Question

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

Answer 1

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`
`=>c^2 = 196`
`=>c = 14`
`:. s = (a+b+c)/2 = (6+10+14)/2 = 15`
Now, `r = Delta/s = (15sqrt3)/15 = sqrt3`
`:. r^2 = 3.`