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Question

The radii r1,r2,r3 of escribed circle of a triangle ABC are in H.P. If its area is 24 cm2 and its perimeter is 24 cm, find the length of its sides.

A
True
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False
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Solution

The correct option is A True
Let a,b,c be the length of the sides of the triangle, s be the semi perimeter and Δ be the area.
r1=Δ(sa)=24(12a)r2=Δ(sb)=Δ(12b)r3=Δsc=24(12c)
Since r1,r2,r3 are in H.P.
1r1,1r2,1r3 are in A.P.
2r1=1r1+1r3 or 2×12b24=12a24+12c24
Or a+c=2b .....(1)
Also, a+b+c=24 ....(2)
From (1) and (2), we get b=8cm and a+c=16cm
Now Δs(sa)(sb)(sc)
24=12(12a)(12b)(12c)24×24=12(12a)×4×(1216+a)12=(12a)(a4)a216a+60=0a=6 or 10.
When a=6,c=10 and when a=10,c=6
Hence sides are 6,8,10 or 10,8,6

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