Problem:
Square ABCD has side length s, a circle centered at E has radius r, and r and s are both rational. The circle passes through D, and D lies on BE. Point F lies on the circle, on the same side of BE as A. Segment AF is tangent to the circle, and AF=9+52ββ. What is r/s?
Answer Choices:
A. 21β
B. 95β
C. 53β
D. 35β
E. 59β Solution:
Let B=(0,0),C=(s,0),A=(0,s),D=(s,s), and E=(s+2βrβ,s+2βrβ). Apply the Pythagorean Theorem to β³AFE to obtain
r2+(9+52β)=(s+2βrβ)2+(2βrβ)2
from which 9+52β=s2+rs2β. Because r and s are rational, it follows that s2=9 and rs=5, so r/s=5/9.
OR
Extend AD past D to meet the circle at Gξ =D. Because E is collinear with B and D,β³EDG is an isosceles right triangle. Thus DG=r2β. By the Power of a Point Theorem,
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