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

Options

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.

Answer

A, C, D

Explanation

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.