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Difference between revisions of "2018 AMC 8 Problems/Problem 22"

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==Solution 3==
 
==Solution 3==
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Extend <math>\overline{AD}</math> and <math>\overline{BE}</math> to meet at <math>X</math>. Drop an altitude from <math>F</math> to <math>\overline{CE}</math> and call it <math>h</math>. Also, call <math>\overline{CE}</math> <math>x</math>. As stated before, we have <math>\triangle ABF \sim \triangle CEF</math>, so the ratio of their heights is in a <math>1:2</math> ratio, making the altitude from <math>F</math> to <math>\overline{AB}</math> <math>2h</math>. Note that this means that the side of the square is <math>3h</math>. In addition, <math>\triangle XDE \sim \triangle XAB</math> by AA Similarity in a <math>1:2</math> ratio. This means that the side length of the square is <math>2x</math>, making <math>3h=2x</math>.
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Extend <math>\overline{AD}</math> and <math>\overline{BE}</math> to meet at <math>X</math>. Drop an altitude from <math>F</math> to <math>\overline{CE}</math> and call it <math>h</math>. Also, call <math>\overline{CE}</math> <math>x</math>. As stated before, we have <math>\triangle ABF \sim \triangle CEF</math>, so the ratio of their heights is in a <math>1:2</math> ratio, making the altitude from <math>F</math> to <math>\overline{AB}</math> <math>2h</math>. Note that this means that the side of the square is <math>3h</math>. In addition, <math>\triangle XDE \sim \triangle XAB</math> by AA Similarity in a <math>1:2</math> ratio. This means that the square's side length is <math>2x</math>, making <math>3h=2x</math>.
  
 
Now, note that <math>[ADEF]=[XAB]-[XDE]-[ABF]</math>. We have <math>[\triangle XAB]=(4x)(2x)/2=4x^2,</math> <math>[\triangle XDE]=(x)(2x)/2=x^2,</math> and <math>[\triangle ABF]=(2x)(2h)/2=(2x)(4x/3)/2=4x^2/3.</math> Subtracting makes <math>[ADEF]=4x^2-x^2-4x^2/3=5x^2/3.</math> We are given that <math>[ADEF]=45,</math> so <math>5x^2/3=45 \Rightarrow x^2=27.</math> Therefore, <math>x= 3 \sqrt{3},</math> so our answer is <math>(2x)^2=4x^2=4(27)=\boxed{\textbf{(B) }108}.</math>  
 
Now, note that <math>[ADEF]=[XAB]-[XDE]-[ABF]</math>. We have <math>[\triangle XAB]=(4x)(2x)/2=4x^2,</math> <math>[\triangle XDE]=(x)(2x)/2=x^2,</math> and <math>[\triangle ABF]=(2x)(2h)/2=(2x)(4x/3)/2=4x^2/3.</math> Subtracting makes <math>[ADEF]=4x^2-x^2-4x^2/3=5x^2/3.</math> We are given that <math>[ADEF]=45,</math> so <math>5x^2/3=45 \Rightarrow x^2=27.</math> Therefore, <math>x= 3 \sqrt{3},</math> so our answer is <math>(2x)^2=4x^2=4(27)=\boxed{\textbf{(B) }108}.</math>  
  
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- moony_eyed
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~ moony_eyed
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~ Minor Edits by WrenMath
  
 
==Solution 4==
 
==Solution 4==
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==Solution 5==
 
==Solution 5==
  
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Since <math>\triangle BEC</math>  is similar to <math>\triangle BAF</math> and the the base of <math>\triangle ECF</math> is half that of <math>\triangle BAF</math>'s, we conclude that the area of <math>\triangle BAF</math> is four times that of <math>\triangle ECF</math>. Using this fact, we make an equation: <math>\triangle ADC + \triangle BEC - \triangle ECF + 4 \cdot \triangle ECF =</math> area of <math>ABCD</math>.
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Since <math>\triangle BEC</math>  is similar to <math>\triangle BAF</math> and the the base of <math>\triangle ECF</math> is half that of <math>\triangle BAF</math>, we conclude that the area of <math>\triangle BAF</math> is four times that of <math>\triangle ECF</math>. Using this fact, we make an equation: <math>\triangle ADC + \triangle BEC - \triangle ECF + 4 \cdot \triangle ECF =</math> area of <math>ABCD</math>.
 
