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The radii r1, r2,r3 of the escribed circ...

The radii `r_1, r_2,r_3` of the escribed circles of the triangle `A B C` are in H.P. If the area of the triangle is `24 c m^2a n d` its perimeter is 24cm, then the length of its largest side is 10 (b) 9 (c) 8 (d) none of these

A

The length of longest side of triangle ABC is equal to 10

B

The radius of circle inscribed in triangle ABC is equal to 4.

C

The circumradius of triangle ABC is equal to 5.

D

The sides of triangle ABC are in A.P.

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
As `r_(1),r_(2),r_(3)` are in H.P.
`rArr (s-a)/(Delta),(s-b)/(Delta),(s-c)/(Delta)` are in A.P.
`rArr s-a, s-b,s-c` are in A.P.
`rArr a,b,c` are in A.P.
Also, `a+b+c=24` (Given)
`rArr b=8, s=12, c=16-a`
Also, `sqrt(s(s-a)(s-b)(s-c))=24` (given)
`rArr 12.(12-a)(12-8).(12-(16-a))=576`
`rArr (12-a)(a-4)=12`
`rArr a=6` or 10
Hence, the sides of triangle are 6,8,10.
Clearly, the length of longest side of triangle is 10.
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