In triangle ABC, `sinA sin B + sin B sin C + sin C sin A = 9//4 and a = 2`, then the value of `sqrt3 Delta`, where `Delta` is the area of triangle, is _______

Answer

3

Explanation

The given equation is
`4 sin A sin B + 4 sin B sin C + 4 sin C sin A = 9`
`rArr 2 cos (A -B) -2 cos (A +B) + 2 cos (B -C)`
`- 2 cos (B +C) + 2 cos (C -A) -2 cos (C + A) = 9`
or `2 [cos (A -B) + cos (B -C) + cos (C -A)]`
`= 9 -2 (cos A + cos B + cos C)`
`ge 9 -2 xx (3)/(2) = 6`
or `cos (A -B) + cos (B -C) + cos (C -A) ge 3`
But `cos (A -B) le1, cos (B -C) le 1, cos (C -A) le 1`
or `cos (A -B) = 1, cos (B -C) =1, cos (C -A) = 1`
Thus, `A = B = C`, i.e., triangle ABC is an equilateral triangle
Hence, `Delta = sqrt3`