AMC 12 2025a 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 $O$ to $A$?

usepackage("mathptmx"); size(6cm); defaultpen(linewidth(0.7)); real r = 1 - pi + sqrt(pi^2 + 1);
pair O = (0, 0), A = r * dir(-30), B = (r + 2pi) * dir(-30), C = (r + 2pi) * dir(30), D = r * dir(30);
draw(arc(O, r, 30, 330)); draw(arc(O, r + 2pi, -30, 30)); draw(O--A, dashed); draw(O--D, dashed); draw(A--B); draw(C--D); dot("$O$", O, dir(135)); dot("$A$", A, dir(270)); dot("$B$", B, dir(0)); dot("$C$", C, dir(0)); dot("$D$", D, 1.5 * dir(70));

\textbf{(A)}1 \textbf{(B)}1 - \pi + \sqrt{\pi^{2} + 1} \textbf{(C)}\frac{\pi}{2} \textbf{(D)}\frac{\sqrt{\pi^{2} + 1}}{2} \textbf{(E)}~2