Question

The sum of the lengths of two sides of a triangle is equal to a and the angle between them is equal to ${30}^{\circ }.$ What must be the lengths of the sides of the triangle for its area to be the greatest?

Answer 1

let the sides of the triangle be $p$ and $q$.

Given $p+q=a$ and the included angle is ${30}^0$. Let S denote the area of the triangle.

$⟹S=\frac{1}{2}pq\sin 30=pq$

$⟹S=p(a-p)=ap=p^2=\frac{a^2}{4}-{(\frac{a}{2}-p)}^2$

This term is maximum when $p=\frac{a}{2}$

Thus the lengths of the sides must be $\frac{a}{2},\frac{a}{2}$ for the triangle to have a maximum area.