Question

The sides of a triangle are in the ratio 13 : 14 : 15 and its perimeter is 84 cm. Find the area of the triangle.

Answer 1

Let the sides of the triangle be 13x cm, 14x cm and 15x cm.
Perimeter = Sum of all sides
or, 84 = 13x + 14x + 15x
or, 42x = 84
or, x = 2
So, sides of the triangle are 13$\times$2, 14$\times$2 and 15$\times$2, i.e., 26 cm, 28 cm and 30 cm.
$$\begin{gathered} \mathrm{Let}: \\ a=26\mathrm{cm},b=28\mathrm{cm}\mathrm{and}c=30\mathrm{cm} \\ s=\frac{84}{2}=42c\mathrm{m} \\ \mathrm{By}\mathrm{Heron}\text{'}\mathrm{s}\mathrm{formula},\mathrm{we}\mathrm{have}: \\ \mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{triangle}=\sqrt{s(s-a)(s-b)(s-c)} \\ =\sqrt{42(42-26)(42-28)(42-30)} \\ =\sqrt{42\times 16\times 14\times 12} \\ =\sqrt{14\times 3\times 4\times 4\times 14\times 3\times 2\times 2} \\ =14\times 3\times 4\times 2 \\ =336{\mathrm{cm}}^2 \end{gathered}$$