The sides of a triangle are in the ratio 5 : 12 : 13 and its perimeter is 150 cm.

(b) $750 \mathrm{~cm}^{2}$

Let the sides of the triangle be 5x cm, 12x cm and 13x cm.
Perimeter = Sum of all sides
or, 150 = 5x + 12x + 13x
or, 30x = 150
or, x = 5
Thus, the sides of the triangle are 5×">××5 cm, 12×">××5 cm and 13×">××5 cm, i.e., 25 cm, 60 cm and 65 cm.

Now

Let :

$a=25 \mathrm{~cm}, b=60 \mathrm{~cm}$ and $c=65 \mathrm{~cm}$

$s=\frac{150}{2}=75 \mathrm{~cm}$

By Heron's formula, we have :

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

$=\sqrt{75(75-25)(75-60)(75-65)}$

$=\sqrt{75 \times 50 \times 15 \times 10}$

$=\sqrt{15 \times 5 \times 5 \times 10 \times 15 \times 10}$

$=15 \times 5 \times 10$

$=750 \mathrm{~cm}^{2}$