EDEXCEL GCSE MATHS · HIGHER
More similarity
Similarity and congruence · Lesson 4 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 6 to 9?
1.5
2. Work out 1.5².
2.25
3. What do corresponding angles on parallel lines do?
They are equal
4. Work out √(27/12).
1.5
5. If k = 3, what is k²?
9
Learning Objectives
1. Find similar triangles created by parallel lines.
2. Find missing lengths using scale factors.
3. Use the area scale factor k².
4. Work backwards from an area ratio to a length ratio.
A Triangle Cut by a Parallel Line
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A line parallel to one side of a triangle makes a smaller similar triangle. |
Triangle ABC with a line DE parallel to BC, showing that the small triangle ADE is similar to the whole triangle ABC.
Parallel Lines
If DE is parallel to BC, then triangle ADE is similar to triangle ABC.
The corresponding angles are equal, and the angle at A is common.
Finding a Length
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In triangle ABC, DE is parallel to BC. AD = 4 cm, DB = 2 cm and DE = 5 cm. Find BC. |
1. AB = AD + DB
4 + 2 = 6
2. Scale factor from ADE to ABC
6/4 = 1.5
3. BC = DE × 1.5
5 × 1.5
Answer: BC = 7.5 cm
Watch the Sides
The whole side, not just the piece, is the corresponding length.
▸ Use AB, not DB. The similar triangles are ADE and ABC, so AD corresponds to AB.
▸ Add the parts. AB = AD + DB.
▸ Label the two triangles. Write them in matching order: ADE and ABC.
Area Scale Factor
If two shapes are similar with length scale factor k, their areas are in the ratio 1 : k².
Doubling all lengths multiplies the area by 4; tripling multiplies it by 9.
Finding an Area
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Two similar rectangles have lengths 12 cm and 18 cm. The area of the smaller is 40 cm². Find the area of the larger. |
1. Length scale factor
18/12 = 1.5
2. Area scale factor
1.5² = 2.25
3. Area of the larger
40 × 2.25
Answer: 90 cm²
From Areas to Lengths
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Two similar triangles have areas 12 cm² and 27 cm². Find the ratio of their lengths in its simplest form. |
1. Ratio of areas
12 : 27 = 4 : 9
2. Take the square root
√4 : √9 = 2 : 3
Answer: The ratio of the lengths is 2 : 3.
Scale Factors
For similar shapes with length scale factor k.
|
Measure |
Scale factor |
Example, k = 3 |
|---|---|---|
|
Lengths |
k |
3 |
|
Areas |
k² |
9 |
|
Volumes |
k³ |
27 |
Key Terms
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Similar triangles Triangles with equal angles and proportional sides. |
Scale factor The multiplier for lengths. |
|
Area scale factor The square of the length scale factor. |
Parallel Lines that never meet. |
|
Corresponding angles Equal angles in matching positions on parallel lines. |
Ratio A comparison of two quantities. |
Your Task: Shadows and Similar Triangles
12 minutes
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A 1.5 m post casts a shadow of 2 m at the same time as a tree casts a shadow of 12 m. Draw two similar right-angled triangles and find the height of the tree. If the post's triangle has area 1.5 m², find the area of the tree's triangle. 1. Draw both triangles. 2. Find the length scale factor. 3. Square it for area. |
A good answer shows: Scale factor 12 ÷ 2 = 6, so the tree is 1.5 × 6 = 9 m tall. The area scale factor is 6² = 36, so the area is 1.5 × 36 = 54 m².
Note: Link to how surveyors measure heights.
Can I...?
☐ Spot similar triangles from parallel lines.
☐ Use the whole side, not part of it.
☐ Find a missing length.
☐ Use k² for areas.
☐ Find an area from a length ratio.
☐ Find a length ratio from an area ratio.
☐ Write the ratio in simplest form.
☐ Explain my reasoning.
Summary
✓ A line parallel to a side makes a similar triangle.
✓ Match corresponding sides in the same order.
✓ Area scale factor = k².
✓ Take a square root to go from area ratio to length ratio.
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EXAM FOCUS In triangle ABC, DE is parallel to BC. AD = 4 cm, DB = 2 cm and DE = 5 cm. Work out the length of BC. (3 marks) The similar triangles are ADE and ABC. Use AB = AD + DB in the scale factor, not DB alone. |
Exam Practice: More Similarity
Answer all questions. Show your working. · 30 minutes
▸ Question 1 · 3 marks · Work out. In triangle ABC, DE is parallel to BC. AD = 4 cm, DB = 2 cm and DE = 5 cm. Work out the length of BC.
