Geometry: Circular Arcs Length

Curve Y subtends $60^\circ$ at its center $Q$ with radius $r_Y$. Length of arc $Y = 2\pi r_Y \cdot \frac{60^\circ}{360^\circ} = \frac{\pi r_Y}{3} = 10\pi \implies r_Y = 30$. The chord length $AB = 2 r_Y \sin(30^\circ) = 30$. Curve X subtends $90^\circ$ at its center $P$ with radius $r_X$. Chord $AB = r_X \sqrt{2} = 30 \implies r_X = \frac{30}{\sqrt{2}}$. Length of arc $X = 2\pi r_X \cdot \frac{90^\circ}{360^\circ} = \frac{\pi r_X}{2} = \frac{\pi (30/\sqrt{2})}{2} = \frac{15\pi}{\sqrt{2}}$.