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Difference between revisions of "2005 AMC 10B Problems/Problem 23"

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In trapezoid <math>ABCD</math> we have <math>\overline{AB}</math> parallel to <math>\overline{DC}</math>, <math>E</math> as the midpoint of <math>\overline{BC}</math>, and <math>F</math> as the midpoint of <math>\overline{DA}</math>. The area of <math>ABEF</math> is twice the area of <math>FECD</math>. What is <math>AB/DC</math>?
 
In trapezoid <math>ABCD</math> we have <math>\overline{AB}</math> parallel to <math>\overline{DC}</math>, <math>E</math> as the midpoint of <math>\overline{BC}</math>, and <math>F</math> as the midpoint of <math>\overline{DA}</math>. The area of <math>ABEF</math> is twice the area of <math>FECD</math>. What is <math>AB/DC</math>?
  
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<math>\mathrm{(A)} 2 \qquad \mathrm{(B)} 3 \qquad \mathrm{(C)} 5 \qquad \mathrm{(D)} 6 \qquad \mathrm{(E)} 8 </math>
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<math>\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 8 </math>
  
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== Solution ==
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== Solution 1==
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Since the height of both trapezoids are equal, and the area of <math>ABEF</math> is twice the area of <math>FECD</math>,  
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Since the heights of both trapezoids are equal, and the area of <math>ABEF</math> is twice the area of <math>FECD</math>,  
−
<math>\frac{AB+EF}{2}=2\left(\frac{CD+EF}{2}\right)</math>.
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−
<math>\frac{AB+EF}{2}=CD+EF</math>, so
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<math>\frac{AB+EF}{2}=2\left(\frac{DC+EF}{2}\right)</math>.
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<math>AB+EF=2CD+2EF</math>.
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<math>EF</math> is exactly halfway between <math>AB</math> and <math>CD</math>, so <math>EF=\frac{AB+CD}{2}</math>. <math>AB+\frac{AB+CD}{2}=2CD+AB+CD</math>, so
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<math>\frac{AB+EF}{2}=DC+EF</math>, so
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<math>\frac{3}{2}AB+\frac{1}{2}CD=3CD+AB</math>, and <math>\frac{1}{2}AB=\frac{5}{2}CD</math>. <math>AB/DC = \boxed{5}</math>.
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 +
<math>AB+EF=2DC+2EF</math>.
 +
 
 +
<math>EF</math> is exactly halfway between <math>AB</math> and <math>DC</math>, so <math>EF=\frac{AB+DC}{2}</math>.  
 +
 
 +
<math>AB+\frac{AB+DC}{2}=2DC+AB+DC</math>, so
 +
 
 +
<math>\frac{3}{2}AB+\frac{1}{2}DC=3DC+AB</math>, and  
 +
 
 +
<math>\frac{1}{2}AB=\frac{5}{2}DC</math>.
 +
 
 +
<math>\frac{AB}{DC} = \boxed{\textbf{(C) }5}</math>.
 +
 
 +
==Solution 2==
 +
 
 +
Mark <math>DC=z</math>, <math>AB=x</math>, and <math>FE=y.</math>
 +
Note that the heights of trapezoids <math>ABEF</math> & <math>FECD</math> are the same. Mark the height to be <math>h</math>.
 +
 
 +
Then, we have that <math>\tfrac{x+y}{2}\cdot h=2(\tfrac{y+z}{2} \cdot h)</math>.
 +
 
 +
From this, we get that <math>x=2z+y</math>.
 +
 
 +
We also get that <math>\tfrac{x+z}{2} \cdot 2h= 3(\tfrac{y+z}{2} \cdot h)</math>.
 +
 
 +
Simplifying, we get that <math>2x=z+3y</math>
 +
 
 +
Notice that we want <math>\tfrac{AB}{DC}=\tfrac{x}{z}</math>.
 +
 
 +
Dividing the first equation by <math>z</math>, we get that <math>\tfrac{x}{z}=2+\tfrac{y}{z}\implies 3(\tfrac{x}{z})=6+3(\tfrac{y}{z})</math>.
 +
 
 +
Dividing the second equation by <math>z</math>, we get that <math>2(\tfrac{x}{z})=1+3(\tfrac{y}{z})</math>.
 +
 
 +
Now, when we subtract the top equation from the bottom, we get that <math>\tfrac{x}{z}=\boxed{\textbf{(C) }5}</math>
 +
 
