AMC 12A 2025 (Problem 10)

In the figure shown below, major arc

\widehat{AD}

and minor arc

\widehat{BC}

have the same center,

O

. Also,

A

lies between

O

and

B

, and

D

lies between

O

and

C

. Major arc

\widehat{AD}

, minor arc

\widehat{BC}

, and each of the two segments

\overline{AB}

and

\overline{CD}

have length

2\pi

. What is the distance from

O

A

\text{(A)}\;1

\text{(B)}\;1-\pi+\sqrt{\pi^2+1}

\text{(C)}\;\frac{\pi}{2}

\text{(D)}\;\frac{\sqrt{\pi^2+1}}{2}

\text{(E)}\;2