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2008 AMC 12A Question 20

AMC12 · 2008 AMC 12A

Question 20

Triangle $ABC$ has $AC = 3,$ $BC = 4,$ and $AB = 5.$ Point $D$ is on $AB,$ and $CD$ bisects the right angle. The inscribed circles of $\triangle ADC$ and $\triangle BCD$ have radii $r_a$ and $r_b,$ respectively. What is $\frac{r_a}{r_b}?$
  1. $\dfrac{1}{28}(10 - \sqrt{2})$
  2. $\dfrac{3}{56}(10 - \sqrt{2})$
  3. $\dfrac{1}{14}(10 - \sqrt{2})$
  4. $\dfrac{5}{56}(10 - \sqrt{2})$
  5. $\dfrac{3}{28}(10 - \sqrt{2})$