GCSE Maths: Similar shapes and scale factors (Higher)
Part of the GCSE Maths transformations and similarity chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Similar shapes and scale factors
Two shapes are similar if one is an enlargement of the other. They have exactly the same shape but can be different sizes (one might also be turned or flipped). For similar shapes:
- all corresponding angles are equal;
- all corresponding sides are in the same ratio. This ratio, large ÷ small, is the scale factor.
Corresponding sides are the ones in the same position in both shapes. For example, the shortest side of one triangle matches the shortest side of the similar triangle. Remember the difference: congruent shapes are the same size and shape; similar shapes are the same shape but not necessarily the same size.
Finding a missing length. (1) Find a pair of corresponding sides that you know both lengths of. (2) Work out the scale factor: larger ÷ smaller. (3) Use it on the side you want: multiply if you are going from the small shape to the large shape, divide if you are going from the large shape to the small one.
Are two shapes similar? Check that all pairs of corresponding sides are in the same ratio. A rectangle 4 cm by 6 cm and a rectangle 6 cm by 9 cm are similar: 6 ÷ 4 = 1.5 and 9 ÷ 6 = 1.5. A 4 cm by 6 cm rectangle and a 6 cm by 8 cm rectangle are not similar, since 6 ÷ 4 = 1.5 but 8 ÷ 6 = 1.33…. (For triangles, equal angles are enough: you do not need to check the sides.)
Similar triangles from parallel lines. Parallel lines create similar triangles. If DE is parallel to BC in triangle ABC, then triangle ADE is similar to triangle ABC. The angle at A is shared, and the corresponding angles ADE and ABC are equal, and so are AED and ACB. Note that the matching side to AD is the whole of AB, not DB. Two triangles that meet at a point (a “bow-tie” shape) with a pair of parallel sides are also similar, because the alternate angles are equal and the vertically opposite angles are equal.
Proving two triangles are similar (Higher). Show that two angles are equal (the third then must be equal as well), and give a reason for each pair: parallel lines (corresponding or alternate angles), vertically opposite angles, a common angle, or angles in a triangle that add up to 180°. Then state: “the triangles are similar (equal angles)”.
Key ideas
- Similar shapes: equal corresponding angles and corresponding sides in the same ratio. The ratio is the scale factor.
- Scale factor = length on the larger shape ÷ matching length on the smaller shape.
- Small to large: multiply by the scale factor. Large to small: divide.
- Parallel lines make similar triangles; look for a common angle, alternate or corresponding angles, or vertically opposite angles.
- Congruent = same size and shape. Similar = same shape, any size.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 6 cm (small shape) and 9 cm (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.9 cm ÷ 6 cm = 32
So the large shape is 32 times as big as the small shape. - Use the scale factor to find x. x is on the large shape and we know the matching side on the small shape. Going from small to large we multiply by the scale factor.x = 5 cm × 32 = 7.5 cm
- Write the answer with its unit. Lengths need a unit.x = 7.5 cm
Check. Ratio check: 7.5 cm ÷ 5 cm = 32, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 4 cm (small shape) and 10 cm (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.10 cm ÷ 4 cm = 52
So the large shape is 52 times as big as the small shape. - Use the scale factor to find x. x is on the small shape and we know the matching side on the large shape. Going from large to small we divide by the scale factor.x = 15 cm ÷ 52 = 6 cm
- Write the answer with its unit. Lengths need a unit.x = 6 cm
Check. Ratio check: 15 cm ÷ 6 cm = 52, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 6 m (small shape) and 12 m (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.12 m ÷ 6 m = 2
So the large shape is 2 times as big as the small shape. - Use the scale factor to find x. x is on the large shape and we know the matching side on the small shape. Going from small to large we multiply by the scale factor.x = 5 m × 2 = 10 m
- Write the answer with its unit. Lengths need a unit.x = 10 m
Check. Ratio check: 10 m ÷ 5 m = 2, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Parallel lines make triangles ADE and ABC similar. Find the scale factor from a pair of sides we know, then use it on the side we need.
- Explain why the triangles are similar. We may only use scale factors if the triangles are similar. Parallel lines give equal corresponding angles.DE is parallel to BC, so angle ADE = angle ABC (corresponding angles) and angle AED = angle ACB (corresponding angles). Angle A is common to both triangles. All three angles match, so triangles ADE and ABC are similar.
