Chapter 5 of 8

All 8 chapters
  1. 1Reflections and rotations, readunderstood
  2. 2Translations and combined moves, readunderstood
  3. 3Enlargements, readunderstood
  4. 4Congruent triangles, readunderstood
  5. 5Similar shapes, readunderstood
  6. 6Similar areas and volumes, readunderstood
  7. 7Arcs and sectors, readunderstood
  8. 8Angle rules in circles, readunderstood

M4.4.5 Chapter 5

Similar shapes

A 3 m pole has a 2 m shadow. At the same time a tree has a 6 m shadow. How tall is the tree?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Show two triangles are similar from their angles
  • Find a missing side using a scale factor

Two shapes are similar when one is an enlargement of the other. The enlargement may also turn or flip the shape. Similar shapes have the same form but not always the same size. Each pair of matching angles is equal. Each pair of matching sides is in the same ratio.

Two similar triangles drawn to the same scale. The small triangle has sides of 3 cm, 4 cm and 5 cm. The large triangle has sides of 6 cm, 8 cm and 10 cm. The matching angles carry the same number of arcs. Corners are named A, B, C, D, E and F.3 cm4 cm5 cmABC6 cm8 cm10 cmDEF
The sides of the large triangle are 2 times the matching sides of the small one.

Take the triangles above. Divide each large side by its matching small side: 6 ÷ 3 = 2, 8 ÷ 4 = 2 and 10 ÷ 5 = 2. All three answers are 2. That number is the scale factor from the small triangle to the large one.

To show two triangles are similar, you only need their angles. The three angles of a triangle add up to 180°. If two angles match, the third is found by taking them from 180°, so it matches too.

Two triangles of different sizes. Each has a 40° angle at corner J and a 60° angle at corner K, marked with one and two arcs. The corners are named J, K, L, P, Q and R.40°60°JKL40°60°PQR
Two angles match, so the third angle is 80° in both. The triangles are similar.

This test works for triangles. For shapes with more sides, equal angles are not enough. A rectangle that is 2 cm by 4 cm and one that is 2 cm by 8 cm have only right angles. Their sides are not in the same ratio, so they are not similar.

Now use the scale factor to find a missing side. Pair up the matching sides. Divide a large side by its small match to get the scale factor. Then multiply any small side by it.

Two similar right angled triangles. Triangle ABC has sides of 3 cm and 4 cm and a hypotenuse of 5 cm. Triangle DEF has a side of 9 cm matching the 3 cm side, a side of 12 cm matching the 4 cm side, and a side marked x. The corners are named A, B, C, D, E and F.3 cm4 cm5 cmABC9 cm12 cmxDEF
Triangle DEF is similar to triangle ABC. Side x matches the 5 cm side.

Worked example

Triangle ABC has sides of 3 cm, 4 cm and 5 cm. Triangle DEF is similar. Its side of 9 cm matches the 3 cm side. How long is the side that matches 5 cm?

  1. Find the scale factor. Divide the large side by its small match: 9 ÷ 3 = 3.
  2. Multiply the 5 cm side by the scale factor: 5 × 3 = 15.
  3. The side that matches 5 cm is 15 cm long.

Answer: 15 cm

Similar triangles solve real problems. On level ground, an upright pole and an upright tree both make a right angle with the ground. At the same time the sun is at the same angle for both. So two angles match, and the shadow triangles are similar.

Two right angled triangles. The pole triangle ABC, on the left, has a pole BC of 3 m standing up and a shadow AC of 2 m lying flat. The tree triangle DEF, on the right, has a tree EF with its height marked h and a shadow DF of 6 m. The right angle is at C and at F, at the bottom left of each triangle.3 m2 mABCh6 mDEF
The pole triangle ABC is on the left and the tree triangle DEF is on the right. The sun makes the same angle with the ground for both.

Worked example

A pole 3 m high has a shadow 2 m long. At the same time a tree has a shadow 6 m long. How high is the tree?

  1. The pole and the tree are both upright, and the sun is at the same angle. The shadow triangles have equal angles, so they are similar.
  2. The scale factor from the pole shadow to the tree shadow is 6 ÷ 2 = 3.
  3. The tree is 3 times as high as the pole: 3 × 3 = 9 m.

Answer: 9 m

See it three ways

With things

Print a photo small and then large. Every person in it looks the same, only bigger. Each length in the large print is the same number of times its length in the small one.

As a picture

Two similar triangles, the large one 1.5 times the small one. The small triangle has sides of 4 cm, 5 cm and 6 cm. The large triangle has sides of 6 cm, 7.5 cm and 9 cm. Corners are named G, H, I, M, N and O.4 cm5 cm6 cmGHI6 cm7.5 cm9 cmMNO

In symbols

Scale factor = 6 ÷ 4 = 1.5. Then 5 × 1.5 = 7.5 and 6 × 1.5 = 9. Every side is multiplied by 1.5.

