Question

If the length of tangents from $A,B,C$ to the incircle of $\Delta ABC$ are $4,6,8$ then which of the following is/are correct:

• A. Area of $\Delta ABC$ is $12\sqrt{6}$
• B. $r_1,r_2,r_3$ are in $HP$
• C. $a,b,c$are in $AP$
• D. $r=\frac{4\sqrt{6}}{3}$

Answer 1

Answer: (B), (C), (D)

$r=\frac{4\sqrt{6}}{3}$

Given: Length of tangents from $A,B,C$ to the incircle are $4,6,8$
We know, length of tangents to incircle are $s-a,s-b,s-c$ which are $4,6,8$ respectively.
Hence on adding, we have: $3s-2s=18\Rightarrow s=18$
$\therefore a=14,b=12$ and $c=10$
Clearly, $a,b,c$ as well as $s-a,s-b,s-c$ are in $AP$
Hence $\frac{\Delta }{s-a},\frac{\Delta }{s-b},\frac{\Delta }{s-c}$ are in $HP$
i.e $r_1,r_2,r_3$ are in $HP$
Now, $\Delta =\sqrt{18\times 4\times 6\times 8}=24\sqrt{6}$
Hence $r=\frac{\Delta }{s}=\frac{4\sqrt{6}}{3}$