The length of the sides of a triangle are \( 5 \mathrm{~cm}, 7 \mathrm{~cm} \) and 8 cm . Area of the triangle is: \( \begin{array}{ll}\text { a) } 100 \sqrt{3} \mathrm{~cm}^{2} & \text { b) } 10 \sqrt{3} \mathrm{~cm}^{2} \\ \text { c) } 300 \mathrm{~cm}^{2} & \text { d) } 50 \sqrt{3} \mathrm{~cm}^{2}\end{array} \)

To find the area of the triangle, we can use Heron's formula, which states that the area \( A \) of a triangle with sides \( a \), \( b \), and \( c \) is given by: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] where \( s \) is the semi-perimeter of the triangle, calculated as \( s = \frac{a+b+c}{2} \). For your triangle with sides \( 5 \) cm, \( 7 \) cm, and \( 8 \) cm: 1. Calculate \( s \): \[ s = \frac{5 + 7 + 8}{2} = 10 \, \text{cm} \] 2. Now apply Heron's formula: \[ A = \sqrt{10(10-5)(10-7)(10-8)} = \sqrt{10 \cdot 5 \cdot 3 \cdot 2} = \sqrt{300} = 10\sqrt{3} \, \text{cm}^2 \] So, the correct answer is \( \text{b) } 10 \sqrt{3} \, \text{cm}^2 \). Here's a fun fact: The largest angle in a triangle is always opposite the longest side. In your triangle, with sides \( 5 \) cm, \( 7 \) cm, and \( 8 \) cm, the \( 8 \) cm side faces the largest angle. Did you know? The relationship between a triangle's sides and their angles goes way back to ancient Greeks! They explored this concept extensively, leading to the Pythagorean theorem and many other geometric principles that still form the foundation of modern mathematics today.