Problem
2006 AMC 12A Problem 17
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 \overline{BE}. Point F lies on the circle, on the same side of \overline{BE} as A. Segment AF is tangent to the circle, and AF=\sqrt{9+5\sqrt{2}}. What is r/s? ![[asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=3; pair B=(0,0), C=(3,0), D=(3,3), A=(0,3); pair Ep=(3+5*sqrt(2)/6,3+5*sqrt(2)/6); pair F=intersectionpoints(Circle(A,sqrt(9+5*sqrt(2))),Circle(Ep,5/3))[0]; pair[] dots={A,B,C,D,Ep,F}; draw(A--F); draw(Circle(Ep,5/3)); draw(A--B--C--D--cycle); dot(dots); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,SW); label("$E$",Ep,E); label("$F$",F,NW); [/asy]](/static/aops/5070473f4545f2932e536bc27857a5203bfe5458.png)
\mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{5}{9}\qquad \mathrm{(C) \ } \frac{3}{5}\qquad \mathrm{(D) \ } \frac{5}{3}\qquad \mathrm{(E) \ } \frac{9}{5}
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