Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m (→‎See Also)
(→‎Solution 2)
 
(74 intermediate revisions by 38 users not shown)
Line 17: Line 17:
  
 
==Solution 1==
 
==Solution 1==
−
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>.
+
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>.  
  
−
By AA 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>:
+
Now, we can see that <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.
  
−
<cmath> 45+4x = \frac{3}{4}\left(90+2x\right) </cmath>
+
The area of the square is then <math>\frac{45}{\frac{5}{12}}=9\cdot12=\boxed{\textbf{(B) } 108}</math>.
−
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>.
 
  
 
==Solution 2==
 
==Solution 2==
−
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>.  
+
<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 <math>\frac{1}{1+2}= \frac{1}{3}</math> the height, so has <math>\dfrac1{12}</math>th of the area of square(<math>\dfrac1{2}</math>*<math>\dfrac1{2}</math>*<math>\dfrac1{3}</math>). Thus, the area of the quadrilateral is <math>1-\frac{1}{2}-\frac{1}{12}=\frac{5}{12}</math> of the area of the square. The area of the square is then <math>45\cdot\dfrac{12}{5}=\boxed{\textbf{(B) } 108}</math>.
 +
 
 +
~minor edit by abirgh
 +
 
 +
==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 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>
 +
 
 +
~ moony_eyed
 +
 
 +
~ Minor Edits by WrenMath
 +
 
 +
==Solution 4==
 +
 
 +
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>
 +
 
 +
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.
 +
 
 +
==Solution 5 (easier)==
 +
 
 +
<math>\triangle ABF \sim \triangle FEC</math>, and the bases <math>AB</math> and <math>EC</math> are in the ratio <math>2:1</math>, so <math>\frac{[\triangle ABF]}{[\triangle FEC]}=\left(\frac{2}{1}\right)^2=\frac{4}{1}</math>, and <math>[\triangle ABF]=4\cdot[\triangle FEC]</math>
 +
 
 +
It is obvious that <math>[AFED]+[\triangle FEC]=45+[\triangle FEC]=\frac{1}{2}\cdot[ABCD]</math>
 +
 
 +
and <math>[AFED]+[\triangle ABF]=45+4\cdot[\triangle FEC]=\frac{3}{4}\cdot[ABCD]</math>
 +
 
 +
Solving the two equations we get <math>[ABCD]=\boxed{\textbf{(B)}108}</math>
 +
 
 +
- bbidev
 +
 
 +
==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
  
−
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.
+
== Video Solution only problem 22's by SpreadTheMathLove==
 +
https://www.youtube.com/watch?v=sOF1Okc0jMc
  
−
The area of the square is then <math>\frac{45}{\frac{5}{12}}=9\cdot12=\boxed{\textbf{(B)}108}</math>.
 
 
=See Also=
 
=See Also=
 
{{AMC8 box|year=2018|num-b=21|num-a=23}}
 
{{AMC8 box|year=2018|num-b=21|num-a=23}}
 
Set s to be the bottom left triangle.
 
Set s to be the bottom left triangle.
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 15:03, 19 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 of the area of square($\dfrac1{2}$*$\dfrac1{2}$*$\dfrac1{3}$). Thus, the area of the quadrilateral is $1-\frac{1}{2}-\frac{1}{12}=\frac{5}{12}$ of 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 (easier)

$\triangle ABF \sim \triangle FEC$, and the bases $AB$ and $EC$ are in the ratio $2:1$, so $\frac{[\triangle ABF]}{[\triangle FEC]}=\left(\frac{2}{1}\right)^2=\frac{4}{1}$, and $[\triangle ABF]=4\cdot[\triangle FEC]$

It is obvious that $[AFED]+[\triangle FEC]=45+[\triangle FEC]=\frac{1}{2}\cdot[ABCD]$

and $[AFED]+[\triangle ABF]=45+4\cdot[\triangle FEC]=\frac{3}{4}\cdot[ABCD]$

Solving the two equations we get $[ABCD]=\boxed{\textbf{(B)}108}$

- bbidev

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2018_AMC_8_Problems/Problem_22&oldid=240608"