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Difference between revisions of "1972 AHSME Problems/Problem 32"
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== Problem ==
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<asy>
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real t=pi/12;real u=8*t;
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real cu=cos(u);real su=sin(u);
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draw(unitcircle);
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draw((cos(-t),sin(-t))--(cos(13*t),sin(13*t)));
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draw((cu,su)--(cu,-su));
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label("A",(cos(13*t),sin(13*t)),W);
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label("B",(cos(-t),sin(-t)),E);
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label("C",(cu,su),N);
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label("D",(cu,-su),S);
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label("E",(cu,sin(-t)),NE);
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label("2",((cu-1)/2,sin(-t)),N);
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label("6",((cu+1)/2,sin(-t)),N);
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label("3",(cu,(sin(-t)-su)/2),E);
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//Credit to Zimbalono for the diagram</asy>
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Chords <math>AB</math> and <math>CD</math> in the circle above intersect at E and are perpendicular to each other.
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If segments <math>AE, EB</math>, and <math>ED</math> have measures <math>2, 3</math>, and <math>6</math> respectively, then the length of the diameter of the circle is
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<math>\textbf{(A) }4\sqrt{5}\qquad
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\textbf{(B) }\sqrt{65}\qquad
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\textbf{(C) }2\sqrt{17}\qquad
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\textbf{(D) }3\sqrt{7}\qquad
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\textbf{(E) }6\sqrt{2}  </math>
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== Solution ==
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<math>\boxed{B}</math>
 
<math>\boxed{B}</math>

Latest revision as of 14:11, 23 June 2021

Problem

[asy] real t=pi/12;real u=8*t; real cu=cos(u);real su=sin(u); draw(unitcircle); draw((cos(-t),sin(-t))--(cos(13*t),sin(13*t))); draw((cu,su)--(cu,-su)); label("A",(cos(13*t),sin(13*t)),W); label("B",(cos(-t),sin(-t)),E); label("C",(cu,su),N); label("D",(cu,-su),S); label("E",(cu,sin(-t)),NE); label("2",((cu-1)/2,sin(-t)),N); label("6",((cu+1)/2,sin(-t)),N); label("3",(cu,(sin(-t)-su)/2),E); //Credit to Zimbalono for the diagram[/asy]

Chords $AB$ and $CD$ in the circle above intersect at E and are perpendicular to each other. If segments $AE, EB$, and $ED$ have measures $2, 3$, and $6$ respectively, then the length of the diameter of the circle is

$\textbf{(A) }4\sqrt{5}\qquad \textbf{(B) }\sqrt{65}\qquad \textbf{(C) }2\sqrt{17}\qquad \textbf{(D) }3\sqrt{7}\qquad \textbf{(E) }6\sqrt{2}$

Solution

$\boxed{B}$

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