Unit 7 Section 3 : Enlargements

In this section we consider enlargements. We look at the use of the terms 'scale factor' and 'centre of enlargement'.

Example 1

The rectangle A, shown below, has been enlarged to give the shapes B, C and D.
Write down the scale factor for each enlargement.

A  to  B  is scale factor  2  because the lengths are doubled.

A  to  C  is scale factor    because the lengths are halved.

A  to  D  is scale factor  2.5  because the lengths are 2.5 times longer.

Example 2

A rectangle has sides of lengths 2 cm and 3 cm. It is enlarged with scale factor 3.
Draw the original rectangle and the enlarged rectangle.

The lengths of the sides of the enlarged rectangle will be:
3 × 2 cm = 6 cm
3 × 3 cm = 9 cm

Examples 3 and 4 show how to use a centre of enlargement when enlarging a shape.

Example 3

The diagram shows the triangle A B C and the point O.
Enlarge the triangle with scale factor 3, using O as the centre of enlargement.

The diagram shows the 2 triangles; the explanation follows.

First draw lines from point O through A, B and C, as shown in the diagram.
Measure the length O A and multiply it by 3 to get the distance from O of the image point A', i.e. O A' = 3 × O A. Mark the point A' on the diagram.
The images B' and C' can then be marked in a similar way and the enlarged triangle A' B' C' can then be drawn.

Example 4

The following diagrams show two shapes that have been enlarged. Determine the centre of enlargement in each case.

To find the centres of enlargement, draw lines through the corresponding corners of each shape. These lines will cross at the centre of enlargement, as shown below. The centres have been marked with the letter O in both diagrams.

Exercises

Question 1

The following diagram shows 5 triangles, A, B, C, D and E:

What scale factor is used for each of the enlargements described below:

(a) B enlarged to A,
(b) C enlarged to E,
(c) D enlarged to E,
(d) D enlarged to A,
(e) B enlarged to C,
(f) B enlarged to D ?
Question 2

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(a)

Draw a rectangle that has sides of lengths 2cm and 4cm.

a cell is 0.5cm × 0.5cm
(b)

Draw enlargements of this rectangle using scale factors

2,
3,
.
Question 3

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(a)

Construct a triangle that has sides of lengths 3cm, 4cm and 5.5cm.

(b)

Draw enlargements of this triangle using scale factors

2,
3.
Question 4

Hannah writes her initials as shown:

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Enlarge her initials with scale factors

2,
.
Question 5

Using (0, 0) as the centre of enlargement, enlarge the shape with scale factor 2 and scale factor 3.

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Question 6

Enlarge the shape with scale factor 2, using first (0, 0) as the centre of enlargement and then (1, 8) as the centre of enlargement.

Draw the points of the centres of enlargements too.
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Question 7

For each of the following enlargements, determine the centre of enlargement.

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Question 8

A triangle has corners at the points with coordinates (1, 2), (3, 3) and (0, 3). It is enlarged to give a triangle with corners at the points (5, 4), (11, 7) and (2, 7).
Determine the scale factor of the enlargement and the coordinates of the centre of enlargement.

Scale factor =

Centre of enlargement is at .

Question 9

A trapezium has corners at the points with coordinates (1, 0), (3, 2), (3, 4) and (1, 5). It is enlarged with scale factor 3, using the point (0, 3) as the centre of enlargement.
Determine the coordinates of the corners of the enlarged trapezium.

Corners of enlarged trapezium are at , , ,

Question 10

A parallelogram has corners at the points with coordinates (5, 1), (9, 3), (11, 9) and (7, 7). Enlarge this shape with scale factor , using the point with coordinates (1, 3) as the centre of enlargement.

Draw the shapes and the centre of enlargement too.

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Question 11

Jill has drawn an original picture of a giraffe for an animal charity. It measures 6.5 cm high by 4 cm wide.
Different-sized copies of the original picture can be made to just fit into various shapes.

(a)

Jill wants to enlarge the original picture so that it just fits inside a rectangle on a carrier bag. The rectangle measures 24 cm high by 12 cm wide.
By what scale factor should she multiply the original picture?

The scale factor for heights must be at most 24 ÷ 6.5 = 3.692 (to 3 d.p.);
the scale factor for widths must be at most 12 ÷ 4 = 3,
so the maximum scale factor Jill can use is 3.
(b)

Jill wants to multiply the original picture by a scale factor so that it just fits inside the square shown below for a badge.

By what scale factor should she multiply the original picture?

(to 3 d.p.)
The scale factor for heights must be at most 2.7 ÷ 6.5 = 0.415 (to 3 d.p.);
the scale factor for widths must be at most 2.7 ÷ 4 = 0.675,
so the maximum scale factor Jill can use is 0.415 (to 3 d.p.).
(c)

The original picture is to be used on a poster. It must fit inside a shape like this.

The shape is to be a semi-circle of radius 6.6 cm.
What would be the perimeter of the shape?

The perimeter = cm (to 2 d.p.)
The perimeter = (π × 6.6) + (2 × 6.6)
= 33.93451151 cm
= 33.93 cm (to 2 d.p.)