- Find AB. AB is the whole side. The small triangle ADE uses AD, and the large triangle ABC uses AB = AD + DB.AB = AD + DB = 4 cm + 6 cm = 10 cm
- Find the scale factor from the small triangle to the large triangle. Scale factor = length in ABC ÷ matching length in ADE. AB matches AD.AB ÷ AD = 10 cm ÷ 4 cm = 52
- Use it to find BC. BC matches DE. Going from the small triangle to the large one, multiply.BC = DE × 52 = 8 cm × 52 = 20 cm
- Write the answer with its unit. x = 20 cm
Check. Ratio check: AD : AB = 4 : 10 and DE : BC = 8 : 20. Both equal 25 when written as a fraction. ✓
Watch out. The side AB of the large triangle is the whole side AD + DB. Do not use DB alone as the matching side for AD.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. The crossing lines make triangles XAB and XCD similar. Find the scale factor from a pair of sides we know, then use it on the side we need.
- Explain why the triangles are similar. Parallel lines give equal alternate angles, and crossing lines give equal vertically opposite angles.AB is parallel to CD, so angle XAB = angle XCD (alternate angles) and angle XBA = angle XDC (alternate angles). Angle AXB = angle CXD (vertically opposite angles). All three angles match, so triangles XAB and XCD are similar.
- Find the scale factor from the small triangle to the large triangle. XA matches XC. Scale factor = XC ÷ XA.4.5 cm ÷ 3 cm = 32
- Use it to find CD. CD matches AB. Multiply by the scale factor.CD = AB × 32 = 5 cm × 32 = 7.5 cm
- Write the answer with its unit. x = 7.5 cm
Check. Ratio check: CD ÷ AB = 7.5 cm ÷ 5 cm = 32. ✓
Watch out. Match the sides by position: XA matches XC, XB matches XD and AB matches CD. The small triangle is the one at the top.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. The man and the tree make similar right-angled triangles with the sun’s rays, because the sun’s rays are parallel and both stand vertically. Use the scale factor from the shadows.
- Explain why the triangles are similar. The man and the tree are both vertical, the ground is horizontal and the rays of the sun are parallel. So the triangles have equal angles.Man’s triangle and tree’s triangle are similar.
- Find the scale factor from the man’s shadow to the tree’s shadow. The shadows match each other. Scale factor = tree shadow ÷ man shadow.10 ÷ 2.4 = 25 ÷ 6 = 256
- Use it to find the height of the tree. The heights match each other. Multiply the man’s height by the scale factor.1.8 × 256 = 45 ÷ 6 = 7.5 m
Check. Check the ratio the other way: 1.8 ÷ 2.4 = 0.75 and 7.5 ÷ 10 = 0.75. ✓
Watch out. Match shadow to shadow and height to height. Do not match the height of one with the shadow of the other.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Write the scale factor two ways: from the lengths we know and from the lengths with x. They must be equal, so we get an equation to solve.
- Find the scale factor from AB to PQ. Scale factor = PQ ÷ AB.9 ÷ 6 = 1.5
- Write QR in terms of BC using the scale factor. QR matches BC, and every length on PQR is 1.5 times the matching length on ABC.x + 4 = 1.5 × x
- Solve the equation. Subtract x from both sides, then divide by 0.5.4 = 1.5x − x = 0.5x
x = 4 ÷ 0.5 = 8
Check. BC = 8 cm and QR = 12 cm, and 12 ÷ 8 = 1.5. ✓
Watch out. Write the scale factor as a number first, then form the equation. Do not mix the two triangles up.
