Question

The sides of a triangle are in the ratio 4 ∶ 3 ∶ 2. The perimeter of the triangle is 54 cm. The area (in cm²) of the triangle is
1. 18√15 cm²
2. 12 cm²
3. 6 cm²
4. 27√15 cm²

Answer 1

Correct Answer - Option 4 : 27√15 cm²

Given:

The sides of a triangle are in the ratio 4 ∶ 3 ∶ 2.

Formula used:

Semi perimeter of a Δ = (a + b + c)/2

Area of Δ = √{s(s - a)(s - b)(s - c)}

Where a, b, c are sides of the Δ.

s → semiperimeter

Calculations:

Let the ratio of sides be 2x, 3x, and 4x.

Then 4x + 3x + 2x = 54

⇒ 9x = 54

⇒ x = 6

So the sides of the Δ are 24 cm, 18 cm, and 12 cm.

Semi perimeter of the Δ = (a + b + c)/2 = (24 + 18 +12)/2 = 27 cm

Area of the Δ = √{s(s - a)(s - b)(s - c)}

⇒ √{s(s - a)(s - b)(s - c)} = √{27(27 - 24)(27 - 18)(27- 12)}

⇒ √(27 × 3 × 9 × 15) = 27√15 cm²

∴ The area of the triangle is 27√15 cm².