Try it

Draw two triangles with the same angles

You need: a ruler, a protractor, a pencil, plain paper.

  1. Draw a line 6 cm long. At one end draw an angle of 40°. At the other end draw an angle of 60°. Extend the two lines until they meet.
  2. Draw a second line, 9 cm long. Use the same two angles, 40° and 60°.
  3. Measure the sides that lie opposite the 40° angle in each triangle. Divide the larger by the smaller.
  4. Do the same for the sides opposite the 60° angle.

What you will see: Both answers are 1.5, the same as 9 ÷ 6. Triangles with the same angles have sides in the same ratio.

If you cannot try it, this is what happens.
Two triangles with a 40° angle at corner A and a 60° angle at corner B, marked with one and two arcs. The first has a base AB of 6 cm and the second a base DE of 9 cm. The corners are named A, B, C, D, E and F.40°60°6 cmABC40°60°9 cmDEF

A wrong idea many people have

If one side grows from 3 cm to 9 cm, then every side grows by 6 cm.

Why it feels right: Adding is how we often compare two numbers. The number 9 is 6 more than 3.

The test that breaks it: Add 6 to the 5 cm side and it becomes 11 cm. The sides would be 9 cm, 10 cm and 11 cm. These are not in the same ratio as 3 cm, 4 cm and 5 cm, so the shape changes.

What is true: Similar shapes grow by multiplying. The scale factor is 9 ÷ 3 = 3, so the 5 cm side becomes 5 × 3 = 15 cm.

Key words

similar
having the same form, with equal matching angles and matching sides in the same ratio. One shape is an enlargement of the other.

Back to your guess

The answer is C: 9 m, because the shadow is 3 times as long.

The pole and the tree both stand upright on level ground, and the sun is at the same angle for both. So their shadow triangles have the same shape. The tree shadow is 3 times the pole shadow, so the tree is 3 times the pole: 9 m.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Two angles are marked in each triangle. Are the triangles similar?

    Two triangles. The first has angles of 50° and 60° marked at corners A and B. The second has angles of 60° and 70° marked at corners D and E. The corners are named A, B, C, D, E and F.50°60°ABC60°70°DEF
    Tap your answer.

    1. Yes, that is right. The first triangle has a third angle of 180° − 50° − 60° = 70°. The second has 180° − 60° − 70° = 50°. Both triangles have the angles 50°, 60° and 70°, so they are similar.

    2. Not this one. You compared only the marked angles. Work out the third angle of each triangle too. The three angles of a triangle add up to 180°. Go to M4.4.5. Try another.

    3. Not this one. Not every pair of triangles is similar. The angles must match, and being triangles is not enough. Go to M4.4.5. Try another.

    See the answer

    Yes. The missing angles are 70° and 50°, so all three angles match.

    The first triangle has a third angle of 180° − 50° − 60° = 70°. The second has 180° − 60° − 70° = 50°. Both triangles have the angles 50°, 60° and 70°, so they are similar.

    • If you chose B: You compared only the marked angles. Work out the third angle of each triangle too. The three angles of a triangle add up to 180°. Go to M4.4.5.
    • If you chose C: Not every pair of triangles is similar. The angles must match, and being triangles is not enough. Go to M4.4.5.

  2. Triangles ABC and DEF are similar. AB = 4 cm matches DE = 10 cm. BC = 6 cm. How long is EF?

    Two similar triangles. Triangle ABC has a side of 6 cm and a side AB of 4 cm. Triangle DEF has a side DE of 10 cm and a side EF marked x. The matching angles carry the same number of arcs. The corners are named A, B, C, D, E and F.6 cm4 cmABCx10 cmDEF
    Tap your answer.

    1. Not this one. You added. DE is 6 cm more than AB, so you added 6 to BC. Similar shapes grow by multiplying by the scale factor. Go to M4.4.5. Try another.

    2. Not this one. You used the scale factor the wrong way round. DEF is the larger triangle, so EF must be longer than BC. Go to M4.4.5. Try another.

    3. Yes, that is right. The scale factor is 10 ÷ 4 = 2.5. Multiply BC by it: 6 × 2.5 = 15 cm.

    See the answer

    15 cm

    The scale factor is 10 ÷ 4 = 2.5. Multiply BC by it: 6 × 2.5 = 15 cm.

    • If you chose A: You added. DE is 6 cm more than AB, so you added 6 to BC. Similar shapes grow by multiplying by the scale factor. Go to M4.4.5.
    • If you chose B: You used the scale factor the wrong way round. DEF is the larger triangle, so EF must be longer than BC. Go to M4.4.5.

  3. Triangle PQR is a smaller copy of triangle ABC. AB = 12 cm matches PQ = 8 cm. BC = 18 cm. How long is QR?

    Tap your answer.

