Congruence and Similarity

Lengths, Areas and Volumes

Finding a missing length

Pick a pair of sides whose lengths are both known, one from each shape, and divide one by the other to get the scale factor
Multiply to move towards the larger shape and divide to move towards the smaller one
Choosing the direction that goes small to large keeps the scale factor above 1, which many find easier to check
Pair the sides correctly before dividing: corresponding sides sit between the same pair of equal angles
Sanity check the answer, since a length on the larger shape must come out larger

Exam tip

Write the two matching sides as an equal pair of ratios before solving anything: that line alone earns the method mark. Check the direction at the end — going to the larger shape the answer must be bigger, and going to the smaller it must be smaller.

Worked example

A missing length on a similar shape

Two triangles are mathematically similar. A side of 8 cm on the smaller corresponds to 12 cm on the larger. Another side of the smaller is 9 cm. Find the corresponding side on the larger.

Solution:

Take the scale factor from the pair of sides that are both known
Scale factor = 12 ÷ 8 = 1.5
The missing side is on the larger shape, so multiply
9 × 1.5 = 13.5 cm
Check: 13.5 ÷ 9 = 1.5, matching the scale factor from the first pair