Similar shapes have the same angles and corresponding sides in the same ratio (scale factor). You can use similarity to find missing lengths in 2D shapes and real-life problems.
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Key facts to remember
1Two shapes are similar if one is an enlargement of the other.
2Corresponding angles in similar shapes are equal.
3Scale factor = (length in image) ÷ (corresponding length in original).
4To find a missing length: identify the scale factor, then multiply or divide.
5Similar triangles often arise when parallel lines cut across a triangle.
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Formulas
Scale factor
SF = new length ÷ original length
Missing length
New length = original length × SF
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Worked examples
Example 1
Two similar triangles have corresponding sides of 6 cm and 9 cm. The shorter triangle has a side of 8 cm. Find the corresponding side in the larger triangle.
Working
Scale factor = 9 ÷ 6 = 1.5
Missing side = 8 × 1.5 = 12 cm
Answer12 cm
Example 2
A 1.8 m tall person casts a shadow of 2.4 m. A tree casts a shadow of 10 m at the same time. Find the height of the tree.
Working
The triangles formed are similar (same sun angle)
Scale factor = 10 ÷ 2.4 = 25/6
Tree height = 1.8 × (25/6) = 7.5 m
Answer7.5 m
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Common mistakes
✗Comparing non-corresponding sides to find the scale factor.
✗Adding the scale factor instead of multiplying.
✗Confusing similarity (same shape, different size) with congruence (identical shape and size).
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Exam tips
✓Mark corresponding vertices with the same letters to identify matching sides correctly.
✓Check: the scale factor should be the same for all pairs of corresponding sides.
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