 +
==Solution 3==
 +
Since the bases of the trapezoids along with the height are the same, the only thing that matters is the second base. Denote the length of the bigger trapezoid <math>x</math>. The area of the smaller trapezoid is <math>A</math> = <math>h \cdot \frac {b_1 + b_2}{2}</math>. The area of the larger trapezoid is <math>A</math> = <math>h \cdot \frac {b_1 + x}{2}</math>. Since this problem asks for proportions, assume that <math>b_1 = 1</math> and <math>b_2 = 2</math>.
 +
 
 +
The smaller trapezoid has area <math>h</math> while the larger trapezoid must have area <math>5h</math>. We have the equation <math>\frac {x}{2} = 5</math>. <math>x</math> = 10, and our answer is <math>\boxed{\textbf{(C) }5}</math>
 +
 
 +
~Arcticturn
 +
 
 +
== Video Solution ==
 +
https://www.youtube.com/watch?v=fsNJbC3hGtk  ~David
  
 
== See Also ==
 
== See Also ==
 
{{AMC10 box|year=2005|ab=B|num-b=22|num-a=24}}
 
{{AMC10 box|year=2005|ab=B|num-b=22|num-a=24}}
 +
{{MAA Notice}}

Latest revision as of 18:22, 17 September 2023

Problem

In trapezoid $ABCD$ we have $\overline{AB}$ parallel to $\overline{DC}$, $E$ as the midpoint of $\overline{BC}$, and $F$ as the midpoint of $\overline{DA}$. The area of $ABEF$ is twice the area of $FECD$. What is $AB/DC$?

$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 8$

Solution 1

Since the heights of both trapezoids are equal, and the area of $ABEF$ is twice the area of $FECD$,

$\frac{AB+EF}{2}=2\left(\frac{DC+EF}{2}\right)$.

$\frac{AB+EF}{2}=DC+EF$, so

$AB+EF=2DC+2EF$.

$EF$ is exactly halfway between $AB$ and $DC$, so $EF=\frac{AB+DC}{2}$.

$AB+\frac{AB+DC}{2}=2DC+AB+DC$, so

$\frac{3}{2}AB+\frac{1}{2}DC=3DC+AB$, and

$\frac{1}{2}AB=\frac{5}{2}DC$.

$\frac{AB}{DC} = \boxed{\textbf{(C) }5}$.

Solution 2

Mark $DC=z$, $AB=x$, and $FE=y.$ Note that the heights of trapezoids $ABEF$ & $FECD$ are the same. Mark the height to be $h$.

Then, we have that $\tfrac{x+y}{2}\cdot h=2(\tfrac{y+z}{2} \cdot h)$.

From this, we get that $x=2z+y$.

We also get that $\tfrac{x+z}{2} \cdot 2h= 3(\tfrac{y+z}{2} \cdot h)$.

Simplifying, we get that $2x=z+3y$

Notice that we want $\tfrac{AB}{DC}=\tfrac{x}{z}$.

Dividing the first equation by $z$, we get that $\tfrac{x}{z}=2+\tfrac{y}{z}\implies 3(\tfrac{x}{z})=6+3(\tfrac{y}{z})$.

Dividing the second equation by $z$, we get that $2(\tfrac{x}{z})=1+3(\tfrac{y}{z})$.

Now, when we subtract the top equation from the bottom, we get that $\tfrac{x}{z}=\boxed{\textbf{(C) }5}$

Solution 3

Since the bases of the trapezoids along with the height are the same, the only thing that matters is the second base. Denote the length of the bigger trapezoid $x$. The area of the smaller trapezoid is $A$ = $h \cdot \frac {b_1 + b_2}{2}$. The area of the larger trapezoid is $A$ = $h \cdot \frac {b_1 + x}{2}$. Since this problem asks for proportions, assume that $b_1 = 1$ and $b_2 = 2$.

The smaller trapezoid has area $h$ while the larger trapezoid must have area $5h$. We have the equation $\frac {x}{2} = 5$. $x$ = 10, and our answer is $\boxed{\textbf{(C) }5}$

~Arcticturn

Video Solution

https://www.youtube.com/watch?v=fsNJbC3hGtk ~David

See Also

2005 AMC 10B (Problems • Answer Key • Resources)
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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