8. $[7]$ A circle inscribed in a square,
Has two chords as shown in a pair.
It has radius 2 ,
And $P$ bisects $T U$.
The chords' intersection is where?

Answer the question by giving the distance of the point of intersection from the center of the circle.
lies between $X$ and $Q$. Then $M N X Q$ is a parallelogram. For, $O B \| N M$ by homothety at $C$ and $P M \| N X$ because $M N X P$ is an isoceles trapezoid. It follows that $Q X=M N$. Considering that the center of the circle together with points $M, C$, and $N$ determines a square of side length 2 , it follows that $M N=2 \sqrt{2}$, so the answer is $2 \sqrt{2}-2$.