Question

The radii $r_1,r_2,r_3$ of the escribed circles of the triangle $ABC$ are in the $H.P.$

If the area of the triangle is $24$ $c{m}^2$ and its perimeter is $24cm$, then the length of its largest side is?

• A. $10$
• B. $9$
• C. $8$
• D. None of the above

Answer 1

The correct option is A $10$

Since, $r_1,r_2,r_3$ are in H.P
Therefore, $\frac{2}{r_2}=\frac{1}{r_1}+\frac{1}{r_3}$
$\Rightarrow \frac{2(s-b)}{△}=\frac{(s-a)}{△}+\frac{(s-c)}{△}$
$\Rightarrow 2b=a+c$
Given that, $2s=a+b+c=24$
$\Rightarrow 3b=24$
$\Rightarrow b=8$ and $a+c=16$ ....(1)
Now, ${△}^2=s(s-a)(s-b)(s-c)$
${24}^2=12(12-a)(12-8)(12-c)$
$\Rightarrow 48=(a-4)(12-a)$ [ from (1) ]
$\Rightarrow a^2-16a+60$
$\Rightarrow a=6,10$
Ans: A