Find the area of a triangle ABC such that a = 16cm, b = 14cm, c = 12cm, leave your answer in surd form

Analyze the question: We are given all three sides of a triangle (a = 16, b = 14, c = 12), so we use Heron's formula.

Concept: Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2

Solution:

1. Semi-perimeter: s = (16 + 14 + 12)/2 = 42/2 = 21

2. Substitute: s − a = 21 − 16 = 5 s − b = 21 − 14 = 7 s − c = 21 − 12 = 9

3. Area = √(21 × 5 × 7 × 9) = √(21 × 5 × 7 × 9)

Multiply: 21 × 5 = 105, 105 × 7 = 735, 735 × 9 = 6615

Factorise: 6615 = 3³ × 5 × 7² = 9 × 49 × 15 (check: 9 × 49 = 441, 441 × 15 = 6615 ✓)

Area = √(9 × 49 × 15) = 3 × 7 × √15 = 21√15 cm²

Answer: Option E, 21√15 cm².

Tip: When using Heron's formula, factorise under the root into perfect squares (like 9 and 49) before simplifying. This makes surd answers quick to get. Also, a quick sanity check: 21√15 ≈ 81.3 cm², which is reasonable for a triangle with sides of 12–16 cm.