Find a length

A standard method from Similar triangles: the steps and worked examples.

Steps

Write the matching sides as a ratio, same triangle on top: AD/AB = DE/BC = AE/AC.
Put in the numbers and solve.
Check: the larger triangle must give the larger length.

Worked examples

Example 1: In △ABC, D is on AB and E is on AC, with DE // BC. AD = 4 cm, DB = 2 cm and DE = 6 cm. Find BC and the ratio of the areas of △ADE and △ABC.

∠ADE = ∠ABC     (corr. ∠s, DE // BC)
∠AED = ∠ACB     (corr. ∠s, DE // BC)
∠A  is common
∴ △ADE ~ △ABC   (AAA)

AB = 4 + 2 = 6
DE/BC = AD/AB
6/BC  = 4/6
BC    = 9 cm

area ratio = (4/6)² = 4 : 9

Example 2: Prove similar, then find a length: AC and BD meet at O, with OA = 4 cm, OC = 6 cm, OB = 6 cm, OD = 9 cm and AB = 5 cm. Find CD.

∠AOB = ∠COD          (vert. opp. ∠s)
OA/OC = 4/6 = 2/3
OB/OD = 6/9 = 2/3
∴ △OAB ~ △OCD        (ratio of 2 sides, inc. ∠)

AB/CD = OA/OC
5/CD  = 2/3
CD    = 7.5 cm

Example 3: Area and perimeter of similar triangles: corresponding sides are 3 cm and 5 cm. The smaller triangle has perimeter 12 cm and area 18 cm². Find the perimeter and area of the larger one.

side ratio = 3 : 5
perimeter  = 12 × 5/3      = 20 cm
area       = 18 × (5/3)²   = 18 × 25/9 = 50 cm²

For the whole topic, read Similar triangles: key points, common mistakes and how the DSE tests it.

Steps

Worked examples

Other methods in this topic