Difference between revisions of "1996 AHSME Problems/Problem 30"

Revision as of 11:47, 30 September 2023

Problem

A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive sides each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 3 and the other with three sides each of length 5, has length equal to $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

$\textbf{(A)}\ 309 \qquad \textbf{(B)}\ 349 \qquad \textbf{(C)}\ 369 \qquad \textbf{(D)}\ 389 \qquad \textbf{(E)}\ 409$

Solution 1

In hexagon $ABCDEF$ , let $AB=BC=CD=3$ and let $DE=EF=FA=5$ . Since arc $BAF$ is one third of the circumference of the circle, it follows that $\angle BCF = \angle BEF=60^{\circ}$ . Similarly, $\angle CBE =\angle CFE=60^{\circ}$ . Let $P$ be the intersection of $\overline{BE}$ and $\overline{CF}$ , $Q$ that of $\overline{BE}$ and $\overline{AD}$ , and $R$ that of $\overline{CF}$ and $\overline{AD}$ . Triangles $EFP$ and $BCP$ are equilateral, and by symmetry, triangle $PQR$ is isosceles and thus also equilateral. $[asy] import olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); real angleUnit = 15; draw(Circle(origin,1)); pair D = dir(22.5); pair C = dir(3*angleUnit + degrees(D)); pair B = dir(3*angleUnit + degrees(C)); pair A = dir(3*angleUnit + degrees(B)); pair F = dir(5*angleUnit + degrees(A)); pair E = dir(5*angleUnit + degrees(F)); draw(A--B--C--D--E--F--cycle); dot("$A$",A,A); dot("$B$",B,B); dot("$C$",C,C); dot("$D$",D,D); dot("$E$",E,E); dot("$F$",F,F); draw(A--D^^B--E^^C--F); label("$3$",D--C,SW); label("$3$",B--C,S); label("$3$",A--B,SE); label("$5$",A--F,NE); label("$5$",F--E,N); label("$5$",D--E,NW); [/asy]$

Furthermore, $\angle BAD$ and $\angle BED$ subtend the same arc, as do $\angle ABE$ and $\angle ADE$ . Hence triangles $ABQ$ and $EDQ$ are similar. Therefore, $\frac{AQ}{EQ}=\frac{BQ}{DQ}=\frac{AB}{ED}=\frac{3}{5}.$ It follows that $\frac{\frac{AD-PQ}{2}}{PQ+5} =\frac{3}{5}\quad \mbox {and}\quad \frac{3-PQ}{\frac{AD+PQ}{2}}=\frac{3}{5}.$ Solving the two equations simultaneously yields $AD=360/49,$ so $m+n=\boxed{409}. \blacksquare$

Solution 2

All angle measures are in degrees. Let the first trapezoid be $ABCD$ , where $AB=BC=CD=3$ . Then the second trapezoid is $AFED$ , where $AF=FE=ED=5$ . We look for $AD$ .

Since $ABCD$ is an isosceles trapezoid, we know that $\angle BAD=\angle CDA$ and, since $AB=BC$ , if we drew $AC$ , we would see $\angle BCA=\angle BAC$ . Anyway, $\widehat{AB}=\widehat{BC}=\widehat{CD}$ ( $\widehat{AB}$ means arc AB). Using similar reasoning, $\widehat{AF}=\widehat{FE}=\widehat{ED}$ .

Let $\widehat{AB}=2\phi$ and $\widehat{AF}=2\theta$ . Since $6\theta+6\phi=360$ (add up the angles), $2\theta+2\phi=120$ and thus $\widehat{AB}+\widehat{AF}=\widehat{BF}=120$ . Therefore, $\angle FAB=\frac{1}{2}\widehat{BDF}=\frac{1}{2}(240)=120$ . $\angle CDE=120$ as well.

Now I focus on triangle $FAB$ . By the Law of Cosines, $BF^2=3^2+5^2-30\cos{120}=9+25+15=49$ , so $BF=7$ . Seeing $\angle ABF=\theta$ and $\angle AFB=\phi$ , we can now use the Law of Sines to get: $\sin{\phi}=\frac{3\sqrt{3}}{14}\;\text{and}\;\sin{\theta}=\frac{5\sqrt{3}}{14}.$

