Problem 142

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Problem 142: (2006 AMC 12A, 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 $r/s$?

$\mathrm{(A) \ } \frac{1}{2}\qquad$ $ \mathrm{(B) \ } \frac{5}{9}\qquad$ $ \mathrm{(C) \ } \frac{3}{5}\qquad$ $ \mathrm{(D) \ } \frac{5}{3}\qquad$ $ \mathrm{(E) \ } \frac{9}{5}$

Solution

Exercise 142: Solution

Square $ABCD$ has side length $s$, a circle centered at $E$ has radius $r$, and $r,s$ are 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 $A$, with $\overline{AF}$ tangent to the circle and $AF=\sqrt{16+6\sqrt2}$. What is $r/s$?

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