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Difference between revisions of "2007 Cyprus MO/Lyceum/Problem 15"

 
 
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==Problem==
 
==Problem==
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<div style="float:right">
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[[Image:2007 CyMO-15.PNG|250px]]
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</div>
 
The reflex angles of the concave octagon <math>ABCDEFGH</math> measure <math>240^\circ</math> each. Diagonals <math>AE</math> and <math>GC</math> are perpendicular, bisect each other, and are both equal to <math>2</math>.
 
The reflex angles of the concave octagon <math>ABCDEFGH</math> measure <math>240^\circ</math> each. Diagonals <math>AE</math> and <math>GC</math> are perpendicular, bisect each other, and are both equal to <math>2</math>.
  
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<math> \mathrm{(A) \ } \frac{6-2\sqrt{3}}{3}\qquad \mathrm{(B) \ } 8\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } \frac{6+2\sqrt{3}}{3}\qquad \mathrm{(E) \ } \mathrm{None\;of\;these}</math>
 
<math> \mathrm{(A) \ } \frac{6-2\sqrt{3}}{3}\qquad \mathrm{(B) \ } 8\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } \frac{6+2\sqrt{3}}{3}\qquad \mathrm{(E) \ } \mathrm{None\;of\;these}</math>
  
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{{image}}
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==Solution==
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The problem statement apparently misses one crucial piece of information: the fact that '''all the sides of the octagon are equal'''. Without this fact the octagon area is not uniquely determined. For example, we could move point <math>B</math> along a suitable arc (the locus of all points <math>X</math> such that <math>AXC</math> is <math>120^\circ</math>), and as this would change the height from <math>B</math> to <math>AC</math>, it would change the area of the triangle <math>ABC</math>, and hence the area of the octagon.
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With this additional assumption we can compute the area of <math>ABCDEFGH</math> as the area of the square <math>ACEG</math> (which is obviously <math>2</math>), minus four times the area of <math>ABC</math>.
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<asy>
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unitsize(4cm);
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defaultpen(0.8);
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pair a=(0,1), c=(1,0), bb=(a+c)/2, b=bb+dir(225)/sqrt(6);
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draw (a -- b -- c -- cycle);
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draw (a -- (0,0) -- c);
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draw (b -- bb);
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label ("$A$", a, N );
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label ("$B$", b, SW );
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label ("$C$", c, S );
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label ("$B'$", bb, NE );
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label ("$1$", a/2, W );
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</asy>
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In the triangle <math>ABC</math>, we have <math>AC=\sqrt 2</math>. Let <math>B'</math> be the foot of the height from <math>B</math> onto <math>AC</math>. As <math>AB=BC</math>, <math>B'</math> bisects <math>AC</math>. As the angle <math>ABC</math> is <math>120^\circ</math>, the angle <math>ABB'</math> is <math>60^\circ</math>.
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We now have <math>\frac{AB'}{BB'} = \tan 60^\circ = \sqrt 3</math>. Hence <math>BB'=\frac{AB'}{\sqrt 3} = \frac{1}{\sqrt 6}</math>.
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Then the area of triangle <math>ABC</math> is <math>\frac{BB' \cdot AC}2 = \frac{\sqrt 2}{2\sqrt 6} = \frac 1{2\sqrt 3} = \frac{\sqrt 3}6</math>.
  
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==Solution==
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Hence the area of the given octogon is <math>2 - 4\cdot \frac{\sqrt 3}6 = \boxed{\frac{6 - 2\sqrt 3}3}</math>.
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{{solution}}
 
  
 
==See also==
 
==See also==
 
{{CYMO box|year=2007|l=Lyceum|num-b=14|num-a=16}}
 
{{CYMO box|year=2007|l=Lyceum|num-b=14|num-a=16}}

Latest revision as of 13:26, 29 January 2009

Problem

2007 CyMO-15.PNG

The reflex angles of the concave octagon $ABCDEFGH$ measure $240^\circ$ each. Diagonals $AE$ and $GC$ are perpendicular, bisect each other, and are both equal to $2$.

The area of the octagon is

$\mathrm{(A) \ } \frac{6-2\sqrt{3}}{3}\qquad \mathrm{(B) \ } 8\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } \frac{6+2\sqrt{3}}{3}\qquad \mathrm{(E) \ } \mathrm{None\;of\;these}$

Solution

The problem statement apparently misses one crucial piece of information: the fact that all the sides of the octagon are equal. Without this fact the octagon area is not uniquely determined. For example, we could move point $B$ along a suitable arc (the locus of all points $X$ such that $AXC$ is $120^\circ$), and as this would change the height from $B$ to $AC$, it would change the area of the triangle $ABC$, and hence the area of the octagon.

With this additional assumption we can compute the area of $ABCDEFGH$ as the area of the square $ACEG$ (which is obviously $2$), minus four times the area of $ABC$.

[asy] unitsize(4cm); defaultpen(0.8); pair a=(0,1), c=(1,0), bb=(a+c)/2, b=bb+dir(225)/sqrt(6); draw (a -- b -- c -- cycle); draw (a -- (0,0) -- c); draw (b -- bb); label ("$A$", a, N ); label ("$B$", b, SW ); label ("$C$", c, S ); label ("$B'$", bb, NE ); label ("$1$", a/2, W ); [/asy]

In the triangle $ABC$, we have $AC=\sqrt 2$. Let $B'$ be the foot of the height from $B$ onto $AC$. As $AB=BC$, $B'$ bisects $AC$. As the angle $ABC$ is $120^\circ$, the angle $ABB'$ is $60^\circ$.

We now have $\frac{AB'}{BB'} = \tan 60^\circ = \sqrt 3$. Hence $BB'=\frac{AB'}{\sqrt 3} = \frac{1}{\sqrt 6}$.

Then the area of triangle $ABC$ is $\frac{BB' \cdot AC}2 = \frac{\sqrt 2}{2\sqrt 6} = \frac 1{2\sqrt 3} = \frac{\sqrt 3}6$.

Hence the area of the given octogon is $2 - 4\cdot \frac{\sqrt 3}6 = \boxed{\frac{6 - 2\sqrt 3}3}$.

See also

2007 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 14 		Followed by
Problem 16
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
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