EDEXCEL GCSE MATHS · HIGHER
More similarity
Similarity and congruence · Lesson 4 of 5
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
|
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.
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. |