Watch out
Match the corresponding sides carefully: the side opposite the same angle, or in the same position. In the nested triangle, the side matching AD is AB (the whole side), not DB. Also remember: from the small shape to the large shape multiply; from the large shape to the small shape divide.Exercise 21I Similar shapes and missing lengths
Non-calculatorKey words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 5 cm (small shape) and 10 cm (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.10 cm ÷ 5 cm = 2
So the large shape is 2 times as big as the small shape. - Use the scale factor to find x. x is on the large shape and we know the matching side on the small shape. Going from small to large we multiply by the scale factor.x = 3 cm × 2 = 6 cm
- Write the answer with its unit. Lengths need a unit.x = 6 cm
Check. Ratio check: 6 cm ÷ 3 cm = 2, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 4 cm (small shape) and 12 cm (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.12 cm ÷ 4 cm = 3
So the large shape is 3 times as big as the small shape. - Use the scale factor to find x. x is on the large shape and we know the matching side on the small shape. Going from small to large we multiply by the scale factor.x = 3 cm × 3 = 9 cm
- Write the answer with its unit. Lengths need a unit.x = 9 cm
Check. Ratio check: 9 cm ÷ 3 cm = 3, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 6 cm (small shape) and 12 cm (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.12 cm ÷ 6 cm = 2
So the large shape is 2 times as big as the small shape. - Use the scale factor to find x. x is on the large shape and we know the matching side on the small shape. Going from small to large we multiply by the scale factor.x = 5 cm × 2 = 10 cm
- Write the answer with its unit. Lengths need a unit.x = 10 cm
Check. Ratio check: 10 cm ÷ 5 cm = 2, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 4 cm (small shape) and 10 cm (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.10 cm ÷ 4 cm = 52
So the large shape is 52 times as big as the small shape. - Use the scale factor to find x. x is on the small shape and we know the matching side on the large shape. Going from large to small we divide by the scale factor.x = 15 cm ÷ 52 = 6 cm
- Write the answer with its unit. Lengths need a unit.x = 6 cm
Check. Ratio check: 15 cm ÷ 6 cm = 52, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 5 cm (small shape) and 15 cm (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.15 cm ÷ 5 cm = 3
So the large shape is 3 times as big as the small shape. - Use the scale factor to find x. x is on the small shape and we know the matching side on the large shape. Going from large to small we divide by the scale factor.x = 12 cm ÷ 3 = 4 cm
- Write the answer with its unit. Lengths need a unit.x = 4 cm
Check. Ratio check: 12 cm ÷ 4 cm = 3, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 6 cm (small shape) and 9 cm (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.9 cm ÷ 6 cm = 32
So the large shape is 32 times as big as the small shape. - Use the scale factor to find x. x is on the large shape and we know the matching side on the small shape. Going from small to large we multiply by the scale factor.x = 4 cm × 32 = 6 cm
- Write the answer with its unit. Lengths need a unit.x = 6 cm
Check. Ratio check: 6 cm ÷ 4 cm = 32, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 8 m (small shape) and 12 m (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.12 m ÷ 8 m = 32
So the large shape is 32 times as big as the small shape. - Use the scale factor to find x. x is on the large shape and we know the matching side on the small shape. Going from small to large we multiply by the scale factor.x = 3 m × 32 = 4.5 m
- Write the answer with its unit. Lengths need a unit.x = 4.5 m
Check. Ratio check: 4.5 m ÷ 3 m = 32, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.
Method. The ratio 2 : 5 tells us that for every 2 units on the small shape there are 5 units on the large shape. Find how many times the ratio fits into 8 cm.
- Find the multiplier. 2 parts of the ratio are 8 cm, so 1 part is 4 cm.8 ÷ 2 = 4 cm per part
- Find the larger side. The larger shape has 5 parts.5 × 4 = 20 cm
Check. 20 ÷ 8 = 2.5 and 5 ÷ 2 = 2.5, so the scale factor is 2.5 either way. ✓
Watch out. Do not add or subtract: 8 + 3 = 11 would be wrong. Similar shapes are linked by multiplying.
Method. Two rectangles are similar if their corresponding sides are in the same ratio.
- Pair up the matching sides. The lengths match each other and the widths match each other.4 cm and 6 cm; 6 cm and 9 cm
- Work out the scale factor for each pair. Larger ÷ smaller. If the shapes are similar, both pairs give the same scale factor.6 ÷ 4 = 32
9 ÷ 6 = 32 - Compare. Similar rectangles need the same scale factor for both pairs of sides.The scale factors are the same, so the rectangles are similar.
Check. As ratios: 4 : 6 = 6 : 9? Both simplify to 2 : 3. ✓
Watch out. All angles in rectangles are 90°, so only the sides need checking.
Method. Two rectangles are similar if their corresponding sides are in the same ratio.
