Off Centre Square Geometry Problem

Question

Square ABCD has side length of 10 cm. The square is placed over a circle, but done so it is off-center. It is offset in such a way that A and D are on the circle and side BC is tangent to the circle at point P (which is exactly half way down side BC). Determine the radius of the circle.

Answer 1

Denote the midpoint of $\overline{AD}$ by $Q$, the centre of the circle by $O$, and the radius of the circle by $r$. Using the Pythagorean theorem, $$ \begin{align} 5^2+(10-r)^2&=r^2\\ 25+100-20r+r^2&=r^2\\ 20r&=125\\ r&=6.25.\\ \end{align} $$