The sides of a triangle are in the ratio 2 : 3 : 4, and its perimeter is 45 cm. Find its area.

Let sides = $2k, 3k, 4k$.

$$\text{Perimeter} = 9k = 45, \text{ so } k = 5.$$

$$\text{Sides: } a = 10, b = 15, c = 20$$

$$s = \frac{45}{2} = 22.5$$

Using Heron's formula:

$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$

$$= \sqrt{22.5(22.5-10)(22.5-15)(22.5-20)}$$

$$= \sqrt{22.5 \times 12.5 \times 7.5 \times 2.5}$$

$\approx 72.6 \text{ cm}^2$