2008 AMC 12A Problem 20

Triangle ABC has AC=3, BC=4, and AB=5. Point D is on \overline{AB}, and \overline{CD} bisects the right angle. The inscribed circles of \triangle ADC and \triangle BCD have radii r_a and r_b, respectively. What is r_a/r_b?

\begin{array}{l} \mathrm{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right)\qquad\mathrm{(B)}\ \frac{3}{56}\left(10-\sqrt{2}\right)\qquad\mathrm{(C)}\ \frac{1}{14}\left(10-\sqrt{2}\right)\qquad\mathrm{(D)}\ \frac{5}{56}\left(10-\sqrt{2}\right)\\\mathrm{(E)}\ \frac{3}{28}\left(10-\sqrt{2}\right) \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.