Question

Consider a triangle $A B C$ and let $a, b$ and $c$ denote the lengths of the sides opposite to vertices $A, B$ and $C$ respectively. Suppose $a=6, b=10$ and the area of the triangle is $15 \sqrt{3}$. If $\angle A C B$ is obtuse and if $r$ denotes the radius of the incircle of the triangle, then $r^{2}$ is equal to ______.

Answer 1

$\Delta=\frac{1}{2} a b \sin C $
$\Rightarrow \sin C=\frac{2 \Delta}{a b}=\frac{2 \times 15 \sqrt{3}}{6 \times 10}=\frac{\sqrt{3}}{2} $
$\Rightarrow C=120^{\circ}$
$\Rightarrow c=\sqrt{a^{2}+b^{2}-2 a b \cos C}$
$=\sqrt{6^{2}+10^{2}-2 \times 6 \times 10 \times \cos 120^{\circ}}=14$
$\therefore r=\frac{\Delta}{s} $
$\Rightarrow r^{2}=\frac{225 \times 3}{\left(\frac{6+10+14}{2}\right)^{2}}=3$