- Pair up the matching sides. The lengths match each other and the widths match each other.3 cm and 6 cm; 5 cm and 9 cm
- Work out the scale factor for each pair. Larger ÷ smaller. If the shapes are similar, both pairs give the same scale factor.6 ÷ 3 = 2
9 ÷ 5 = 95 - Compare. Similar rectangles need the same scale factor for both pairs of sides.The scale factors are different, so the rectangles are not similar.
Check. As ratios: 3 : 5 = 6 : 9? They do not simplify to the same ratio. ✓
Watch out. All angles in rectangles are 90°, so only the sides need checking.
Method. Two rectangles are similar if their corresponding sides are in the same ratio.
- Pair up the matching sides. The lengths match each other and the widths match each other.2 cm and 4 cm; 5 cm and 10 cm
- Work out the scale factor for each pair. Larger ÷ smaller. If the shapes are similar, both pairs give the same scale factor.4 ÷ 2 = 2
10 ÷ 5 = 2 - Compare. Similar rectangles need the same scale factor for both pairs of sides.The scale factors are the same, so the rectangles are similar.
Check. As ratios: 2 : 5 = 4 : 10? Both simplify to 2 : 5. ✓
Watch out. All angles in rectangles are 90°, so only the sides need checking.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Parallel lines make triangles ADE and ABC similar. Find the scale factor from a pair of sides we know, then use it on the side we need.
- Explain why the triangles are similar. We may only use scale factors if the triangles are similar. Parallel lines give equal corresponding angles.DE is parallel to BC, so angle ADE = angle ABC (corresponding angles) and angle AED = angle ACB (corresponding angles). Angle A is common to both triangles. All three angles match, so triangles ADE and ABC are similar.
- Find AB. AB is the whole side. The small triangle ADE uses AD, and the large triangle ABC uses AB = AD + DB.AB = AD + DB = 4 cm + 6 cm = 10 cm
- Find the scale factor from the small triangle to the large triangle. Scale factor = length in ABC ÷ matching length in ADE. AB matches AD.AB ÷ AD = 10 cm ÷ 4 cm = 52
- Use it to find BC. BC matches DE. Going from the small triangle to the large one, multiply.BC = DE × 52 = 8 cm × 52 = 20 cm
- Write the answer with its unit. x = 20 cm
Check. Ratio check: AD : AB = 4 : 10 and DE : BC = 8 : 20. Both equal 25 when written as a fraction. ✓
Watch out. The side AB of the large triangle is the whole side AD + DB. Do not use DB alone as the matching side for AD.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Parallel lines make triangles ADE and ABC similar. Find the scale factor from a pair of sides we know, then use it on the side we need.
- Explain why the triangles are similar. We may only use scale factors if the triangles are similar. Parallel lines give equal corresponding angles.DE is parallel to BC, so angle ADE = angle ABC (corresponding angles) and angle AED = angle ACB (corresponding angles). Angle A is common to both triangles. All three angles match, so triangles ADE and ABC are similar.
- Find AB. AB is the whole side. The small triangle ADE uses AD, and the large triangle ABC uses AB = AD + DB.AB = AD + DB = 3 cm + 2 cm = 5 cm
- Find the scale factor from the small triangle to the large triangle. Scale factor = length in ABC ÷ matching length in ADE. AB matches AD.AB ÷ AD = 5 cm ÷ 3 cm = 53
- Use it to find DE. DE matches BC. Going from the large triangle to the small one, divide.DE = BC ÷ 53 = 15 cm ÷ 53 = 9 cm
- Write the answer with its unit. x = 9 cm
Check. Ratio check: DE : BC = 9 : 15 = 35 (as a fraction), the same as AD : AB = 3 : 5. ✓
Watch out. The side AB of the large triangle is the whole side AD + DB. Do not use DB alone as the matching side for AD.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Parallel lines make triangles ADE and ABC similar. Find the scale factor from a pair of sides we know, then use it on the side we need.
- Explain why the triangles are similar. We may only use scale factors if the triangles are similar. Parallel lines give equal corresponding angles.DE is parallel to BC, so angle ADE = angle ABC (corresponding angles) and angle AED = angle ACB (corresponding angles). Angle A is common to both triangles. All three angles match, so triangles ADE and ABC are similar.
