The sides of a triangle have lengths 7 cm, 24 cm, 25 cm. Find the area of the triangle in two different ways.

Method 1: Using Pythagoras theorem

Check:

$$7^2 + 24^2 = 49 + 576 = 625$$

$$25^2 = 625$$

So, it is a right-angled triangle with legs 7 and 24, hypotenuse 25.

$$\text{Area} = \frac{1}{2} \times 7 \times 24 = 84 \text{ cm}^2$$

Method 2: Using Heron's formula:

$$\begin{aligned} A &= \sqrt{s(s-a)(s-b)(s-c)} \\ &= \sqrt{28(28-7)(28-24)(28-25)} \\ &= \sqrt{28 \times 21 \times 4 \times 3} \\ &= \sqrt{7056} = 84 \text{ cm}^2 \end{aligned}$$