Difference between revisions of "2003 AMC 10B Problems/Problem 23"

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{{duplicate|[[2003 AMC 12B Problems|2003 AMC 12B #15]] and [[2003 AMC 10B Problems|2003 AMC 10B #23]]}}
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==Problem==
 
==Problem==
  
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<math> \textbf{(A)}\ 1-\frac{\sqrt2}{2}\qquad\textbf{(B)}\ \frac{\sqrt2}{4}\qquad\textbf{(C)}\ \sqrt2-1\qquad\textbf{(D)}\ \frac{1}2\qquad\textbf{(E)}\ \frac{1+\sqrt2}{4} </math>
 
<math> \textbf{(A)}\ 1-\frac{\sqrt2}{2}\qquad\textbf{(B)}\ \frac{\sqrt2}{4}\qquad\textbf{(C)}\ \sqrt2-1\qquad\textbf{(D)}\ \frac{1}2\qquad\textbf{(E)}\ \frac{1+\sqrt2}{4} </math>
  
==Solution==
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==Video Solution==
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https://www.youtube.com/watch?v=LREcUjK-56U&feature=youtu.be
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 +
 
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==Solution 1==
  
An easy way to look at this:
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Here is an easy way to look at this, where <math>p</math> is the perimeter, and <math>a</math> is the [[apothem]]:
  
 
Area of Octagon: <math> \frac{ap}{2}=1 </math>.
 
Area of Octagon: <math> \frac{ap}{2}=1 </math>.
  
Area of Rectangle: <math> \frac{p}{8}\times 2a=\frac{ap}{4} </math>.
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Area of Rectangle: <math> \frac{p}{8}\times 2a=\dfrac{2ap}{8}=\frac{ap}{4} </math>.
  
 
You can see from this that the octagon's area is twice as large as the rectangle's area is <math>\boxed{\textbf{(D)}\ \frac{1}{2}}</math>.
 