We can turn this equation into <math>3 \cdot \triangle ECF=1/4 ABCD</math>, so the area of <math>\triangle ECF</math> is <math>\frac{1}{12}</math> of <math>ABCD</math>.
 
We can turn this equation into <math>3 \cdot \triangle ECF=1/4 ABCD</math>, so the area of <math>\triangle ECF</math> is <math>\frac{1}{12}</math> of <math>ABCD</math>.
 
Since <math>\triangle ECF</math> is <math>\frac{1}{12}</math> of the area of <math>ABCD</math> and combined with <math>ADFE</math>, which has area <math>45</math> and is half the area of <math>ABCD</math>, <math>\frac{5}{12} ABCD=45</math>.
 
Since <math>\triangle ECF</math> is <math>\frac{1}{12}</math> of the area of <math>ABCD</math> and combined with <math>ADFE</math>, which has area <math>45</math> and is half the area of <math>ABCD</math>, <math>\frac{5}{12} ABCD=45</math>.

Latest revision as of 20:50, 9 January 2025

Problem 22

Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$

[asy] size(5cm); draw((0,0)--(6,0)--(6,6)--(0,6)--cycle); draw((0,6)--(6,0)); draw((3,0)--(6,6)); label("$A$",(0,6),NW); label("$B$",(6,6),NE); label("$C$",(6,0),SE); label("$D$",(0,0),SW); label("$E$",(3,0),S); label("$F$",(4,2),E); [/asy]

$\textbf{(A) } 100 \qquad \textbf{(B) } 108 \qquad \textbf{(C) } 120 \qquad \textbf{(D) } 135 \qquad \textbf{(E) } 144$

Solution 1

We can use analytic geometry for this problem.

Let us start by giving $D$ the coordinate $(0,0)$, $A$ the coordinate $(0,1)$, and so forth. $\overline{AC}$ and $\overline{EB}$ can be represented by the equations $y=-x+1$ and $y=2x-1$, respectively. Solving for their intersection gives point $F$ coordinates $\left(\frac{2}{3},\frac{1}{3}\right)$.

Now, we can see that $\triangle$$EFC$’s area is simply $\frac{\frac{1}{2}\cdot\frac{1}{3}}{2}$ or $\frac{1}{12}$. This means that pentagon $ABCEF$’s area is $\frac{1}{2}+\frac{1}{12}=\frac{7}{12}$ of the entire square, and it follows that quadrilateral $AFED$’s area is $\frac{5}{12}$ of the square.

The area of the square is then $\frac{45}{\frac{5}{12}}=9\cdot12=\boxed{\textbf{(B) } 108}$.

Solution 2

$\triangle ABC$ has half the area of the square. $\triangle FEC$ has base equal to half the square side length, and by AA Similarity with $\triangle FBA$, it has $\frac{1}{1+2}= \frac{1}{3}$ the height, so has $\dfrac1{12}$th area of square($\dfrac1{2}$*$\dfrac1{2}$*$\dfrac1{3}$). Thus, the area of the quadrilateral is $1-\frac{1}{2}-\frac{1}{12}=\frac{5}{12}$ th the area of the square. The area of the square is then $45\cdot\dfrac{12}{5}=\boxed{\textbf{(B) } 108}$.

~minor edit by abirgh

Solution 3

Extend $\overline{AD}$ and $\overline{BE}$ to meet at $X$. Drop an altitude from $F$ to $\overline{CE}$ and call it $h$. Also, call $\overline{CE}$ $x$. As stated before, we have $\triangle ABF \sim \triangle CEF$, so the ratio of their heights is in a $1:2$ ratio, making the altitude from $F$ to $\overline{AB}$ $2h$. Note that this means that the side of the square is $3h$. In addition, $\triangle XDE \sim \triangle XAB$ by AA Similarity in a $1:2$ ratio. This means that the square's side length is $2x$, making $3h=2x$.

