Area and perimeter
A standard method from Similar triangles: the steps and worked examples.
Steps
- Find the side ratio from one pair of matching sides.
- Perimeter ratio = side ratio.
- Area ratio = (side ratio)².
Why: area uses two lengths multiplied, so the scale factor is used twice.
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 : 9Example 2: 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 cm²For the whole topic, read Similar triangles: key points, common mistakes and how the DSE tests it.