smc

Let $a$ be radius of larger circle, and $b$ be radius of smaller circle. The area of the ring can be determined by subtracting the area of the smaller circle from the area of the larger circle, so $$\pi a^2-\pi b^2=\frac{25\pi }{2}$$ $$a^2-b^2=\frac{25}{2}$$ From the diagram, by using the Pythagorean Theorem, half of the chord is $\frac{5}{\sqrt{2}}=\frac{5\sqrt{2}}{2}$ units long, so the the length of the entire chord is $\boxed{5\sqrt{2}}$.