
(c)
(d)
(e)
(f)

Find the coordinates of the centre of enlargement.

(, )

A triangle with vertices at the points with coordinates (2, 1) (7, 1) and (7, 6) is enlarged to give the triangle with coordinates at the points (6, 3), (21, 3) and (21, 18).

(a)

What is the scale factor of the enlargement?

(b)

What are the coordinates of the centre of enlargement?

(, )

On a set of axes draw a triangle with vertices at (2, 0), (4, 0) and (3, 3).

Enlarge the triangle with scale factor 2 using the point (0, 0) as the centre of enlargement.

(a)

Write down the coordinates of both triangles.

(2,0)   →   (, )

(4,0)   →   (, )

(3,3)   →   (, )

(b)

What would you expect to be the coordinates of your triangle if it were to be enlarged with scale factor 3 using (0, 0) as the centre of enlargement?

(2,0)   →   (, )

(4,0)   →   (, )

(3,3)   →   (, )

(c)

Enlarge your original triangle with a different centre. Is there a simple relationship between the coordinates of the original and the enlargement, when the centre of enlargement is used?

The shaded square has sides of length 1 cm. It is enlarged a number of times as shown.

(a)

Complete the table below

Length of Side of Square1 cm2 cm3 cm4 cm
Perimeter of Square4 cm8 cm12 cm cm
Area of Square1 cm24 cm2 cm216 cm2

The shaded square continues to be enlarged.
(b)

Complete the following table.

Length of Side of Square cm
Perimeter of Square cm
Area of Square64 cm2

For each pair of objects, state the scale factor of an enlargement which produces the smaller image from the larger one.

(a)
(b)
(c)
(d)
(e)
(f)

In each example below, the smaller shape has been obtained from the larger shape by an enlargement. For each example, state the scale factor and the coordinates of the centre of enlargement.

(a)
, centre (, )
(b)
, centre (, )
(c)
, centre (, )

The larger triangle shown below is reduced in size by using a photocopier to give the smaller triangle.

(a)

What is the scale factor of the enlargement which took place?

(b)

What are the lengths of A′C′ and B′C′ ?

A′C′ cm

B′C′ cm