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 `15sqrt3`. If `angleACB` is obtuse and if r denotes the radius of the in circle of the triangle , then `r^(2)` is equal to

Options

A. 2
B. 4
C. 3
D. 6

Explanation

We have,
Area=`1/2absinC`
`rArr15sqrt3=1/2xx6xx10xx` sinC
`rArrsinC=(sqrt3)/2rArrC=120^(@)`
`thereforec^(2)=a^(2)+b^(2)-2abcosC`
`c^(2)=36+100-120xx(-1/2)=196rArrc=14`
So, 2s = a + b + c `rArr` 2s = 30 `rArr` s = 15
`thereforer=(Delta)/s=(15sqrt3)/15=sqrt3rArrr^(2)=3`