EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Similarity
Similarity and congruence · Lesson 3 of 5
Teacher copy - includes the notes for whoever is teaching from it.
Warm-up
Answer each one, then check.
1. What is the scale factor from 4 cm to 10 cm?
2.5
2. Write 3:5 as a fraction of the total.
⅜ and ⅝
3. Work out 9 × 1.5.
13.5
4. What does congruent mean?
Same shape and size
5. What do the angles in a triangle add up to?
180°
Learning Objectives
1. Explain what similar means.
2. Find scale factors between similar shapes.
3. Find missing lengths in similar shapes.
4. Show that two triangles are similar using equal angles.
Similar
Two shapes are similar if they have the same shape but not necessarily the same size: their angles are equal and their lengths are in the same ratio.
The scale factor is the ratio of corresponding lengths.
Similar Triangles
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Every side is multiplied by the same scale factor. |
A small triangle with sides 3, 4 and 5 and a larger similar triangle with sides 6, 8 and 10, joined by a scale factor of 2.
Facts About Similar Shapes
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Angles All corresponding angles are equal. |
Lengths All corresponding lengths are in the same ratio. |
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Scale factor The number you multiply by to go from one shape to the other. |
Congruent is a special case A scale factor of 1. |
Finding a Missing Length
Use the scale factor.
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1 Match the corresponding sides Find a pair where both lengths are known |
2 Work out the scale factor Larger ÷ smaller |
3 Multiply or divide Depending on the direction you go |
4 Check The answer should look sensible |
A Missing Side
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Triangles ABC and DEF are similar. AB = 6 cm, BC = 10 cm and AC = 8 cm. DE = 9 cm. Find EF and DF. |
1. Scale factor
DE/AB = 9/6 = 1.5
2. EF corresponds to BC
10 × 1.5 = 15
3. DF corresponds to AC
8 × 1.5 = 12
Answer: EF = 15 cm and DF = 12 cm.
Similar Rectangles
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Rectangle A is 4 cm by 6 cm. Rectangle B is similar to A and 10 cm long on its shorter side. Find the other side of B. |
1. Scale factor
10 ÷ 4 = 2.5
2. Other side
6 × 2.5
Answer: 15 cm
Showing Triangles Are Similar
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In triangle PQR, angle P = 50° and angle Q = 60°. In triangle XYZ, angle X = 50° and angle Z = 70°. Show that the triangles are similar. |
1. Find angle R
180 − 50 − 60 = 70°
2. Find angle Y
180 − 50 − 70 = 60°
3. All three angles match
50°, 60°, 70°
Answer: The triangles have equal angles, so they are similar.
Similar or Congruent?
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SIMILAR |
CONGRUENT |
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▸ Same shape, possibly different size. ▸ Equal angles, sides in the same ratio. ▸ Scale factor may be any number. |
▸ Same shape and same size. ▸ Equal angles and equal sides. ▸ Scale factor 1. |
Key Terms
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Similar The same shape but a different size. |
Scale factor The multiplier that takes one length to the corresponding length. |
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Corresponding sides Sides in matching positions. |
Ratio A comparison of two quantities. |
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Enlargement A transformation that produces similar shapes. |
Equiangular Having equal angles. |
Your Task: Photo Enlargement
10 minutes
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A photograph is 12 cm by 8 cm. It is enlarged to fit a frame that is 30 cm wide. Find the height of the enlargement and the scale factor. Then say whether a 30 cm by 18 cm frame would suit the photograph. 1. Find the scale factor. 2. Apply it to the other side. 3. Compare ratios. |
A good answer shows: Scale factor 30 ÷ 12 = 2.5, so the height is 8 × 2.5 = 20 cm. A 30 by 18 frame is not similar to 12 by 8, because 18 ÷ 8 = 2.25 ≠ 2.5, so the picture would be distorted.
Note: Link to aspect ratio on screens.
Can I...?
☐ Say what similar means.
☐ Find a scale factor.
☐ Find a missing length.
☐ Use scale factors greater than 1.
☐ Use scale factors less than 1.
☐ Show triangles are similar by angles.
☐ Identify corresponding sides.
☐ Explain the difference from congruent.
Summary
✓ Similar shapes have equal angles and proportional sides.
✓ Scale factor = new length ÷ old length.
✓ Match corresponding sides carefully.
✓ Two equal angles are enough to show triangles are similar.
