Question

In triangle ABC, $\angle A=\pi /3$ and its incircle is of unit radius. If the radius of the circle touching the
sides AB, AC internally and incircle externally is $x$, then the value of $x$ is

• A. $1/2$
• B. $1/4$
• C. $1/3$
• D. none of these

Answer 1

The correct option is C $1/3$

From the figure,
in right angled triangles APO1 and ABO2
$(x+y)sin\left(30\right)=1$
$x+y=2$
also
$xsin30=r$
$x=2r$
and $y=1+r$
solving this
we get
$r=\frac{1}{3}$