Question

If the radius of the incircle of a triangle with its sides $5k,6k$ and $5k$ is $6$, then $k$ is equal to:

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is B $4$

Given: Radius of the incircle of a triangle with its sides $5k,6k$ and $5k$ is $r=6$
$2s=a+b+c=5k+6k+5k=16k$
So, we have $s=8k$
Also we have,
$\Delta =\sqrt{s(s-a)(s-b)(s-c)}$
$\Delta =\sqrt{8k\left(3k\right)\left(2k\right)\left(3k\right)}=12k^2$
We know, $r=\frac{\Delta }{s}$
$\Rightarrow 6=\frac{12k^2}{8k}$
$\Rightarrow k=4$