Now I focus on triangle $AFD$ . $\angle AFD=3\phi$ and $\angle ADF=\theta$ , and we are given that $AF=5$ , so $\frac{\sin{\theta}}{5}=\frac{\sin{3\phi}}{AD}.$ We know $\sin{\theta}=\frac{5\sqrt{3}}{14}$ , but we need to find $\sin{3\phi}$ . Using various identities, we see $\begin{align*}\sin{3\phi}&=\sin{(\phi+2\phi)}=\sin{\phi}\cos{2\phi}+\cos{\phi}\sin{2\phi}\\ &=\sin{\phi}(1-2\sin^2{\phi})+2\sin{\phi}\cos^2{\phi}\\ &=\sin{\phi}\left(1-2\sin^2{\phi}+2(1-\sin^2{\phi})\right)\\ &=\sin{\phi}(3-4\sin^2{\phi})\\ &=\frac{3\sqrt{3}}{14}\left(3-\frac{27}{49}\right)=\frac{3\sqrt{3}}{14}\left(\frac{120}{49}\right)=\frac{180\sqrt{3}}{343} \end{align*}$ Returning to finding $AD$ , we remember $\frac{\sin{\theta}}{5}=\frac{\sin{3\phi}}{AD}\;\text{so}\;AD=\frac{5\sin{3\phi}}{\sin{\theta}}.$ Plugging in and solving, we see $AD=\frac{360}{49}$ . Thus, the answer is $360 + 49 = 409$ , which is answer choice $\boxed{\textbf{(E)}}$ .

Solution 3

Let $x$ be the desired length. One can use Parameshvara's circumradius formula, which states that for a cyclic quadrilateral with sides $a, b, c, d$ the circumradius $R$ satisfies $R^2=\frac{1}{16}\cdot\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)},$ where $s$ is the semiperimeter. Applying this to the trapezoid with sides $3, 3, 3, x$ , we see that many terms cancel and we are left with $R^2=\frac{27}{9-x}$ Similar canceling occurs for the trapezoid with sides $5, 5, 5, x$ , and since the two quadrilaterals share the same circumradius, we can equate: $\frac{27}{9-x}=\frac{125}{15-x}$ Solving for $x$ gives $x=\frac{360}{49}$ , so the answer is $\fbox{(E) 409}$ .

Solution 4

Note that minor arc $\overarc{AB}$ is a third of the circumference, therefore, $\angle AOB = 120^{\circ}$ . Major arc $\overarc{AB}$ $=240^{\circ}$ , $\angle ACB = 120^{\circ}$

By the Law of Cosine, $AB = \sqrt{ 3^2 + 5^2 - 2 \cdot 3 \cdot 5 \cdot \cos 120^{\circ}} = \sqrt{ 9 + 25 + 15 } = 7$

$\frac{\angle AOB}{2} = 60^{\circ}$ , therefore, $r = \frac{\frac{AB}{2}}{\sin 60^{\circ}} = \frac{\frac{7}{2}}{ \frac{\sqrt{3}}{2} } = \frac{7\sqrt{3}}{3}$

$\sin \frac{\theta}{2} = \frac{\frac32}{r} = \frac{3}{2r} = \frac{3}{2 \cdot \frac{7\sqrt{3}}{3}} = \frac{3 \sqrt{3}}{14}$

Let $x$ be the length of the chord, $\sin \frac{3 \theta}{2} = \frac{\frac{x}{2}}{r}$

By the triple angle formula, $\sin \frac{3 \theta}{2} = 3 \cdot \sin \frac{\theta}{2} - 4 \cdot \sin(\frac{ \theta}{2})^3 = 3 \cdot \frac{3 \sqrt{3}}{14} - 4 \cdot (\frac{3 \sqrt{3}}{14})^3$

$x = 2 \cdot \frac{7\sqrt{3}}{3} \cdot [3 \cdot \frac{3 \sqrt{3}}{14} - 4 \cdot (\frac{3 \sqrt{3}}{14})^3] = 2 \cdot \frac{7\sqrt{3}}{3} \cdot (\frac{9\sqrt{3}}{14} - \frac{82\sqrt{3}}{2 \cdot 7^3}) = 9 - \frac{81}{49} = \frac{360}{49}$

Therefore, the answer is $\boxed{\textbf{(E) } 409}$ .

~isabelchen

Solution 5

Note that minor arc $\overarc{AB}$ is a third of the circumference, therefore, $\angle AOB = 120^{\circ}$ .

$\sin \frac{\alpha}{2} = \frac{\frac32}{r}$ , $\sin \frac{\alpha}{2} = \frac{3}{2r}$

$\sin \frac{120^{\circ}-\alpha}{2} = \frac{\frac52}{r}$ , $\sin (60^{\circ} - \frac{\alpha}{2}) = \frac{5}{2r}$

$\frac{\sin \frac{\alpha}{2}}{\sin (60^{\circ} - \frac{\alpha}{2}) } = \frac{\frac{3}{2r}}{\frac{5}{2r}} = \frac35$