    1. Not this one. You used the scale factor from the small triangle to the large one. PQR is smaller, so QR is shorter than BC. Go to M4.4.5. Try another.

    2. Yes, that is right. The scale factor from ABC to PQR is 8 ÷ 12 = 2/3. Multiply BC by it: 18 × 2/3 = 12 cm.

    3. Not this one. You took 4 cm away because 8 is 4 less than 12. Similar shapes shrink by multiplying, not by taking away. Go to M4.4.5. Try another.

    See the answer

    12 cm

    The scale factor from ABC to PQR is 8 ÷ 12 = 2/3. Multiply BC by it: 18 × 2/3 = 12 cm.

    • If you chose A: You used the scale factor from the small triangle to the large one. PQR is smaller, so QR is shorter than BC. Go to M4.4.5.
    • If you chose C: You took 4 cm away because 8 is 4 less than 12. Similar shapes shrink by multiplying, not by taking away. Go to M4.4.5.

  4. Which pair of shapes must be similar?

    Tap your answer.

    1. Not this one. Two rectangles have equal angles, but their sides need not be in the same ratio. A 2 cm by 4 cm rectangle and a 2 cm by 8 cm rectangle are not similar. Go to M4.4.5. Try another.

    2. Not this one. One equal angle is not enough. The other angles can differ, so the triangles can have different forms. Go to M4.4.5. Try another.

    3. Yes, that is right. Every square has four right angles and four equal sides. Any two squares have equal angles, and their sides are in the same ratio, so they are similar.

    4. Not this one. Kites can be long and thin or short and fat. Two kites need not have equal angles or sides in the same ratio. Go to M4.4.5. Try another.

    See the answer

    Two squares

    Every square has four right angles and four equal sides. Any two squares have equal angles, and their sides are in the same ratio, so they are similar.

    • If you chose A: Two rectangles have equal angles, but their sides need not be in the same ratio. A 2 cm by 4 cm rectangle and a 2 cm by 8 cm rectangle are not similar. Go to M4.4.5.
    • If you chose B: One equal angle is not enough. The other angles can differ, so the triangles can have different forms. Go to M4.4.5.
    • If you chose D: Kites can be long and thin or short and fat. Two kites need not have equal angles or sides in the same ratio. Go to M4.4.5.

  5. A 2 m stick has a shadow 3 m long. At the same time a building has a shadow 24 m long. How tall is the building?

    Tap your answer.

    1. Not this one. You used the scale factor the wrong way round, 3 ÷ 2. The building shadow is 8 times the stick shadow, so use 24 ÷ 3. Go to M4.4.5. Try another.

    2. Yes, that is right. The scale factor is 24 ÷ 3 = 8. The building is 8 times as tall as the stick: 2 × 8 = 16 m.

    3. Not this one. You added. The building shadow is 21 m longer, and you added 21 to the stick. Multiply by the scale factor instead. Go to M4.4.5. Try another.

    See the answer

    16 m

    The scale factor is 24 ÷ 3 = 8. The building is 8 times as tall as the stick: 2 × 8 = 16 m.

    • If you chose A: You used the scale factor the wrong way round, 3 ÷ 2. The building shadow is 8 times the stick shadow, so use 24 ÷ 3. Go to M4.4.5.
    • If you chose C: You added. The building shadow is 21 m longer, and you added 21 to the stick. Multiply by the scale factor instead. Go to M4.4.5.

  6. From an earlier chapter

    Triangle ABC has AB = AC and angle B = 65°. What is angle A, in degrees?

    See the answer

    50

    The base angles of an isosceles triangle are equal, so angle C is also 65°. Angle A = 180° − 65° − 65° = 50°.

  7. Triangle ABC has sides of 5 cm and 7 cm. A similar triangle has a side of 15 cm that matches the 5 cm side. How long is the side that matches 7 cm, in centimetres?

    See the answer

    21

    The scale factor is 15 ÷ 5 = 3. Multiply 7 by it: 7 × 3 = 21 cm.

  8. A triangle has sides of 12 cm and 15 cm. A similar, smaller triangle has a side of 4 cm that matches the 12 cm side. How long is the side that matches 15 cm, in centimetres?

    See the answer

    5

    The scale factor is 4 ÷ 12 = 1/3. Multiply 15 by it: 15 ÷ 3 = 5 cm.

  9. A photo is 12 cm wide and 8 cm high. It is enlarged so that it is 30 cm wide. How high is the enlarged photo, in centimetres?

    See the answer

    20

    An enlarged photo is similar to the original. The scale factor is 30 ÷ 12 = 2.5. Multiply 8 by it: 8 × 2.5 = 20 cm.

Keep it

Copy this into your exercise book.

Similar shapes have equal angles and matching sides in the same ratio. Two equal angles make triangles similar. Scale factor = a side of one ÷ the matching side of the other. Multiply.

Draw: A small and a large triangle with the same angle arcs, and the scale factor written between them.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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