Properties of Triangles 3 Question 20
20. 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. $a=6, b=10$ 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……
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Solution:
- $\sin C=\frac{\sqrt{3}}{2}$ and $C$ is given to be obtuse.
$$ \begin{aligned} \Rightarrow C & =\frac{2 \pi}{3}=\sqrt{a^{2}+b^{2}-2 a b \cos C} \\ & =\sqrt{6^{2}+10^{2}-2 \times 6 \times 10 \times \cos \frac{2 \pi}{3}}=14 \\ \therefore \quad r & =\frac{\Delta}{s} \Rightarrow r^{2}=\frac{225 \times 3}{\frac{6+10+14^{2}}{2}}=3 \end{aligned} $$
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