$5 \cdot \sin \frac{\alpha}{2} = 3 \cdot \sin (60^{\circ} - \frac{\alpha}{2})$

$5 \cdot \sin \frac{\alpha}{2} = 3 ( \sin 60^{\circ} \cos \frac{\alpha}{2} - \sin \frac{\alpha}{2} \xos 60^{\circ} = 3 ( \frac{\sqrt{3}}{2} \cdot \cos \frac{\alpha}{2} - \frac12 \cdot \sin \frac{\alpha}{2})$ (Error compiling LaTeX. Unknown error_msg)13 \cdot \sin \frac{\alpha}{2} = \frac{3\sqrt{3}}{2} \cdot \cos \frac{\alpha}{2} $Let$ \sin \frac{\alpha}{2} = a $,$ \cos \frac{\alpha}{2} = \sqrt{1-a^2}$$ (Error compiling LaTeX. Unknown error_msg)13a = \frac{3\sqrt{3}}{2} \cdot \sqrt{1-a^2}$$ (Error compiling LaTeX. Unknown error_msg)169a^2 = 27-27a^2 $,$ 196a^2=27 $,$ \sin \frac{\alpha}{2} = a = \sqrt{\frac{27}{196}} = \frac{3 \sqrt{3}}{14} $Let$ x $be the length of the chord,$ \sin \frac{3 \theta}{2} = \frac{\frac{x}{2}}{r} $By the triple angle formula,$ \sin \frac{3 \theta}{2} = 3 \cdot \sin \frac{\theta}{2} - 4 \cdot \sin(\frac{ \theta}{2})^3 = 3 \cdot \frac{3 \sqrt{3}}{14} - 4 \cdot (\frac{3 \sqrt{3}}{14})^3$$ (Error compiling LaTeX. Unknown error_msg)x = 2 \cdot \frac{7\sqrt{3}}{3} \cdot [3 \cdot \frac{3 \sqrt{3}}{14} - 4 \cdot (\frac{3 \sqrt{3}}{14})^3] = 2 \cdot \frac{7\sqrt{3}}{3} \cdot (\frac{9\sqrt{3}}{14} - \frac{82\sqrt{3}}{2 \cdot 7^3}) = 9 - \frac{81}{49} = \frac{360}{49} $Therefore, the answer is$ \boxed{\textbf{(E) } 409}$.

~isabelchen

@@ Line 71: / Line 71: @@
 Therefore, the answer is <math>\boxed{\textbf{(E) } 409}</math>.
+~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
+==Solution 5==
+Note that minor arc <math>\overarc{AB}</math> is a third of the circumference, therefore, <math>\angle AOB = 120^{\circ}</math>.
+<math>\sin \frac{\alpha}{2} = \frac{\frac32}{r}</math>, <math>\sin \frac{\alpha}{2} = \frac{3}{2r}</math>
+<math>\sin \frac{120^{\circ}-\alpha}{2} = \frac{\frac52}{r}</math>, <math>\sin (60^{\circ} - \frac{\alpha}{2}) = \frac{5}{2r}</math>
+<math>\frac{\sin \frac{\alpha}{2}}{\sin (60^{\circ} - \frac{\alpha}{2}) } = \frac{\frac{3}{2r}}{\frac{5}{2r}} = \frac35</math>
+<math>5 \cdot \sin \frac{\alpha}{2} = 3 \cdot \sin (60^{\circ} - \frac{\alpha}{2})</math>
+<math>5 \cdot \sin \frac{\alpha}{2} = 3 ( \sin 60^{\circ} \cos \frac{\alpha}{2} - \sin \frac{\alpha}{2} \xos 60^{\circ} = 3 ( \frac{\sqrt{3}}{2} \cdot \cos \frac{\alpha}{2} - \frac12 \cdot \sin \frac{\alpha}{2})
+</math>13 \cdot \sin \frac{\alpha}{2} = \frac{3\sqrt{3}}{2} \cdot \cos \frac{\alpha}{2}<math>
+Let </math>\sin \frac{\alpha}{2} = a<math>, </math>\cos \frac{\alpha}{2} = \sqrt{1-a^2}<math>
+</math>13a = \frac{3\sqrt{3}}{2} \cdot  \sqrt{1-a^2}<math>
+</math>169a^2 = 27-27a^2<math>, </math>196a^2=27<math>, </math>\sin \frac{\alpha}{2} = a = \sqrt{\frac{27}{196}} = \frac{3 \sqrt{3}}{14}<math>
+Let </math>x<math> be the length of the chord, </math>\sin \frac{3 \theta}{2} = \frac{\frac{x}{2}}{r}<math>
+By the triple angle formula, </math>\sin \frac{3 \theta}{2} = 3 \cdot \sin \frac{\theta}{2} - 4 \cdot \sin(\frac{ \theta}{2})^3 = 3 \cdot \frac{3 \sqrt{3}}{14} - 4 \cdot (\frac{3 \sqrt{3}}{14})^3<math>
+</math>x = 2 \cdot \frac{7\sqrt{3}}{3} \cdot [3 \cdot \frac{3 \sqrt{3}}{14} - 4 \cdot (\frac{3 \sqrt{3}}{14})^3] = 2 \cdot \frac{7\sqrt{3}}{3} \cdot (\frac{9\sqrt{3}}{14} - \frac{82\sqrt{3}}{2 \cdot 7^3}) = 9 - \frac{81}{49} = \frac{360}{49}<math>
+Therefore, the answer is </math>\boxed{\textbf{(E) } 409}$.
 ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]

1996 AHSME (Problems • Answer Key • Resources)
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