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Angle Bisector+Incircles

Source: AMC 12 2008A Problem 20

February 17, 2008
geometryratiotrigonometryincenterinradiusanalytic geometryangle bisector

Problem Statement

Triangle ABC ABC has AC\equal{}3, BC\equal{}4, and AB\equal{}5. Point D D is on AB‾ \overline{AB}, and CD‾ \overline{CD} bisects the right angle. The inscribed circles of △ADC \triangle ADC and △BCD \triangle BCD have radii ra r_a and rb r_b, respectively. What is ra/rb r_a/r_b? (A)\ \frac{1}{28}\left(10\minus{}\sqrt{2}\right) \qquad (B)\ \frac{3}{56}\left(10\minus{}\sqrt{2}\right) \qquad (C)\ \frac{1}{14}\left(10\minus{}\sqrt{2}\right) \qquad (D)\ \frac{5}{56}\left(10\minus{}\sqrt{2}\right) \\ (E)\ \frac{3}{28}\left(10\minus{}\sqrt{2}\right)
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