- Find AB and the scale factor. AB = AD + DB. The scale factor from ADE to ABC is AB ÷ AD.AB = 6 cm + 3 cm = 9 cm
Scale factor = 9 cm ÷ 6 cm = 32 - Find the whole side AC. AC matches AE. Multiply by the scale factor.AC = AE × 32 = 8 cm × 32 = 12 cm
- Find EC. EC is the rest of AC after AE.EC = AC − AE = 12 cm − 8 cm = 4 cm
- Write the answer with its unit. x = 4 cm
Check. Check the ratio AE : EC = 8 : 4 and AD : DB = 6 : 3: both simplify to 2. ✓
Watch out. The side AB of the large triangle is the whole side AD + DB. Do not use DB alone as the matching side for AD.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. The crossing lines make triangles XAB and XCD similar. Find the scale factor from a pair of sides we know, then use it on the side we need.
- Explain why the triangles are similar. Parallel lines give equal alternate angles, and crossing lines give equal vertically opposite angles.AB is parallel to CD, so angle XAB = angle XCD (alternate angles) and angle XBA = angle XDC (alternate angles). Angle AXB = angle CXD (vertically opposite angles). All three angles match, so triangles XAB and XCD are similar.
- Find the scale factor from the small triangle to the large triangle. XA matches XC. Scale factor = XC ÷ XA.4.5 cm ÷ 3 cm = 32
- Use it to find CD. CD matches AB. Multiply by the scale factor.CD = AB × 32 = 5 cm × 32 = 7.5 cm
- Write the answer with its unit. x = 7.5 cm
Check. Ratio check: CD ÷ AB = 7.5 cm ÷ 5 cm = 32. ✓
Watch out. Match the sides by position: XA matches XC, XB matches XD and AB matches CD. The small triangle is the one at the top.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. The crossing lines make triangles XAB and XCD similar. Find the scale factor from a pair of sides we know, then use it on the side we need.
- Explain why the triangles are similar. Parallel lines give equal alternate angles, and crossing lines give equal vertically opposite angles.AB is parallel to CD, so angle XAB = angle XCD (alternate angles) and angle XBA = angle XDC (alternate angles). Angle AXB = angle CXD (vertically opposite angles). All three angles match, so triangles XAB and XCD are similar.
- Find the scale factor from the small triangle to the large triangle. AB matches CD. Scale factor = CD ÷ AB.12 cm ÷ 4 cm = 3
- Use it to find XA. XA is in the small triangle and XC is in the large triangle. Going from large to small, divide.XA = XC ÷ 3 = 6 cm ÷ 3 = 2 cm
- Write the answer with its unit. x = 2 cm
Check. Ratio check: XC ÷ XA = 6 cm ÷ 2 cm = 3. ✓
Watch out. Match the sides by position: XA matches XC, XB matches XD and AB matches CD. The small triangle is the one at the top.
Method. The enlarged photograph is similar to the original, so the scale factor is the same for the width and the height.
- Find the scale factor. The width goes from 12 cm to 20 cm. Scale factor = new ÷ old.20 ÷ 12 = 53
- Multiply the height by the scale factor. The height is the 18 cm side.18 × 53 = 30 cm
Check. Check the ratio: 12 : 18 = 2 : 3 and 20 : 30 = 2 : 3. ✓
Watch out. Use the same scale factor for both measurements.
Method. The scale 1 : 24 means every length on the real car is 24 times the matching length on the model. To get the model we divide.
- Change the length to centimetres. 1 m = 100 cm.4.8 m = 4.8 × 100 = 480 cm
- Divide by 24. The model is smaller, so divide the real length by 24.480 ÷ 24 = 20 cm
Check. 20 × 24 = 480 cm = 4.8 m. ✓
Watch out. A scale of 1 : 24 makes the model smaller. Divide, do not multiply.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. The pole and the building make similar triangles with the sun’s rays. Match shadow with shadow and height with height.
- Find the scale factor from the pole to the building. Scale factor = building shadow ÷ pole shadow.18 ÷ 1.5 = 12
- Multiply the pole’s height. Heights match each other.2 × 12 = 24 m
Check. 2 ÷ 1.5 = 1.33… and 24 ÷ 18 = 1.33…. ✓
Watch out. Match shadow with shadow and height with height.
