Similarity

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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