Question

In an equilateral triangle, $R:r:r_2$ is equal to

• A. $1:1:1$
• B. $1:2:3$
• C. $2:1:3$
• D. $3:2:4$

Answer 1

The correct option is C $2:1:3$

$\because A=B=C={60}^0$
$\because △ABC$ is equilateral.
If length of side$=\lambda$
$\therefore R=\frac{\lambda }{2\sin {60}^0}$
$=\frac{\lambda }{2\times \frac{\sqrt{3}}{2}}$
$=\frac{\lambda }{\sqrt{3}}$
$r=(s-a)\tan \left(\frac{{60}^0}{2}\right)$
$=(\frac{3\lambda }{2}-\lambda )\tan {30}^0$
$=\left(\frac{3\lambda -2\lambda }{2}\right)\frac{1}{\sqrt{3}}$
$=\frac{\lambda }{2}\frac{1}{\sqrt{3}}$
$=\frac{\lambda }{2\sqrt{3}}$
and $r_1=s\tan \frac{A}{2}$
$=\frac{3\lambda }{2}\tan \frac{{60}^0}{2}$
$=\frac{3\lambda }{2}\tan {30}^0$
$=\frac{3\lambda }{2}\frac{1}{\sqrt{3}}$
$=\frac{3\lambda }{2\sqrt{3}}$
$\therefore R:r:r_1=\frac{\lambda }{\sqrt{3}}:\frac{\lambda }{2\sqrt{3}}:\frac{3\lambda }{2\sqrt{3}}$
$=1:\frac{1}{2}:\frac{3}{2}$
$=2:1:3$