SATHEE: Properties of Triangles 3 Question 20

Properties of Triangles 3 Question 20

20. Consider a $△ A B C$ 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 $∠ A C B$ 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.

$\begin{aligned} \Rightarrow C & = \frac{2 π}{3} = \sqrt{a^{2} + b^{2} - 2 a b \cos C} \\ = \sqrt{6^{2} + 10^{2} - 2 \times 6 \times 10 \times \cos \frac{2 π}{3}} = 14 \\ ∴ r & = \frac{Δ}{s} \Rightarrow r^{2} = \frac{225 \times 3}{\frac{6 + 10 + 14^{2}}{2}} = 3 \end{aligned}$

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Properties of Triangles 3 Question 20

20. Consider a △ABC 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 153. If ∠ACB is obtuse and if r denotes the radius of the incircle of the triangle, then r2 is equal to……

Solution:

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