Problem:
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 △ADC and △BCD have radii ra and rb, respectively. What is ra/rb?
Answer Choices:
A. 281(10−2)
B. 563(10−2)
C. 141(10−2)
D. 565(10−2)
E. 283(10−2) Solution:
By the Angle Bisector Theorem,
AD=5⋅3+43=715 and BD=5⋅3+44=720
To determine CD, start with the relation Area (△ADC)+Area(△BCD)= Area (△ABC) to get
223⋅CD+224⋅CD=23⋅4
This gives CD=7122. Now use the fact that the area of a triangle is given by rs, where r is the radius of the inscribed circle and s is half the perimeter of the triangle. The ratio of the area of △ADC to the area of △BCD is the ratio of the altitudes to their common base CD, which is BDAD=43. Hence