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The radii r1,r2,r3 of escribed circles of trisngle ABC are in HP . if area =24, perimeter =24, find a,b,c

The radii r1,r2,r3 of escribed circles of trisngle ABC are in HP . if area =24, perimeter =24, find a,b,c

Grade:11

1 Answers

Arun
25763 Points
4 years ago
Area = 24
perimeter = 24, hence semi perimeter (s) = 12
r1, r2, r3 are in H.P.
hence 1/r1, 1/r2, 1/r3 will be in A.P.
1/r2 – 1/r1 = 1/r3 – 1/r2
2/r2 = 1/r1 + 1/r3
2(s – b)/ \Delta = (s –a)/ \Delta + (s –c)/ \Delta
on solving 
2s – a –c = 2(s –b)
a+b+c –a –c = 2(s –b)
b = 2(s –b)
3 b = 2s
b = 24/3
b = 8
as we know,
a+b+c = 24
a+c = 16
c= 16 – a
when applying heron’s formula,
we get,
(a –6) (a –10) = 0
a= 6  or  a= 10
if a =6 , c = 10
if a =10, c= 6
hence three sides are 6,8, 10

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