Question

The radii $r_1,r_2,r_3$ escribed circle of the traingle $ABC$ are in H.P. If its area is $24$ square cm and its perimeter is $24$ cm, then lengths of its sides are

• A. $4,6,8$
• B. $3,9,11$
• C. $6,8,10$
• D. None of these

Answer 1

Answer: (C)

$6,8,10$

$r_1,r_2,r_3$ are in H.P.

$\Rightarrow \frac{s-a}{S},\frac{s-b}{S},\frac{s-c}{S}$ are in A.P.

$\Rightarrow s-a,s-b,s-c$ are in A.P

$\Rightarrow a,b,c$ are in A.P

Now, $a+b+c=24\Rightarrow b=8,s=12,c=4-a$

$\therefore 12(12-a)(12-8)(12-c)={\left(24\right)}^2=576\Rightarrow a^2-16a+60=0\Rightarrow a=10,6$

If $a=10,b=6$ and if $a=6,b=10$

$\therefore$ sides of the triangle are $6,8$ and $10$ cm