Consider a triangle `A B C` and let `a , ba n dc` denote the lengths of the sides opposite to vertices `A , B ,a n dC` , respectively. Suppose `a=6,b=10 ,` and the area of triangle is `15sqrt(3)dot` If `/_A C B` is obtuse and if `r` denotes the radius of the incircle of the triangle, then the value of `r^2` is

Explanation

We know, `Delta = 1/2ab sinC`

`:. 15sqrt3 = 1/2*6*10*sinC`

`=>sinC = sqrt3/2`

As, `C` is obtuse, so `C = 120^@.`

Now, we know,

`cosC = (a^2+b^2 - c^2)/(2ab)`

`cos120^@ = (36+100 - c^2)/(120)`

`=>-1/2(120) = 136-c^2`

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