You can see from this that the octagon's area is twice as large as the rectangle's area is <math>\boxed{\textbf{(D)}\ \frac{1}{2}}</math>.
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==Solution 2==
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<asy>
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unitsize(1cm);
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defaultpen(linewidth(.8pt)+fontsize(8pt));
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pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5);
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draw(A--B--C--D--E--F--G--H--cycle);
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label("$A$",A,NNW);
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label("$B$",B,NNE);
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label("$C$",C,ENE);
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label("$D$",D,ESE);
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label("$E$",E,SSE);
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label("$F$",F,SSW);
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label("$G$",G,WSW);
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label("$H$",H,WNW);
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draw(A--E--F--B--C--G--H--D);
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draw(A--E--F--B--A,blue);
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draw(A--F--E--B--A,red);
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</asy>
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Here is a less complicated way than that of the user below. If you draw a line segment from each vertex to the center of the octagon and draw the rectangle ABEF (in red), you can see that <math>2</math> of the triangles (in blue) share the same base and height with <math>\dfrac{1}{2}</math> the rectangle. Therefore, the rectangle's area is the same as <math>2\cdot2</math> of the <math>8</math> triangles, and is  <math>\boxed{\textbf{(D)}\ \frac{1}{2}}</math> the area of the octagon.
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==Solution 3==
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<asy>
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unitsize(1cm);
 +
defaultpen(linewidth(.8pt)+fontsize(8pt));
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pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5);
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draw(A--B--C--D--E--F--G--H--cycle);
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label("$A$",A,NNW);
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label("$B$",B,NNE);
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label("$C$",C,ENE);
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label("$D$",D,ESE);
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label("$E$",E,SSE);
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label("$F$",F,SSW);
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label("$G$",G,WSW);
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label("$H$",H,WNW);
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draw(A--D--E--H--G--B--C--F);
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</asy>
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Drawing lines <math>AD</math>, <math>BG</math>, <math>CF</math>, and <math>EH</math>, we can see that the octagon is comprised of <math>1</math> square, <math>4</math> rectangles, and <math>4</math> triangles. The triangles each are <math>45-45-90</math> triangles, and since their diagonal is length <math>x</math>, each of their sides is <math>\frac{\sqrt{2}}{2}x</math>. The area of the entire figure is, likewise, <math>x^2</math> (the square)<math>+4x^2\frac{\sqrt{2}}{2}</math> (the 4 rectangles)<math> +2\cdot(\frac{\sqrt{2}}{2}x)^2</math> (the triangles), which simplifies to <math>2x^2 + 2\sqrt{2}x^2</math>. The area of <math>ABEF</math> is just <math>x(x+\frac{2\sqrt{2}}{2}x)</math>, or <math>x^2</math> + <math>x^2\sqrt{2}</math>, which we can see is the area of <math>\frac{ABCDEFGH}{2} =  \boxed{\textbf{(D)}\ \frac{1}{2}}</math> the area of the octagon.
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==Solution 4==
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First, we are going to divide the diagram. Now we need to find the ratio of the area of the rectangle to the area of the trapezoid.
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 +
<asy> unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); draw(A--B--C--D--E--F--G--H--cycle); label("$A$",A,NNW); label("$B$",B,NNE); label("$C$",C,ENE); label("$D$",D,ESE); label("$E$",E,SSE); label("$F$",F,SSW); label("$G$",G,WSW); label("$H$",H,WNW);draw(A--F);draw(B--E);draw(D--G);draw(C--H);</asy>
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The area of a trapezoid is <math>\frac{b_1 + b_2}{2}h</math>
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Note that the trapezoid is made out of 2 45-45-90 triangles and a rectangle, and that <math>AH=FG=1</math>. By realizing that, the area of the trapezoid is <math>(\frac{2+\sqrt{2}}{2})(\frac{\sqrt{2}}{2}</math>). To make this product easier, note there is two trapezoids, so the new product is now this, <math>(\frac{2+\sqrt{2}}{2})(\sqrt{2}) = \sqrt{2} + 1</math>
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Notice how the rectangle has side lengths <math>\sqrt{2}+1</math> and <math>1</math>, so it's area is also <math>\sqrt{2}+1</math>.
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The ratio of the area of rectangle <math>ABEF</math> to the two trapezoids is <math>1:1</math>, meaning they share half the area of the octogon. Since the area “op0;op9o;lopof the octogon is 1, <math>\therefore</math> the area of the rectangle is <math>\boxed{\textbf{(D)}\ \frac{1}{2}}</math>.
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~ghfhgvghj10
  
 
==See Also==
 
==See Also==
 +
{{AMC12 box|year=2003|ab=B|num-b=14|num-a=16}}
 
{{AMC10 box|year=2003|ab=B|num-b=22|num-a=24}}
 
{{AMC10 box|year=2003|ab=B|num-b=22|num-a=24}}
 +
{{MAA Notice}}

Latest revision as of 19:07, 10 April 2023

The following problem is from both the 2003 AMC 12B #15 and 2003 AMC 10B #23, so both problems redirect to this page.

Problem

A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ABEF$?

[asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); draw(A--B--C--D--E--F--G--H--cycle); label("$A$",A,NNW); label("$B$",B,NNE); label("$C$",C,ENE); label("$D$",D,ESE); label("$E$",E,SSE); label("$F$",F,SSW); label("$G$",G,WSW); label("$H$",H,WNW);[/asy]

$\textbf{(A)}\ 1-\frac{\sqrt2}{2}\qquad\textbf{(B)}\ \frac{\sqrt2}{4}\qquad\textbf{(C)}\ \sqrt2-1\qquad\textbf{(D)}\ \frac{1}2\qquad\textbf{(E)}\ \frac{1+\sqrt2}{4}$

Video Solution

https://www.youtube.com/watch?v=LREcUjK-56U&feature=youtu.be


Solution 1

Here is an easy way to look at this, where $p$ is the perimeter, and $a$ is the apothem:

Area of Octagon: $\frac{ap}{2}=1$.

Area of Rectangle: $\frac{p}{8}\times 2a=\dfrac{2ap}{8}=\frac{ap}{4}$.

