Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?

**Answer Choices**

A. \dfrac{1+\sqrt{2}}{4}

B. \dfrac{\sqrt{5}-1}{2}

C. \dfrac{\sqrt{3}+1}{4}

D. \dfrac{2 \sqrt{3}}{5}

E. \dfrac{\sqrt{5}}{3}