Congruent shapes
Two shapes are congruent if they are exactly the same shape and the same size. One can be placed on top of the other after a rotation, reflection or translation. Their corresponding sides are equal and their corresponding angles are equal.
Remember
- Congruence tests for triangles: SSS, SAS, ASA (AAS) and RHS.
- RHS applies only to right-angled triangles: equal Right angle, Hypotenuse and one Side.
Similar shapes
Two shapes are similar if they have the same shape but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are in the same ratio. This constant ratio is the scale factor k.
Key idea
Length scale factor = k → Area scale factor = k² → Volume scale factor = k³.
Working with the scale factor
To find a missing length, first find k by dividing a known pair of corresponding sides, then multiply.
Worked example
Triangle A has sides 3 cm, 4 cm, 5 cm. Similar triangle B has shortest side 9 cm. Then k = 9 ÷ 3 = 3, so the longest side of B = 5 × 3 = 15 cm. If the area of A is 6 cm², the area of B = 6 × k² = 6 × 9 = 54 cm².
The same idea extends to solids: if two similar solids have length scale factor 2, their volumes are in the ratio 1 : 8.
Take care not to confuse the two ideas: congruent figures are a special case of similar figures in which the scale factor is exactly 1. In an exam, always list the equal sides or angles you use, and quote the correct congruence condition (SSS, SAS, ASA or RHS) as your reason.