5. 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
Thus, x = 10
So, the sides of the triangle are 12 × 10cm, 17 × 10cm, 25 × 10cm i.e., 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 = $\sqrt{270\left(270?120\right)\left(270?170\right)\left(270?250\right)}$ ...(Since, Heron's formula [area = $\sqrt{s\left(s?a\right)\left(s?b\right)\left(s?c\right)}$])
= $\sqrt{270×150×100×20}$
= $\sqrt{27×10×3×15×10×100×20}$
= $100\sqrt{27×15×10×2}{cm}^{2}$
= $100\sqrt{9×3×3×5×10×2}{cm}^{2}$
= $100\sqrt{9×3×3×10×10}{cm}^{2}$
= 100 × 10 × 9
Therefore, area of the given triangle is 9000${cm}^{2}$