Consider a triangle ABC and let a, b and c denote the lengths of the sides opposite to vertices A, B and C, respectivelu. Suppose `a = 6, b = 10` and the triangle is `15 sqrt3`. If `angle ACB` is obtus and if r denotes than radius of the incircle of the triangle, then the value of `r^(2)` is _____

Answer

3

Explanation

`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`