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Difference between revisions of "2013 AIME II Problems/Problem 8"

(Problem 8)
 
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A hexagon that is inscribed in a circle has side lengths <math>22</math>, <math>22</math>, <math>20</math>, <math>22</math>, <math>22</math>, and <math>20</math> in that order. The radius of the circle can be written as <math>p+\sqrt{q}</math>, where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>.
 
A hexagon that is inscribed in a circle has side lengths <math>22</math>, <math>22</math>, <math>20</math>, <math>22</math>, <math>22</math>, and <math>20</math> in that order. The radius of the circle can be written as <math>p+\sqrt{q}</math>, where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>.
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==Solution==
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Let us call the hexagon <math>ABCDEF</math>, where <math>AB=CD=DE=AF=22</math>, and <math>BC=EF=20</math>.
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We can just consider one half of the hexagon, <math>ABCD</math>, to make matters simpler.
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Draw a line from the center of the circle, <math>O</math>, to the midpoint of <math>BC</math>, <math>E</math>. Now, draw a line from <math>O</math> to the midpoint of <math>AB</math>, <math>F</math>. Clearly, <math>\angle BEO=90^{\circ}</math>, because <math>BO=CO</math>, and <math>\angle BFO=90^{\circ}</math>, for similar reasons. Also notice that <math>\angle AOE=90^{\circ}</math>.
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Let us call <math>\angle BFO=\theta</math>. Therefore, <math>\angle AOB=2\theta</math>, and so <math>\angle AOE=90-2\theta</math>. Let us label the radius of the circle <math>r</math>. This means <cmath>\sin{\theta}=\frac{BF}{r}=\frac{11}{r}</cmath> <cmath>\sin{90-2\theta}=\frac{BE}{r}=\frac{10}{r}</cmath>
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Now we can use simple trigonometry to solve for <math>r</math>.
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Recall that  <math>\sin{90-\alpha}=\cos(\alpha)</math>: That means <math>\sin{90-2\theta}=\cos{2\theta}=\frac{10}{r}</math>
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Recall that <math>\cos{2\alpha}=1-2\sin^2{\alpha}</math>: That means <math>\cos{2\theta}=1-2\sin^2{\theta}=\frac{10}{r}</math>.
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Let <math>\sin{\theta}=x</math>.
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Substitute to get <math>x=\frac{11}{r}</math> and <math>1-2x^2=\frac{10}{r}</math>
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Now substitute the first equation into the second equation: <math>1-2\left(\frac{11}{r}\right)^2=\frac{10}{r}</math>
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Multiplying both sides by <math>r^2</math> and reordering gives us the quadratic <cmath>r^2-10r-242=0</cmath>
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Using the quadratic equation to solve, we get that <math>r=10+\sqrt{267}</math>, so the answer is <math>10+267=\boxed{277}</math>

Revision as of 20:31, 4 April 2013

A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.

Solution

Let us call the hexagon $ABCDEF$, where $AB=CD=DE=AF=22$, and $BC=EF=20$. We can just consider one half of the hexagon, $ABCD$, to make matters simpler. Draw a line from the center of the circle, $O$, to the midpoint of $BC$, $E$. Now, draw a line from $O$ to the midpoint of $AB$, $F$. Clearly, $\angle BEO=90^{\circ}$, because $BO=CO$, and $\angle BFO=90^{\circ}$, for similar reasons. Also notice that $\angle AOE=90^{\circ}$. Let us call $\angle BFO=\theta$. Therefore, $\angle AOB=2\theta$, and so $\angle AOE=90-2\theta$. Let us label the radius of the circle $r$. This means \[\sin{\theta}=\frac{BF}{r}=\frac{11}{r}\] \[\sin{90-2\theta}=\frac{BE}{r}=\frac{10}{r}\] Now we can use simple trigonometry to solve for $r$. Recall that $\sin{90-\alpha}=\cos(\alpha)$: That means $\sin{90-2\theta}=\cos{2\theta}=\frac{10}{r}$ Recall that $\cos{2\alpha}=1-2\sin^2{\alpha}$: That means $\cos{2\theta}=1-2\sin^2{\theta}=\frac{10}{r}$. Let $\sin{\theta}=x$. Substitute to get $x=\frac{11}{r}$ and $1-2x^2=\frac{10}{r}$ Now substitute the first equation into the second equation: $1-2\left(\frac{11}{r}\right)^2=\frac{10}{r}$ Multiplying both sides by $r^2$ and reordering gives us the quadratic \[r^2-10r-242=0\] Using the quadratic equation to solve, we get that $r=10+\sqrt{267}$, so the answer is $10+267=\boxed{277}$

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