Now, note that $[ADEF]=[XAB]-[XDE]-[ABF]$. We have $[\triangle XAB]=(4x)(2x)/2=4x^2,$ $[\triangle XDE]=(x)(2x)/2=x^2,$ and $[\triangle ABF]=(2x)(2h)/2=(2x)(4x/3)/2=4x^2/3.$ Subtracting makes $[ADEF]=4x^2-x^2-4x^2/3=5x^2/3.$ We are given that $[ADEF]=45,$ so $5x^2/3=45 \Rightarrow x^2=27.$ Therefore, $x= 3 \sqrt{3},$ so our answer is $(2x)^2=4x^2=4(27)=\boxed{\textbf{(B) }108}.$

~ moony_eyed

~ Minor Edits by WrenMath

Solution 4

Solution with Cartesian and Barycentric Coordinates:

We start with the following:

Claim: Given a square $ABCD$, let $E$ be the midpoint of $\overline{DC}$ and let $BE\cap AC = F$. Then $\frac {AF}{FC}=2$.

Proof: We use Cartesian coordinates. Let $D$ be the origin, $A=(0,1),C=(0,1),B=(1,1)$. We have that $\overline{AC}$ and $\overline{EB}$ are governed by the equations $y=-x+1$ and $y=2x-1$, respectively. Solving, $F=\left(\frac{2}{3},\frac{1}{3}\right)$. The result follows. $\square$

Now, we apply Barycentric Coordinates w.r.t. $\triangle ACD$. We let $A=(1,0,0),D=(0,1,0),C=(0,0,1)$. Then $E=(0,\tfrac 12,\tfrac 12),F=(\tfrac 13,0,\tfrac 23)$.

In the barycentric coordinate system, the area formula is $[XYZ]=\begin{vmatrix} x_{1} &y_{1} &z_{1} \\ x_{2} &y_{2} &z_{2} \\ x_{3}& y_{3} & z_{3} \end{vmatrix}\cdot [ABC]$ where $\triangle XYZ$ is a random triangle and $\triangle ABC$ is the reference triangle. Using this, we find that\[\frac{[FEC]}{[ACD]}=\begin{vmatrix} 0&0&1\\ 0&\tfrac 12&\tfrac 12\\ \tfrac 13&0&\tfrac 23 \end{vmatrix}=\frac16.\] Let $[FEC]=x$ so that $[ACD]=45+x$. Then, we have $\frac{x}{x+45}=\frac 16 \Rightarrow x=9$, so the answer is $2(45+9)=\boxed{108}$.

Note: Please do not learn Barycentric Coordinates for the AMC 8.


Solution 5

Since $\triangle BEC$ is similar to $\triangle BAF$ and the the base of $\triangle ECF$ is half that of $\triangle BAF$, we conclude that the area of $\triangle BAF$ is four times that of $\triangle ECF$. Using this fact, we make an equation: $\triangle ADC + \triangle BEC - \triangle ECF + 4 \cdot \triangle ECF =$ area of $ABCD$. We can turn this equation into $3 \cdot \triangle ECF=1/4 ABCD$, so the area of $\triangle ECF$ is $\frac{1}{12}$ of $ABCD$. Since $\triangle ECF$ is $\frac{1}{12}$ of the area of $ABCD$ and combined with $ADFE$, which has area $45$ and is half the area of $ABCD$, $\frac{5}{12} ABCD=45$. Solving this gives an area of $\boxed{108}$.


-Sixth Grader

Video Solution by OmegaLearn

https://youtu.be/FDgcLW4frg8?t=4038

- pi_is_3.14

Video Solutions

https://youtu.be/c4_-h7DsZFg

- Happytwin

https://youtu.be/EJ-eFP3KHWg

~savannahsolver

Video Solution only problem 22's by SpreadTheMathLove

https://www.youtube.com/watch?v=sOF1Okc0jMc

See Also

2018 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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 AJHSME/AMC 8 Problems and Solutions

Set s to be the bottom left triangle. The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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