▸ Question 2 · 3 marks · Work out. Two similar rectangles have lengths of 12 cm and 18 cm. The area of the smaller rectangle is 40 cm². Work out the area of the larger…
▸ Question 3 · 3 marks · Work out. Two similar triangles have areas of 12 cm² and 27 cm². Work out the ratio of the lengths of their corresponding sides. Give your answer in…
▸ Question 4 · 3 marks · Work out. Two similar shapes have corresponding lengths in the ratio 3 : 5. The area of the smaller shape is 36 cm². Work out the area of the larger…
▸ Question 5 · 3 marks · Work out. A 1.5 m post casts a shadow 2 m long at the same time as a tree casts a shadow 12 m long. Work out the height of the tree.
▸ Question 6 · 3 marks · Explain. Explain why triangle ADE is similar to triangle ABC when DE is parallel to BC.
Question 1 · 3 marks · Work out
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In triangle ABC, DE is parallel to BC. AD = 4 cm, DB = 2 cm and DE = 5 cm. Work out the length of BC. (3 marks) |
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Question 1 · mark scheme
3 marks available. Award a mark for each point made.
▸ Similar triangles ADE and ABC. M1
▸ Scale factor 6/4. M1
▸ 7.5. A1
▸ Model answer. Triangles ADE and ABC are similar. AB = 6, so the scale factor is 6/4 = 1.5. BC = 5 × 1.5 = 7.5 cm.
Question 2 · 3 marks · Work out
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“Two similar rectangles have lengths of 12 cm and 18 cm. The area of the smaller rectangle is 40 cm². Work out the area of the larger rectangle.” |
HOW TO ANSWER IT Command word: Work out. Worth 3 marks, so plan before writing.
Question 2 · mark scheme
3 marks available. Award a mark for each point made.
▸ Length scale factor 1.5. M1
▸ 1.5² = 2.25. M1
▸ 90. A1
▸ Model answer. The length scale factor is 18/12 = 1.5, so the area scale factor is 1.5² = 2.25. The area is 40 × 2.25 = 90 cm².
Question 3 · 3 marks · Work out
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“Two similar triangles have areas of 12 cm² and 27 cm². Work out the ratio of the lengths of their corresponding sides. Give your answer in the form 1 : n or in its simplest form.” |
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.
▸ 4 : 9. M1
▸ Square roots. M1
▸ 2 : 3. A1
▸ Model answer. The area ratio is 12 : 27 = 4 : 9. The length ratio is √4 : √9 = 2 : 3.
Question 4 · 3 marks · Work out
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“Two similar shapes have corresponding lengths in the ratio 3 : 5. The area of the smaller shape is 36 cm². Work out the area of the larger shape.” |
HOW TO ANSWER IT Command word: Work out. Worth 3 marks, so plan before writing.
Question 4 · mark scheme
3 marks available. Award a mark for each point made.
▸ 9 : 25. M1
▸ 36 × 25/9. M1
▸ 100. A1
▸ Model answer. The area ratio is 3² : 5² = 9 : 25. The area is 36 × 25/9 = 100 cm².
Question 5 · 3 marks · Work out
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“A 1.5 m post casts a shadow 2 m long at the same time as a tree casts a shadow 12 m long. Work out the height of the tree.” |
HOW TO ANSWER IT Command word: Work out. Worth 3 marks, so plan before writing.
Question 5 · mark scheme
3 marks available. Award a mark for each point made.
▸ Scale factor 6. M1
▸ 1.5 × 6. M1
▸ 9. A1
▸ Model answer. The triangles are similar. The scale factor is 12 ÷ 2 = 6. The tree is 1.5 × 6 = 9 m tall.
Question 6 · 3 marks · Explain
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“Explain why triangle ADE is similar to triangle ABC when DE is parallel to BC.” |
HOW TO ANSWER IT Command word: Explain. Worth 3 marks, so plan before writing.
Question 6 · mark scheme
3 marks available. Award a mark for each point made.
▸ Corresponding angles on parallel lines. M1
▸ Angle A common. M1
▸ All angles equal, so similar. C1
▸ Model answer. Angle ADE = angle ABC and angle AED = angle ACB (corresponding angles on parallel lines). Angle A is common. All three angles are equal, so the triangles are similar.