Method. Use the two conditions for similar shapes: equal angles and sides in the same ratio.
- Squares. A square has four 90° angles and all four sides equal. For two squares with sides a and b, every pair of matching sides is in the ratio b : a.Angles equal, sides in one ratio: similar.
- Rectangles. All rectangles have four 90° angles, but the sides can have different ratios. A 2 cm by 3 cm rectangle and a 2 cm by 6 cm rectangle have the same widths but their lengths are 3 and 6.2 ÷ 2 = 1 but 6 ÷ 3 = 2, so the scale factors differ.
- Conclude. Squares are always similar; rectangles are not always similar.
Check. A 2 by 3 rectangle and a 4 by 6 rectangle are similar (both scale factors are 2), so some rectangles are similar. ✓
Watch out. Equal angles are not enough for shapes with more than three sides; the sides must also be in proportion.
Method. Triangles are similar if their angles are equal. Work out the third angle of each triangle.
- Third angle of ABC. The angles in a triangle add up to 180°.180 − 40 − 60 = 80°, so ABC has angles 40°, 60°, 80°.
- Third angle of PQR. The same rule.180 − 60 − 80 = 40°, so PQR has angles 60°, 80°, 40°.
- Compare. Both triangles have angles of 40°, 60° and 80°.The angles are equal, so the triangles are similar.
Check. Two equal angles are enough (the third must be equal). Here we have three. ✓
Watch out. You do not need the side lengths to show triangles are similar, but you do need them to find the scale factor.
Method. The scale factor is the same for both pairs of sides. Write it two ways and make an equation.
- Write the ratio of matching sides in each triangle. Larger ÷ smaller gives the same scale factor for both pairs.x ÷ 6 = (x + 6) ÷ 8
- Multiply both sides by 6 and by 8 (cross-multiply). This removes the fractions.8x = 6(x + 6)
- Expand the bracket. 6 × x = 6x and 6 × 6 = 36.8x = 6x + 36
- Solve. Subtract 6x from both sides, then divide by 2.2x = 36
x = 18
Check. The larger triangle has sides 18 and 24. 18 ÷ 6 = 3 and 24 ÷ 8 = 3. ✓
Watch out. Write the equation so that both sides compare the same triangles (larger over smaller).
Method. Show that all three angles of triangle ADE equal the angles of triangle ABC, giving a reason for each.
- Angle at A. Both triangles contain the same angle at A.Angle DAE = angle BAC (common angle)
- Angle ADE. DE is parallel to BC, and AB crosses both lines, so these are corresponding angles.Angle ADE = angle ABC (corresponding angles)
- Angle AED. AC also crosses both parallel lines.Angle AED = angle ACB (corresponding angles)
- Conclude. All three pairs of angles are equal.Triangles ADE and ABC are similar (equal angles).
Check. Using the angle sum of 180°, if two pairs of angles are equal the third pair must be equal as well. ✓
Watch out. Give a reason for every pair of equal angles, such as “corresponding angles” or “common angle”.
Method. Check that the scale factor is the same for both pairs of matching sides. Match long side with long side, and short side with short side.
- Long sides. A4 long side ÷ A5 long side.297 ÷ 210 = 1.414… = 1.41 (3 s.f.)
- Short sides. A4 short side ÷ A5 short side.210 ÷ 148.5 = 1.414… = 1.41 (3 s.f.)
- Compare. The scale factors agree (to 3 significant figures).A4 and A5 are similar, with scale factor about 1.41.
Check. In fact the sides of A-series paper are in the ratio 1 : √2 = 1 : 1.414…, which makes every size similar to the next. ✓
Watch out. Match long side to long side. Dividing long by short on one sheet and comparing with the other also works.
Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.
Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.
- Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.Known pair: 10 cm (small shape) and 25 cm (large shape).
The side we want is the one marked x. - Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.25 cm ÷ 10 cm = 52
So the large shape is 52 times as big as the small shape. - Use the scale factor to find x. x is on the large shape and we know the matching side on the small shape. Going from small to large we multiply by the scale factor.x = 8 cm × 52 = 20 cm
- Write the answer with its unit. Lengths need a unit.x = 20 cm
Check. Ratio check: 20 cm ÷ 8 cm = 52, the same scale factor as before. ✓
Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.