Question

Let $ABCD$ be a unit square. Draw a quadrant of a circle with $A$ as centre and $B,D$ as endpoints of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as endpoints of the arc. Inscribe a circle $\Gamma$ touching the arcs $AC$ and $BD$ both externally and also touching the side $CD.$ Find the radius of the circle $\Gamma$.

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $\frac{1}{16}$

Answer 1

The correct option is D $\frac{1}{16}$

Let O be the centre of $\Gamma$
By symmetry, O is on the perpendicular bisector of CD
Draw $OL\perp CD$ and $OK\perp BC$.
From this $OK=CL=\frac{CD}{2}=\frac{1}{2}$
If $r$ is the radius of T, we see that $BK=1-r$, and $OE=r$
Using Pythagoras theorem
$(1+r{)}^2=(1-r{)}^2+{\left(\frac{1}{2}\right)}^2$
Thus $r=1/16$.