The radii r 1 , r 2 , r 3 of escribed circles of a triangle ABC are in harmonic progression. If its area is 24 sq. cm and its perimeter is 24 cm, find the lengths of its sides.
Text Solution
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6, 8, 10 cm
Sol. Δ = 24 sq. cm .... (i)
2 s = 24 ⇒ s = 12 .... ( ii )
r 1 , r 2 , r 3 are in H.P.
∴ 
are in A.P.
∴ 
, 
, 
are in A.P.
∴ a, b, c are in A.P. ⇒ 2b = a + c
2s = 24
∴ a + b + c = 24
3b = 24
∴ b = 8 ⇒ a + c = 16
But Δ = 
⇒ Δ = 
⇒ 24 × 24 = 12 × (12 – a) × 4 × (12 – c)
⇒ 2 × 6 = 144 – 12 (a + c) + ac ⇒ 12 = 144 – 192 + ac
∴ ac = 60 and a + c = 16 ∴ a = 10, c = 6 or a = 6, c = 10 and b = 8
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