Question

Let $ABCD$ be a square of side length $1,$ and $\Gamma$ a circle passing through $B$ and $C,$ and touching $AD.$ The radius of $\Gamma$ is

• A. $\frac{3}{8}$
• B. $\frac{1}{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{5}{8}$

Answer 1

The correct option is D $\frac{5}{8}$


$\text{Let}O$ be the centre and $r$ be the radius of the circle.
$\text{Then}ON=OC=r$
$MN=1\Rightarrow OM=1-r$
$BC=1\Rightarrow MC=\frac{1}{2}$

$\text{In right triangle}OMC,$
$(1-r{)}^2+{\left(\frac{1}{2}\right)}^2=r^2$
$\Rightarrow r=\frac{5}{8}$