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Consider a triangle A B C and let a , ba...

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

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Verified by Experts

The correct Answer is:
3
`Delta = (1)/(2) ab sin C rArr sin C = (2Delta)/(ab) = (2 xx 15 sqrt3)/(6 xx 10) = (sqrt3)/(2)`
`rArr C = 120^(@)`
`rArr c = sqrt(a^(2) + b^(2) - 2ab cos C)`
`= sqrt(6^(2) + 10^(2) - 2 xx 6 xx 10 xx cos 120^(@)) = 14`
Now `r = (Delta)/(s)`
`rArr r^(2) = (225 xx 3)/(((6 + 10 + 14)/(2))^(2)) = 3`
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