We are given a triangle with the following details:
We need to find the area of the triangle.
The perimeter is the sum of all three sides. Let the third side be \( x \). The perimeter is given by: \[ \text{Perimeter} = 18 + 10 + x = 42 \] Simplifying: \[ 28 + x = 42 \] Solving for \( x \): \[ x = 42 - 28 = 14 \, \text{cm} \] So, the third side is \( 14 \, \text{cm} \).
To find the area of the triangle, we use Heron's formula. The semi-perimeter \( s \) is given by: \[ s = \frac{\text{Perimeter}}{2} = \frac{42}{2} = 21 \, \text{cm} \] Heron's formula for the area \( A \) of a triangle is: \[ A = \sqrt{s(s - a)(s - b)(s - c)} \] where: - \( a = 18 \, \text{cm} \), - \( b = 10 \, \text{cm} \), - \( c = 14 \, \text{cm} \), - \( s = 21 \, \text{cm} \). Substituting the values into Heron's formula: \[ A = \sqrt{21(21 - 18)(21 - 10)(21 - 14)} \] Simplifying: \[ A = \sqrt{21 \times 3 \times 11 \times 7} \] \[ A = \sqrt{21 \times 231} = \sqrt{4851} \] \[ A = 69.7 \, \text{cm}^2 \]
The area of the triangle is \( \boxed{69.7 \, \text{cm}^2} \).