[Solution] Draw $O F \perp A B, O G \perp C D$ (as shown in the figure). In the right $\triangle A O F$, by the Pythagorean theorem, we have
$$
r=\sqrt{\left(\frac{3+4}{2}\right)^{2}+2^{2}}=\frac{\sqrt{65}}{2} \text {, }
$$

thus $2 r=\sqrt{65}$.
Therefore, the answer is $(E)$.
20.14 Two perpendicular chords in a circle, one chord is divided into segments of length 3 and 4, the other chord is divided into segments of length 6 and 2, then the diameter of this circle is
(A) $\sqrt{89}$.
(B) $\sqrt{56}$.
(C) $\sqrt{61}$.
(D) $\sqrt{75}$.
(E) $\sqrt{65}$.
(8th American High School Mathematics Examination, 1957)