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Difference between revisions of "2003 AIME II Problems/Problem 14"

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(yay i can do geometry! :D)
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== Solution ==
 
== Solution ==
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From this image, we can see that the oordinate of F is 4, and from this, we can gather that the oordinates of E, D, and C, respectively, are 8, 10, and 6.
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{{image}}
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In this image, we have drawn perpendiculars to the <math>x</math>-axis from F and B, and have labeled the angle between the <math>x</math>-axis and segment <math>AB</math> <math>x</math>. Since <math>ABCDEF</math> is a regular hexagon, <math>\sin{(60-x)}=2\sin{x}</math>. Expanding, we have
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<center><math>\sin{60}\cos{x}-\cos{60}\sin{x}=\frac{\sqrt{3}\cos{x}}{2}-\frac{\sin{x}}{2}=2\sin{x}</math></center>
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Isolating <math>\sin{x}</math> we see that <math>\frac{\sqrt{3}\cos{x}}{2}=\frac{5\sin{x}}{2}</math>, or <math>\cos{x}=\frac{5}{\sqrt{3}}\sin{x}</math>. Using the fact that <math>\sin^2{x}+\cos^2{x}=1</math>, we have <math>\frac{28}{3}\sin^2{x}=1</math>, or <math>\sin{x}=\sqrt{\frac{3}{28}}</math>. Letting the side length of the hexagon be <math>y</math>, we have <math>\frac{2}{y}=\sqrt{\frac{3}{28}}</math>. After simplification we see that <math>y=\frac{4\sqrt{21}}{3}</math>. The area of the hexagon is <math>\frac{3y^2\sqrt{3}}{2}</math>, so the area of the hexagon is <math>\frac{3*\frac{16*21}{3^2}*\sqrt{3}}{2}=56\sqrt{3}</math>, or <math>m+n=\boxed{059}</math>.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2003|n=II|num-b=13|num-a=15}}
 
{{AIME box|year=2003|n=II|num-b=13|num-a=15}}
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[[Category:Intermediate Geometry Problems]]

Revision as of 10:49, 4 January 2009

Problem

Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,2,4,6,8,10\}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where $m$ and $n$ are positive integers and n is not divisible by the square of any prime. Find $m + n.$

Solution

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

From this image, we can see that the oordinate of F is 4, and from this, we can gather that the oordinates of E, D, and C, respectively, are 8, 10, and 6.

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

In this image, we have drawn perpendiculars to the $x$-axis from F and B, and have labeled the angle between the $x$-axis and segment $AB$ $x$. Since $ABCDEF$ is a regular hexagon, $\sin{(60-x)}=2\sin{x}$. Expanding, we have

$\sin{60}\cos{x}-\cos{60}\sin{x}=\frac{\sqrt{3}\cos{x}}{2}-\frac{\sin{x}}{2}=2\sin{x}$

Isolating $\sin{x}$ we see that $\frac{\sqrt{3}\cos{x}}{2}=\frac{5\sin{x}}{2}$, or $\cos{x}=\frac{5}{\sqrt{3}}\sin{x}$. Using the fact that $\sin^2{x}+\cos^2{x}=1$, we have $\frac{28}{3}\sin^2{x}=1$, or $\sin{x}=\sqrt{\frac{3}{28}}$. Letting the side length of the hexagon be $y$, we have $\frac{2}{y}=\sqrt{\frac{3}{28}}$. After simplification we see that $y=\frac{4\sqrt{21}}{3}$. The area of the hexagon is $\frac{3y^2\sqrt{3}}{2}$, so the area of the hexagon is $\frac{3*\frac{16*21}{3^2}*\sqrt{3}}{2}=56\sqrt{3}$, or $m+n=\boxed{059}$.

See also

2003 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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