Question 10In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$$\dfrac{\sqrt{5}}{12}$$\dfrac{\sqrt{5}}{10}$$\dfrac{\sqrt{5}}{9}$$\dfrac{\sqrt{5}}{8}$$\dfrac{2\sqrt{5}}{15}$