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EXAM FOCUS Triangles ABC and DEF are similar. AB = 6 cm, BC = 10 cm, DE = 9 cm. Work out the length of EF. (3 marks) Find the scale factor from a pair of sides you know, then use it on the side you need. Make sure the sides really correspond. |
Exam Practice: Similarity
Answer all questions. Show your working. · 25 minutes
▸ Question 1 · 3 marks · Work out. The diagram shows two similar triangles, ABC and DEF. AB = 6 cm, BC = 10 cm, AC = 8 cm and DE = 9 cm. Work out the length of EF.
▸ Question 2 · 2 marks · Work out. For the triangles in the last question, work out the length of DF.
▸ Question 3 · 3 marks · Work out. Rectangle A is 4 cm by 6 cm. Rectangle B is similar to rectangle A. The shorter side of rectangle B is 10 cm. Work out the length of the…
▸ Question 4 · 3 marks · Show that. In triangle PQR, angle P = 50° and angle Q = 60°. In triangle XYZ, angle X = 50° and angle Z = 70°. Show that triangles PQR and XYZ are…
▸ Question 5 · 2 marks · Explain. A photograph is 12 cm by 8 cm. It is enlarged so that its width is 30 cm. Work out the height of the enlarged photograph.
▸ Question 6 · 2 marks · Decide. Are a 3 cm by 5 cm rectangle and a 9 cm by 14 cm rectangle similar? Give a reason.
Question 1 · 3 marks · Work out
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The diagram shows two similar triangles, ABC and DEF. AB = 6 cm, BC = 10 cm, AC = 8 cm and DE = 9 cm. Work out the length of EF. (3 marks) |
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Question 1 · mark scheme
3 marks available. Award a mark for each point made.
▸ Scale factor 1.5. M1
▸ 10 × 1.5. M1
▸ 15. A1
▸ Model answer. The scale factor is 9/6 = 1.5. EF = 10 × 1.5 = 15 cm.
Question 2 · 2 marks · Work out
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“For the triangles in the last question, work out the length of DF.” |
HOW TO ANSWER IT Command word: Work out. Worth 2 marks, so plan before writing.
Question 2 · mark scheme
2 marks available. Award a mark for each point made.
▸ 8 × 1.5. M1
▸ 12. A1
▸ Model answer. DF = 8 × 1.5 = 12 cm.
Question 3 · 3 marks · Work out
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“Rectangle A is 4 cm by 6 cm. Rectangle B is similar to rectangle A. The shorter side of rectangle B is 10 cm. Work out the length of the longer side of rectangle B.” |
HOW TO ANSWER IT Command word: Work out. Worth 3 marks, so plan before writing.
Question 3 · mark scheme
3 marks available. Award a mark for each point made.
▸ Scale factor 2.5. M1
▸ 6 × 2.5. M1
▸ 15. A1
▸ Model answer. The scale factor is 10 ÷ 4 = 2.5. The longer side is 6 × 2.5 = 15 cm.
Question 4 · 3 marks · Show that
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“In triangle PQR, angle P = 50° and angle Q = 60°. In triangle XYZ, angle X = 50° and angle Z = 70°. Show that triangles PQR and XYZ are similar.” |
HOW TO ANSWER IT Command word: Show that. Worth 3 marks, so plan before writing.
Question 4 · mark scheme
3 marks available. Award a mark for each point made.
▸ Finding a third angle. M1
▸ Finding the other third angle. M1
▸ All angles equal, so similar. C1
▸ Model answer. Angle R = 180 − 50 − 60 = 70° and angle Y = 180 − 50 − 70 = 60°. Both triangles have angles 50°, 60° and 70°, so they are similar.
Question 5 · 2 marks · Explain
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“A photograph is 12 cm by 8 cm. It is enlarged so that its width is 30 cm. Work out the height of the enlarged photograph.” |
HOW TO ANSWER IT Command word: Explain. Worth 2 marks, so plan before writing.
Question 5 · mark scheme
2 marks available. Award a mark for each point made.
▸ 30 ÷ 12. M1
▸ 20. A1
▸ Model answer. The scale factor is 30 ÷ 12 = 2.5. The height is 8 × 2.5 = 20 cm.
Question 6 · 2 marks · Decide
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“Are a 3 cm by 5 cm rectangle and a 9 cm by 14 cm rectangle similar? Give a reason.” |
HOW TO ANSWER IT Command word: Decide. Worth 2 marks, so plan before writing.
Question 6 · mark scheme
2 marks available. Award a mark for each point made.
▸ Comparing the ratios. M1
▸ Not similar with a reason. C1
▸ Model answer. 9 ÷ 3 = 3 but 14 ÷ 5 = 2.8. The scale factors are different, so the rectangles are not similar.