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Question:

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 15√3. If ∠ACB is obtuse and if r denotes the radius of the incircle of the triangle, then r² is equal to?

3

4

5

6

Solution:

Δ=1/2absinC ⇒sinC=2Δ/ab=2×15√3/6×10=√3/2 ⇒C=120° ⇒c=√a²+b²-2abcosC=√6²+10²-2×6×10×cos120°=14
∴r=Δ/s ⇒r²=225/(6+10+14/2)²=3.