Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area.

Suppose that the sides in cm, are 12x, 17x and 25x.

Then, we know that

12x + 17x + 25x = 540
(Given, Perimeter of the triangle)

54x = 540
\(\Rightarrow \) x = 10

So, the sides of the triangle are 12 × 10cm, 17 × 10cm, 25 × 10cm \(\Rightarrow \) 120cm, 170cm and 250cm.

Now, we know that,
\(s = \frac{a + b + c}{2}\)
\(\therefore \) \(s = \frac{120 + 170 + 250}{2} = \frac{540}{2} = 270cm\)

Now, Area of triangle
= \(\sqrt{270(270 - 120)(270 - 170)(270 - 250)}\)
(Since, Heron's formula [area = \(\sqrt{s(s - a)(s - b)(s - c)}\)])

= \(\sqrt{270 × 150 × 100 × 20}\)
= \(\sqrt{27 × 10 × 3 × 15 × 10 × 100 × 20}\)
= \(100\sqrt{27 × 15 × 10 × 2} {cm}^2\)
= \(100\sqrt{9 × 3 × 3 × 5 × 10 × 2} {cm}^2\)
= \(100\sqrt{9 × 3 × 3 × 10 × 10} {cm}^2\)
= 100 × 10 × 9

Therefore, area of the given triangle is 9000\({cm}^2\)