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Square and circle

Source: AMC 12 2006A, Problem 17

February 5, 2006
trigonometryquadraticsPythagorean Theoremgeometrytrig identitiesLaw of Cosinesalgebra

Problem Statement

Square ABCD ABCD has side length s s, a circle centered at E E has radius r r, and r r and s s are both rational. The circle passes through D D, and D D lies on BE‾ \overline{BE}. Point F F lies on the circle, on the same side of BE‾ \overline{BE} as A A. Segment AF AF is tangent to the circle, and AF \equal{} \sqrt {9 \plus{} 5\sqrt {2}}. What is r/s 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("AA",A,NW); label("BB",B,SW); label("CC",C,SE); label("DD",D,SW); label("EE",Ep,E); label("FF",F,NW);[/asy]<spanclass=′latex−bold′>(A)</span>12<spanclass=′latex−bold′>(B)</span>59<spanclass=′latex−bold′>(C)</span>35<spanclass=′latex−bold′>(D)</span>53<spanclass=′latex−bold′>(E)</span>95 <span class='latex-bold'>(A) </span> \frac {1}{2}\qquad <span class='latex-bold'>(B) </span> \frac {5}{9}\qquad <span class='latex-bold'>(C) </span> \frac {3}{5}\qquad <span class='latex-bold'>(D) </span> \frac {5}{3}\qquad <span class='latex-bold'>(E) </span> \frac {9}{5}
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