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Difference between revisions of "1957 AHSME Problems/Problem 46"

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\textbf{(E)}\ \sqrt{65} </math>   
 
\textbf{(E)}\ \sqrt{65} </math>   
  
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== Solution ==
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== Solution 1 ==
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<math>\boxed{\textbf{(E) }\sqrt{65}}</math>.
+
<asy>
 +
 
 +
import geometry;
 +
 
 +
point A = (0,0);
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point B = (6,3);
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point C = (8,0);
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point D, P;
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 +
circle c = circumcircle(A,B,C);
 +
 
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// Triangle ABC w/ Circumcircle
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draw(triangle(A,B,C));
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dot(A);
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label("A",A,W);
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dot(B);
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label("B",B,N);
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dot(C);
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label("C",C,E);
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draw(c);
 +
 
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// Segment BD, Triangle ADC
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pair[] d = intersectionpoints(perpendicular(B,line(A,C)),c);
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D = d[0];
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dot(D);
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label("D",D,S);
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draw(B--D);
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draw(triangle(A,D,C));
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pair[] p = intersectionpoints(B--D,A--C);
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P = p[0];
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dot(P);
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label("P",P,SW);
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// Right angle mark
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markscalefactor = 0.0577;
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draw(rightanglemark(A,P,B));
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// Length Labels
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label("$3$", midpoint(B--P), W);
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label("$4$", midpoint(P--D), W);
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label("$6$", midpoint(A--P), S);
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label("$2$", midpoint(P--C), S);
 +
 
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</asy>
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 +
Let the chords intersect the circle at points <math>A,B,C,</math> and <math>D</math> to form simple polygon <math>ABCD</math>. Further, let the chords intersect at point <math>P</math> with <math>AP=6,PC=2,BP=3,</math> and <math>PD=4</math>, as in the diagram. Then, because <math>\overline{AC} \perp \overline{BD}</math>, by the [[Pythagorean Theorem]], <math>AB=3\sqrt5</math> and <math>BC=\sqrt{13}</math>. Because a circle is determined by three coplanar points, the [[circumcircle]] of <math>\triangle ABC</math> will be the circumcircle of <math>ABCD</math>, so the circumdiameter of <math>\triangle ABC</math> will be our desired answer. We know that <math>[\triangle ABC]=\tfrac{(6+2) \cdot 3}2=12</math>. Furthermore, we know that we can express this area as <math>\tfrac{abc}{4R}</math>, where <math>a,b,</math> and <math>c</math> are <math>\triangle ABC</math>'s side lengths and <math>R</math> is its circumradius. Setting this expression equal to <math>12</math>, we can now solve for <math>R</math>:
 +
\begin{align*}
 +
\frac{abc}{4R} &= 12 \\
 +
\frac{\sqrt{13} \cdot 8 \cdot 3\sqrt{15}}{4R} &= 12 \\
 +
\frac{2 \cdot 3\sqrt{65}}R &= 12 \\
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6\sqrt{65} &= 12R \\
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R &= \frac{\sqrt{65}} 2
 +
\end{align*}
 +
Because the problem asks for the diameter of the circle, our answer is <math>2R=2 \cdot \frac{\sqrt{65}}2=\boxed{\textbf{(E) }\sqrt{65}}</math>.
 +
 
 +
== Solution 2 (Answer choices, intution) ==
 +
Because one of the chords decribed in the problem has length <math>6+2=8</math>, we know that the diameter, the largest chord in the circle, must have a length <math>\geq 8</math>. This fact eliminates options (B) and (C). Furthermore, because the chord of length <math>8</math> is close to bisecting the other perpendicular chord (being only <math>0.5</math> off of its midpoint), it should be rather close to the length of the diameter. Because the circle's tangent lines are perpendicular to the diameter, moving a small distance away from the diameter will not rapidly decrease the length of the chord, so <math>AC</math> is only slightly less than the diameter's length. This fact guides us towards answer <math>\boxed{\textbf{(E) }\sqrt{65}}</math>.
  
 
== See Also ==
 
== See Also ==

Latest revision as of 12:54, 27 July 2024

Problem

Two perpendicular chords intersect in a circle. The segments of one chord are $3$ and $4$; the segments of the other are $6$ and $2$. Then the diameter of the circle is:

$\textbf{(A)}\ \sqrt{89}\qquad \textbf{(B)}\ \sqrt{56}\qquad \textbf{(C)}\ \sqrt{61}\qquad \textbf{(D)}\ \sqrt{75}\qquad \textbf{(E)}\ \sqrt{65}$

Solution 1

[asy] import geometry; point A = (0,0); point B = (6,3); point C = (8,0); point D, P; circle c = circumcircle(A,B,C); // Triangle ABC w/ Circumcircle draw(triangle(A,B,C)); dot(A); label("A",A,W); dot(B); label("B",B,N); dot(C); label("C",C,E); draw(c); // Segment BD, Triangle ADC pair[] d = intersectionpoints(perpendicular(B,line(A,C)),c); D = d[0]; dot(D); label("D",D,S); draw(B--D); draw(triangle(A,D,C)); pair[] p = intersectionpoints(B--D,A--C); P = p[0]; dot(P); label("P",P,SW); // Right angle mark markscalefactor = 0.0577; draw(rightanglemark(A,P,B)); // Length Labels label("$3$", midpoint(B--P), W); label("$4$", midpoint(P--D), W); label("$6$", midpoint(A--P), S); label("$2$", midpoint(P--C), S); [/asy]

Let the chords intersect the circle at points $A,B,C,$ and $D$ to form simple polygon $ABCD$. Further, let the chords intersect at point $P$ with $AP=6,PC=2,BP=3,$ and $PD=4$, as in the diagram. Then, because $\overline{AC} \perp \overline{BD}$, by the Pythagorean Theorem, $AB=3\sqrt5$ and $BC=\sqrt{13}$. Because a circle is determined by three coplanar points, the circumcircle of $\triangle ABC$ will be the circumcircle of $ABCD$, so the circumdiameter of $\triangle ABC$ will be our desired answer. We know that $[\triangle ABC]=\tfrac{(6+2) \cdot 3}2=12$. Furthermore, we know that we can express this area as $\tfrac{abc}{4R}$, where $a,b,$ and $c$ are $\triangle ABC$'s side lengths and $R$ is its circumradius. Setting this expression equal to $12$, we can now solve for $R$: \begin{align*} \frac{abc}{4R} &= 12 \\ \frac{\sqrt{13} \cdot 8 \cdot 3\sqrt{15}}{4R} &= 12 \\ \frac{2 \cdot 3\sqrt{65}}R &= 12 \\ 6\sqrt{65} &= 12R \\ R &= \frac{\sqrt{65}} 2 \end{align*} Because the problem asks for the diameter of the circle, our answer is $2R=2 \cdot \frac{\sqrt{65}}2=\boxed{\textbf{(E) }\sqrt{65}}$.

Solution 2 (Answer choices, intution)

Because one of the chords decribed in the problem has length $6+2=8$, we know that the diameter, the largest chord in the circle, must have a length $\geq 8$. This fact eliminates options (B) and (C). Furthermore, because the chord of length $8$ is close to bisecting the other perpendicular chord (being only $0.5$ off of its midpoint), it should be rather close to the length of the diameter. Because the circle's tangent lines are perpendicular to the diameter, moving a small distance away from the diameter will not rapidly decrease the length of the chord, so $AC$ is only slightly less than the diameter's length. This fact guides us towards answer $\boxed{\textbf{(E) }\sqrt{65}}$.

See Also

1957 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 45 		Followed by
Problem 47
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

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