Enlargements: Edexcel GCSE Foundation maths revision notes

What an enlargement is

An enlargement changes the size of a shape but keeps its shape. Every length is multiplied by the same number, called the scale factor, and every angle stays the same. The new shape is called the image.

In the exam you may be asked to draw an enlargement on a grid with a whole-number scale factor, such as 3, for 2 marks, or to describe fully the single transformation that maps one shape onto another, also for 2 marks. You may also be asked to write down the scale factor, to mark the centre of enlargement with a cross, or to say whether an angle changes.

The scale factor

Every length on the image is the matching length on the original multiplied by the scale factor. With scale factor 3, a side of 5 cm becomes 5\times3=15 cm.

\text{new length}=\text{original length}\times\text{scale factor}

To find the scale factor, divide a length on the image by the matching length on the original.

\text{scale factor}=\frac{\text{new length}}{\text{original length}}

To go back from the image to the original, divide by the scale factor: a side of 30 cm after scale factor 5 was 30\div5=6 cm on the original.

Angles do not change. The image is the same shape as the original, so a 40^\circ angle is still 40^\circ after any enlargement. Only the lengths change.

Worked example 1: Lengths and angles after an enlargement

Shape B is an enlargement of shape A with scale factor 3. One side of shape A is 4 cm long, and one of its angles is 50^\circ. Work out the length of the matching side of shape B, and write down the size of the matching angle.

Every length is multiplied by the scale factor, 3.
Your turn. Work out the length of the matching side of shape B.

Your answer

4\times3=12 cm
Angles do not change in an enlargement, so the matching angle is still 50^\circ, not 150^\circ.
Answer: 12 cm and 50^\circ

Worked example 2: Finding the scale factor from two lengths

Shape Q is an enlargement of shape P. One side of P is 7 cm long, and the matching side of Q is 21 cm long. Work out the scale factor of the enlargement. Another side of Q is 18 cm long. Work out the length of the matching side of P.

Scale factor = new length \div original length =21\div7
21\div7=3, so the scale factor is 3. Q is bigger than P, so a scale factor bigger than 1 makes sense.
Your turn. Work out the length of the matching side of P.

Your answer

To go back from Q to P, divide by the scale factor: 18\div3=6 cm
Answer: scale factor 3, and the side of P is 6 cm

Enlarging a shape on a grid

When no centre is given, you can draw the image anywhere on the grid. Count the squares along each side of the shape, multiply by the scale factor, and draw the new sides.

For a sloping side, count how many squares it goes across and how many it goes up or down, then multiply both counts. With scale factor 2, a side that goes 3 across and 1 up becomes 6 across and 2 up.

Check your image: it should be the same shape, the same way up, with every side multiplied by the scale factor. Use a ruler.

Worked example 3: Drawing an enlargement with no centre

Enlarge the trapezium by scale factor 2

Count each side. The bottom is 2 squares, the left side is 2 squares tall and the top is 1 square.
Multiply each one by 2: the bottom becomes 4 squares, the left side 4 squares tall and the top 2 squares.
Your turn. The sloping side goes 1 square left and 2 squares up. How many squares up does the enlarged sloping side go?

Your answer

Multiply both counts by 2: the enlarged sloping side goes 2 squares left and 4 squares up.
Draw the bottom and the left side first, then the top. The sloping side should then join the ends exactly.
Answer: the image has a bottom of 4 squares, a left side of 4 squares, a top of 2 squares, and a sloping side 2 left and 4 up.

Using a centre of enlargement

A centre of enlargement fixes where the image goes. Each point moves along a straight line from the centre, and its distance from the centre is multiplied by the scale factor.

Count from the centre to a vertex: across, then up or down. Multiply both counts by the scale factor, then count that far from the centre. Here the centre is (1,1), and P(3,2) is 2 right and 1 up from it. With scale factor 2, P′ is 4 right and 2 up from the centre, at (5,3)

When the centre is (0,0), this is the same as multiplying the coordinates by the scale factor: (3,2) with scale factor 2 goes to (6,4). With any other centre, count from the centre, not from the origin.

When the shape is to the left of the centre or below it, count left or down from the centre. After the multiplying, count the same way, left or down, from the centre again.

Worked example 4: Enlarging from a centre

Triangle DEF has vertices D(2,4), E(4,4) and F(4,5). Enlarge triangle DEF by scale factor 2, centre (1,2).

Count from the centre (1,2) to each vertex, then double both counts.
D is 1 right and 2 up from the centre. Doubled, that is 2 right and 4 up, so D′ is at (3,6)
E is 3 right and 2 up from the centre. Doubled, that is 6 right and 4 up, so E′ is at (7,6)
Your turn. Write down the coordinates of F′.

Your answer

Write your answer as (x, y)

F is 3 right and 3 up from the centre. Doubled, that is 6 right and 6 up, so F′ is at (7,8)
Answer: D′(3,6), E′(7,6) and F′(7,8)

Worked example 5: Enlarging by scale factor 3 from a centre

Triangle RST has vertices R(7,6), S(6,6) and T(7,5). Enlarge triangle RST by scale factor 3, centre (8,7).

The centre (8,7) is above and to the right of the triangle, so count left and down from it. Multiply both counts by 3.
R is 1 left and 1 down from the centre. Times 3, that is 3 left and 3 down, so R′ is at (5,4)
S is 2 left and 1 down from the centre. Times 3, that is 6 left and 3 down, so S′ is at (2,4)
Your turn. Write down the coordinates of T′.

Your answer

Write your answer as (x, y)

T is 1 left and 2 down from the centre. Times 3, that is 3 left and 6 down, so T′ is at (5,1)
Answer: R′(5,4), S′(2,4) and T′(5,1)

Finding the centre and describing an enlargement

To find the scale factor from two shapes, pick a side on the image and the matching side on the original. Divide the image's length by the original length.

To find the centre, draw a straight line through a vertex of the image and the matching vertex of the original, and carry it on past the smaller shape. Do this for two or three vertices. The point where the lines meet is the centre of enlargement.

To describe fully an enlargement, give three things: the word enlargement, the scale factor and the centre of enlargement, for example "enlargement, scale factor 2, centre (0,3)". Give only one transformation.

Worked example 6: Describing an enlargement fully

Describe fully the single transformation that maps triangle A onto triangle B.

B is the same shape as A, but bigger, so the transformation is an enlargement. To find the scale factor, compare the bottom side of A with the matching side of B.
Your turn. Write down the scale factor of the enlargement.

Your answer

The bottom side of A is 1 square and the matching side of B is 3 squares: 3\div1=3
Draw a straight line through each vertex of B and the matching vertex of A, and carry it on past A.
Your turn. Write down the coordinates of the point where the lines meet.

Your answer

Write your answer as (x, y)

The lines meet at (1,1), so this is the centre of enlargement.
Answer: enlargement, scale factor 3, centre (1,1)

Common mistakes

Adding instead of multiplying. With scale factor 2, a side of 3 squares becomes 6 squares, not 5.
Changing the angles. An enlargement keeps every angle the same size; only the lengths change.
Drawing a sloping side wrongly. Multiply both the across and the up counts by the scale factor.
Counting from the origin instead of the centre. Count across and up from the centre of enlargement, multiply, then count from the centre again.
Putting the centre in the wrong place. Draw lines through matching vertices; the centre is where they meet, not between the shapes.
Leaving the centre out of a description. Give the word enlargement, the scale factor and the centre, and only one transformation.