The sides of a triangle are in the ratio 2: 3: 4, and its perimeter is 45 cm. Find its area.

Step-by-step solution

Idea: Find the sides from the ratio, then use Heron’s formula. Working with halves (452, 252 …) keeps the square root exact.

1. 2x + 3x + 4x = 45 ⇒ x = 5. Sides: 10, 15, 20 cm. s = 22.5 cm.1 mark
2. s − a = 12.5, s − b = 7.5, s − c = 2.5. Area = √(452 × 252 × 152 × 52) = √843754.1 mark
3. 84375 = 5625 × 15 = 75² × 15, so area = 75√154 ≈ 72.62 cm².1 mark

Check: 22.5 × 12.5 × 7.5 × 2.5 = 5273.4…, and 72.62² ≈ 5273.6 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Taking x = 45 ÷ 3 = 15.
• Rounding the factors early (e.g. 22.5 → 23), which changes the answer noticeably.

How this can come in the exam

Try one yourself

The sides of a triangle are in the ratio 7 : 15 : 20 and its perimeter is 126 cm. Find its area.

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