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MATH / 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 …

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problem

Square ABCDABCD has side length ss, a circle centered at EE has radius rr, and rr and ss are both rational. The circle passes through DD, and DD lies on BE¯\overline{BE}. Point FF lies on the circle, on the same side of BE¯\overline{BE} as AA. Segment AFAF is tangent to the circle, and AF=9+52AF=\sqrt{9+5\sqrt{2}}. What is r/sr/s? [asy] pair A,B,C,D,I,F; A=(0,10); B=(0,0); C=(10,0); D=(10,10); I=(14,13); F=(11,17); draw(A--B--C--D--cycle,linewidth(0.7)); draw(Circle(I,5),linewidth(0.7)); draw(A--F,linewidth(0.7)); label("AA",A,NW); label("BB",B,SW); label("CC",C,SE); label("DD",D,SW); label("FF",F,N); label("EE",I,E); dot(I); [/asy]
Plain-text mathematical notation (without MathML)
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+5√(2)). What is r/s?

[asy]
pair A,B,C,D,I,F;
A=(0,10); B=(0,0); C=(10,0); D=(10,10);

I=(14,13); F=(11,17);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(Circle(I,5),linewidth(0.7));
draw(A--F,linewidth(0.7));
label("A",A,NW);
label("B",B,SW);
label("C",C,SE);
label("D",D,SW);
label("F",F,N);
label("E",I,E);
dot(I);
[/asy]
Original LaTeX notation
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]
pair A,B,C,D,I,F;
A=(0,10); B=(0,0); C=(10,0); D=(10,10);

I=(14,13); F=(11,17);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(Circle(I,5),linewidth(0.7));
draw(A--F,linewidth(0.7));
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,SW);
label("$F$",F,N);
label("$E$",I,E);
dot(I);
[/asy]

level

Level 5

type

Geometry

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Source and history

task_id
4739d646-ed3c-5370-a83d-1f37ef4a1404
task_key
geometry--test--4739d646-ed3c-5370-a83d-1f37ef4a1404
task_revision_id
1
config / split
geometry / test

Official source

initial import