Question

The length of sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm. The area of the triangle is:

• A. 764 cm\(^2\)
• B. 864 cm\(^2\)
• C. 664 cm\(^2\)
• D. 684 cm\(^2\)

Hint:

For a triangle with known sides, use Heron's formula to calculate the area when the sides are given, and the perimeter is known.

Answer 1

The Correct Option is C

Let the sides of the triangle be \( 3x \), \( 4x \), and \( 5x \). The perimeter of the triangle is given by: \[ 3x + 4x + 5x = 144 \] \[ 12x = 144 \quad \Rightarrow \quad x = 12 \] Thus, the sides of the triangle are \( 36 \, \text{cm} \), \( 48 \, \text{cm} \), and \( 60 \, \text{cm} \). Now, we can use Heron's formula to find the area of the triangle. The semi-perimeter \( s \) is: \[ s = \frac{36 + 48 + 60}{2} = 72 \] Using Heron's formula: \[ A = \sqrt{s(s - 36)(s - 48)(s - 60)} \] \[ A = \sqrt{72(72 - 36)(72 - 48)(72 - 60)} = \sqrt{72 \times 36 \times 24 \times 12} \] \[ A = \sqrt{62208} \approx 664 \, \text{cm}^2 \] Thus, the area of the triangle is \( 664 \, \text{cm}^2 \).