IIT-JEE 2010 · Paper 2 · Mathematics Q27

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MathematicsInteger+3Question 27 of 57

Consider a triangle ABC 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 $15\sqrt3$ . If $\angle ACB$ is obtuse and if $r$ denotes the radius of the incircle of the triangle, then $r^2$ is equal to

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Answer and worked solution

Official answer: 3

Angle C. With $a=6$ , $b=10$ : $\frac12\cdot60\sin C=15\sqrt3$ , so $\sin C=\frac{\sqrt3}2$ ; C is obtuse, so $C=120°$ .

Third side. $c^2=36+100-2\cdot60\cos120°=196$ , $c=14$ .

In-radius. $s=\frac{6+10+14}2=15$ , $r=\frac\Delta s=\frac{15\sqrt3}{15}=\sqrt3$ , so $r^2=3$ .

Answer: 3

Topics:Properties of triangles

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IIT-JEE 2010 · Paper 2, Mathematics question 27