We are given a triangle with sides in the ratio 12:17:25 and its perimeter is 540 cm. We need to find the area of the triangle.
Let the sides of the triangle be \( 12x \), \( 17x \), and \( 25x \), where \( x \) is the common factor. The perimeter is the sum of the three sides: \[ \text{Perimeter} = 12x + 17x + 25x = 540 \] Simplifying: \[ 54x = 540 \] Solving for \( x \): \[ x = \frac{540}{54} = 10 \] Now, we can find the lengths of the sides: - Side 1 = \( 12x = 12 \times 10 = 120 \, \text{cm} \) - Side 2 = \( 17x = 17 \times 10 = 170 \, \text{cm} \) - Side 3 = \( 25x = 25 \times 10 = 250 \, \text{cm} \)
To find the area of the triangle, we use Heron's formula. The semi-perimeter \( s \) is given by: \[ s = \frac{\text{Perimeter}}{2} = \frac{540}{2} = 270 \, \text{cm} \] Heron's formula for the area \( A \) of a triangle is: \[ A = \sqrt{s(s - a)(s - b)(s - c)} \] where: - \( a = 120 \, \text{cm} \), - \( b = 170 \, \text{cm} \), - \( c = 250 \, \text{cm} \), - \( s = 270 \, \text{cm} \). Substituting the values into Heron's formula: \[ A = \sqrt{270(270 - 120)(270 - 170)(270 - 250)} \] Simplifying: \[ A = \sqrt{270 \times 150 \times 100 \times 20} \] \[ A = \sqrt{270 \times 3000000} \] \[ A = \sqrt{810000000} \] \[ A = 28460.4 \, \text{cm}^2 \]
The area of the triangle is \( \boxed{28460.4 \, \text{cm}^2} \).