You can see from this that the octagon's area is twice as large as the rectangle's area is $\boxed{\textbf{(D)}\ \frac{1}{2}}$.

Solution 2

[asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); draw(A--B--C--D--E--F--G--H--cycle); label("$A$",A,NNW); label("$B$",B,NNE); label("$C$",C,ENE); label("$D$",D,ESE); label("$E$",E,SSE); label("$F$",F,SSW); label("$G$",G,WSW); label("$H$",H,WNW);  draw(A--E--F--B--C--G--H--D); draw(A--E--F--B--A,blue); draw(A--F--E--B--A,red); [/asy]

Here is a less complicated way than that of the user below. If you draw a line segment from each vertex to the center of the octagon and draw the rectangle ABEF (in red), you can see that $2$ of the triangles (in blue) share the same base and height with $\dfrac{1}{2}$ the rectangle. Therefore, the rectangle's area is the same as $2\cdot2$ of the $8$ triangles, and is $\boxed{\textbf{(D)}\ \frac{1}{2}}$ the area of the octagon.

Solution 3

[asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); draw(A--B--C--D--E--F--G--H--cycle); label("$A$",A,NNW); label("$B$",B,NNE); label("$C$",C,ENE); label("$D$",D,ESE); label("$E$",E,SSE); label("$F$",F,SSW); label("$G$",G,WSW); label("$H$",H,WNW);  draw(A--D--E--H--G--B--C--F); [/asy] Drawing lines $AD$, $BG$, $CF$, and $EH$, we can see that the octagon is comprised of $1$ square, $4$ rectangles, and $4$ triangles. The triangles each are $45-45-90$ triangles, and since their diagonal is length $x$, each of their sides is $\frac{\sqrt{2}}{2}x$. The area of the entire figure is, likewise, $x^2$ (the square)$+4x^2\frac{\sqrt{2}}{2}$ (the 4 rectangles)$+2\cdot(\frac{\sqrt{2}}{2}x)^2$ (the triangles), which simplifies to $2x^2 + 2\sqrt{2}x^2$. The area of $ABEF$ is just $x(x+\frac{2\sqrt{2}}{2}x)$, or $x^2$ + $x^2\sqrt{2}$, which we can see is the area of $\frac{ABCDEFGH}{2} =  \boxed{\textbf{(D)}\ \frac{1}{2}}$ the area of the octagon.

Solution 4

First, we are going to divide the diagram. Now we need to find the ratio of the area of the rectangle to the area of the trapezoid.

[asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); draw(A--B--C--D--E--F--G--H--cycle); label("$A$",A,NNW); label("$B$",B,NNE); label("$C$",C,ENE); label("$D$",D,ESE); label("$E$",E,SSE); label("$F$",F,SSW); label("$G$",G,WSW); label("$H$",H,WNW);draw(A--F);draw(B--E);draw(D--G);draw(C--H);[/asy]

The area of a trapezoid is $\frac{b_1 + b_2}{2}h$ Note that the trapezoid is made out of 2 45-45-90 triangles and a rectangle, and that $AH=FG=1$. By realizing that, the area of the trapezoid is $(\frac{2+\sqrt{2}}{2})(\frac{\sqrt{2}}{2}$). To make this product easier, note there is two trapezoids, so the new product is now this, $(\frac{2+\sqrt{2}}{2})(\sqrt{2}) = \sqrt{2} + 1$

Notice how the rectangle has side lengths $\sqrt{2}+1$ and $1$, so it's area is also $\sqrt{2}+1$.

The ratio of the area of rectangle $ABEF$ to the two trapezoids is $1:1$, meaning they share half the area of the octogon. Since the area “op0;op9o;lopof the octogon is 1, $\therefore$ the area of the rectangle is $\boxed{\textbf{(D)}\ \frac{1}{2}}$.

~ghfhgvghj10

See Also

2003 AMC 12B (ProblemsAnswer KeyResources)
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
All AMC 12 Problems and Solutions
2003 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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
All AMC 10 Problems and Solutions

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