Enlargements

Every length is multiplied by the scale factor.

Compare the length of a side of B with the matching side of A.

An enlargement changes the size of a shape, but not its shape.

With centre O, the distances across and up from O are both multiplied by the scale factor.

A scale factor of one half makes every length half as long.

The centre of enlargement is the point where the lines meet.

1. The lines through matching vertices all pass through the centre of enlargement.
2. The three lines meet at the point (1,2).
3. The centre of enlargement is (1,2).

With centre (0,0), multiply the distances across and up from the origin by the scale factor.

Part (a)

1. P is at (2,1): 2 across and 1 up from the origin.
2. Multiply both by 3: 6 across and 3 up.
3. P′ is (6,3).

Part (b)

1. Q is at (2,2).
2. 2\times3=6 and 2\times3=6.
3. Q′ is (6,6).

Part (c)

1. R is at (1,2).
2. 1\times3=3 and 2\times3=6.
3. R′ is (3,6).

A scale factor of one half makes the distances from the centre half as long.

1. A is 6 across and 4 up from the origin.
2. Halve both distances: 6\div2=3 and 4\div2=2.
3. The image of A is (3,2).

The enlarged side is 4 times as long as the original side.

1. The enlarged length = original length \times4.
2. So the original length =28\div4=7.
3. The side is 7 cm long on the original shape.

A full description of an enlargement needs two things. Check where lines through matching vertices meet.

1. The base of A is 1 square long and the base of B is 2 squares long, so the scale factor is 2.
2. Lines through matching vertices, such as (2,2) to (3,3) and (3,2) to (5,3), meet at (1,1).
3. The transformation is an enlargement, scale factor 2, centre (1,1).

Count how far P is across and up from the centre, not from the origin.

1. From the centre (2,2) to P (4,3) is 2 across and 1 up.
2. Multiply by 3: 6 across and 3 up.
3. Start at the centre: 2+6=8 and 2+3=5.
4. The image of P is (8,5).

Find how far each vertex is from the centre, then halve those distances.

Part (a)

1. From (1,1) to P (5,3) is 4 across and 2 up.
2. Halve them: 2 across and 1 up.
3. 1+2=3 and 1+1=2, so P′ is (3,2).

Part (b)

1. From (1,1) to Q (9,3) is 8 across and 2 up.
2. Halve them: 4 across and 1 up.
3. 1+4=5 and 1+1=2, so Q′ is (5,2).

Part (c)

1. From (1,1) to R (5,9) is 4 across and 8 up.
2. Halve them: 2 across and 4 up.
3. 1+2=3 and 1+4=5, so R′ is (3,5).

Every length of U is one third of the matching length of T.

Part (a)

1. Each length of U is one third of the matching length of T.
2. So each length of T is 3 times the matching length of U.
3. 5\times3=15, so the longest side of T is 15 cm.

Part (b)

1. Multiply by \frac{1}{3}, which is the same as dividing by 3.
2. 12\div3=4, so the shortest side of U is 4 cm.

Work out how far the vertex of B is from the centre. The matching vertex of A is half as far.

1. From the centre (1,0) to (7,4) is 6 across and 4 up.
2. The vertex of A is half as far from the centre: 3 across and 2 up.
3. 1+3=4 and 0+2=2.
4. The vertex of A is at (4,2).

Compare the lengths of Sam's sides with T. Then look at how far his triangle is from the centre.

Part (a)

1. Sam's sides are 4 and 2 squares long, twice the sides of T, so his scale factor is right.
2. But his triangle starts at the same corner as T. He has not used the centre of enlargement.

Part (b)

1. From the centre (1,1) to (2,2) is 1 across and 1 up.
2. Multiply by 2: 2 across and 2 up.
3. 1+2=3 and 1+2=3, so the point is (3,3).

Compare matching sides for the scale factor. For the centre, imagine lines through matching vertices and find where they meet.

Part (a)

1. The base of A is 1 square long and the base of B is 3 squares long.
2. 3\div1=3, so the scale factor is 3.

Part (b)

1. The line through (5,4) and (3,2) goes 1 down for every 1 left. Carry it on from (3,2) and it reaches (2,1).
2. The line through (8,4) and (4,2) goes 1 down for every 2 left. Carry it on from (4,2) and it also reaches (2,1).
3. Check: from (2,1), the vertex (3,2) is 1 across and 1 up, and (5,4) is 3 across and 3 up, three times as far.
4. The centre of enlargement is (2,1).

Compare the distance of P′ from the origin with the distance of P from the origin.

Part (a)

1. P is 4 across and 6 up from the origin. P′ is 2 across and 3 up.
2. The distances have been halved, so the scale factor is \frac{1}{2}.

Part (b)

1. Halve both distances from the origin: 8\div2=4 and 2\div2=1.
2. Q′ is (4,1).

Part (c)

1. R′ is half as far from the origin as R.
2. So R is twice as far: 1\times2=2 and 5\times2=10.
3. R is (2,10).

For each vertex, count across and up from the centre, multiply by 3, then count from the centre again.

Part (a)

1. From (1,1) to R (3,3) is 2 across and 2 up.
2. Multiply by 3: 6 across and 6 up.
3. 1+6=7 and 1+6=7, so R′ is (7,7).

Part (b)

1. From (1,1) to S (2,3) is 1 across and 2 up.
2. Multiply by 3: 3 across and 6 up.
3. 1+3=4 and 1+6=7, so S′ is (4,7).

Use the longest sides to find the scale factor first.

1. The longest side goes from 6 cm to 15 cm, so the scale factor is 15\div6=2.5.
2. The other sides become 4\times2.5=10 cm and 5\times2.5=12.5 cm.
3. Perimeter =10+12.5+15=37.5 cm.

Work out where each vertex should go: count across and up from the centre and multiply by 3.

Part (a)

1. P (2,3) is 1 across and 1 up from the centre. Multiplied by 3: 3 across and 3 up, which is (4,5). P′ is correct.
2. Q (3,3) is 2 across and 1 up. Multiplied by 3: 6 across and 3 up, which is (7,5). Q′ is correct.
3. R (2,4) is 1 across and 2 up. Multiplied by 3: 3 across and 6 up, which is (4,8), not (4,7). R′ is wrong.

Part (b)

1. From the centre (1,2), R is 1 across and 2 up.
2. Multiply these by 3: 3 across and 6 up.
3. 1+3=4 and 2+6=8, so R′ should be at (4,8).