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

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==Solution 1==
 
==Solution 1==
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Let the area of <math>\triangle CEF</math> be <math>x</math>. Thus, the area of triangle <math>\triangle ACD</math> is <math>45+x</math> and the area of the square is <math>2(45+x) = 90+2x</math>.
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We can use analytic geometry for this problem.
  
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By AAA similarity, <math>\triangle CEF \sim \triangle ABF</math> with a 1:2 ratio, so the area of triangle <math>\triangle ABF</math> is <math>4x</math>. Now consider trapezoid <math>ABED</math>. Its area is <math>45+4x</math>, which is three-fourths the area of the square. We set up an equation in <math>x</math>:
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Let us start by giving <math>D</math> the coordinate <math>(0,0)</math>, <math>A</math> the coordinate <math>(0,1)</math>, and so forth. <math>\overline{AC}</math> and <math>\overline{EB}</math> can be represented by the equations <math>y=-x+1</math> and <math>y=2x-1</math>, respectively. Solving for their intersection gives point <math>F</math> coordinates <math>\left(\frac{2}{3},\frac{1}{3}\right)</math>.
  
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<cmath> 45+4x = \frac{3}{4}\left(90+2x\right) </cmath>
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Now, <math>\triangle</math><math>EFC</math>’s area is simply <math>\frac{\frac{1}{2}\cdot\frac{1}{3}}{2}</math> or <math>\frac{1}{12}</math>. This means that pentagon <math>ABCEF</math>’s area is <math>\frac{1}{2}+\frac{1}{12}=\frac{7}{12}</math> of the entire square, and it follows that quadrilateral <math>AFED</math>’s area is <math>\frac{5}{12}</math> of the square.  
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Solving, we get <math>x = 9</math>. The area of square <math>ABCD</math> is <math>90+2x = 90 + 2 \cdot 9 = \boxed{\textbf{(B)} 108}</math>.
 
  
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==Solution 1.5==
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The area of the square is then <math>\frac{45}{\frac{5}{12}}=9\cdot12=\boxed{\textbf{(B) } 108}</math>.
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We notice some similar triangles. This is good, but you want to know the ratios in order to begin using them. Say that 45 is the area of half the triangle subtracted by <math>a</math>, so then the area of the whole square would be <math>90+2a</math>.
 
  
 
==Solution 2==
 
==Solution 2==
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We can use analytic geometry for this problem.
 
  
−
Let us start by giving <math>D</math> the coordinate <math>(0,0)</math>, <math>A</math> the coordinate <math>(0,1)</math>, and so forth. <math>\overline{AC}</math> and <math>\overline{EB}</math> can be represented by the equations <math>y=-x+1</math> and <math>y=2x-1</math>, respectively. Solving for their intersection gives point <math>F</math> coordinates <math>\left(\frac{2}{3},\frac{1}{3}\right)</math>.  
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<math>\triangle ABC</math> has half the area of the square.
 +
<math>\triangle FEC</math> has base equal to half the square side length, and by AA Similarity with <math>\triangle FBA</math>, it has 1/(1+2)= 1/3 the height, so has <math>\dfrac1{12}</math>th area of square. Thus, the area of the quadrilateral is <math>1-1/2-1/12=5/12</math> th the area of the square. The area of the square is then <math>45\cdot\dfrac{12}{5}=\boxed{\textbf{(B) } 108}</math>.
 +
 
 +
==Solution 3==
 +
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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 +
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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 +
==Solution 4==
 +
 
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Solution with Cartesian and Barycentric Coordinates:
 +
 
 +
We start with the following:
 +
 
 +
Claim: Given a square <math>ABCD</math>, let <math>E</math> be the midpoint of <math>\overline{DC}</math> and let <math>BE\cap AC = F</math>. Then <math>\frac {AF}{FC}=2</math>.
 +
 
 +
Proof: We use Cartesian coordinates. Let <math>D</math> be the origin, <math>A=(0,1),C=(0,1),B=(1,1)</math>. We have that <math>\overline{AC}</math> and <math>\overline{EB}</math> are governed by the equations <math>y=-x+1</math> and <math>y=2x-1</math>, respectively. Solving, <math>F=\left(\frac{2}{3},\frac{1}{3}\right)</math>. The result follows. <math>\square</math>
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 +
Now, we apply Barycentric Coordinates w.r.t. <math>\triangle ACD</math>. We let <math>A=(1,0,0),D=(0,1,0),C=(0,0,1)</math>. Then <math>E=(0,\tfrac 12,\tfrac 12),F=(\tfrac 13,0,\tfrac 23)</math>.
 +
 
 +
In the barycentric coordinate system, the area formula is <math>[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]</math> where <math>\triangle XYZ</math> is a random triangle and <math>\triangle ABC</math> is the reference triangle. Using this, we find that<cmath>\frac{[FEC]}{[ACD]}=\begin{vmatrix} 0&0&1\\ 0&\tfrac 12&\tfrac 12\\ \tfrac 13&0&\tfrac 23 \end{vmatrix}=\frac16.</cmath> Let <math>[FEC]=x</math> so that <math>[ACD]=45+x</math>. Then, we have <math>\frac{x}{x+45}=\frac 16 \Rightarrow x=9</math>, so the answer is <math>2(45+9)=\boxed{108}</math>.
 +
 
 +
Note: Please do not learn Barycentric Coordinates for the AMC 8.
 +
 
 +
==Video Solution by OmegaLearn==
 +
https://youtu.be/FDgcLW4frg8?t=4038
 +
 
 +
- pi_is_3.14
 +
 
 +
==Video Solutions==
 +
https://youtu.be/c4_-h7DsZFg
 +
 
 +
- Happytwin
  
−
Now, <math>\triangle</math><math>EFC</math>’s area is simply <math>\frac{\frac{1}{2}\cdot\frac{1}{3}}{2}</math> or <math>\frac{1}{12}</math>. This means that pentagon <math>ABCEF</math>’s area is <math>\frac{1}{2}+\frac{1}{12}=\frac{7}{12}</math> of the entire square, and it follows that quadrilateral <math>AFED</math>’s area is <math>\frac{5}{12}</math> of the square.
+
https://youtu.be/EJ-eFP3KHWg
  
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The area of the square is then <math>\frac{45}{\frac{5}{12}}=9\cdot12=\boxed{\textbf{(B)}108}</math>.
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~savannahsolver
  
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==See Also==
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=See Also=
 
{{AMC8 box|year=2018|num-b=21|num-a=23}}
 
{{AMC8 box|year=2018|num-b=21|num-a=23}}
−
 
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Set s to be the bottom left triangle.
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 13:28, 14 October 2024

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, $\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 1/(1+2)= 1/3 the height, so has $\dfrac1{12}$th area of square. Thus, the area of the quadrilateral is $1-1/2-1/12=5/12$ th the area of the square. The area of the square is then $45\cdot\dfrac{12}{5}=\boxed{\textbf{(B) } 108}$.

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 side length of the square 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

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.

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

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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