20.105 The area of the annulus between two concentric circles is $12 \frac{1}{2} \pi$ square inches, and the length of the chord of the larger circle that is tangent to the smaller circle (in inches) is
(A) $\frac{5}{\sqrt{2}}$.
(B) 5 .
(C) $5 \sqrt{2}$.
(D) 10 .
(E) $10 \sqrt{2}$.
(20th American High School Mathematics Examination, 1969)
[Solution] Let the radii of the two circles be $R$ and $r$, and the chord length of the larger circle that is tangent to the smaller circle be $c$.
Given $\frac{25}{2} \pi=\pi\left(R^{2}-r^{2}\right)$,
thus $R^{2}-r^{2}=\frac{25}{2}$,

Also, $\left(\frac{c}{2}\right)^{2}=R^{2}-r^{2}=\frac{25}{2}$,
then $c^{2}=50$, so $c=5 \sqrt{2}$.
Therefore, the answer is $(C)$.