Question

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

Answer 1

Let the sides of the triangle be 5x m, 12x m and 13x m.
We know:
Perimeter = Sum of all sides
or, 150 = 5x + 12x + 13x
or, 30x = 150
or, x = 5
Thus, we obtain the sides of the triangle.
5$\times$5 = 25 m
12$\times$5 = 60 m
13$\times$5 = 65 m

Now,
$$\begin{gathered} \mathrm{Let}: \\ a=25\mathrm{m},b=60\mathrm{m}\mathrm{and}c=65\mathrm{m} \\ \therefore s=\frac{150}{2}=75\mathrm{m} \\ \mathrm{By}\mathrm{Heron}\text{'}\mathrm{s}\mathrm{formula},\mathrm{we}\mathrm{have}: \\ \mathrm{Area}\mathrm{of}\mathrm{triangle}=\sqrt{s(s-a)(s-b)(s-c)} \\ =\sqrt{75(75-25)(75-60)(75-65)} \\ =\sqrt{75\times 50\times 15\times 10} \\ =\sqrt{15\times 5\times 5\times 10\times 15\times 10} \\ =15\times 5\times 10 \\ =750{\mathrm{m}}^2 \end{gathered}$$