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 "2003 AMC 10B Problems/Problem 20"

(→‎Problem)
m (→‎Solution 2)
 
(16 intermediate revisions by 8 users not shown)
Line 6: Line 6:
 
unitsize(8mm);
 
unitsize(8mm);
 
defaultpen(linewidth(.8pt)+fontsize(10pt));
 
defaultpen(linewidth(.8pt)+fontsize(10pt));
−
dotfactor=4;
+
dotfactor=0;
  
 
pair A=(0,0), B=(5,0), C=(5,3), D=(0,3);
 
pair A=(0,0), B=(5,0), C=(5,3), D=(0,3);
Line 31: Line 31:
 
<math>\textbf{(A) } 10 \qquad\textbf{(B) } \frac{21}{2} \qquad\textbf{(C) } 12 \qquad\textbf{(D) } \frac{25}{2} \qquad\textbf{(E) } 15</math>
 
<math>\textbf{(A) } 10 \qquad\textbf{(B) } \frac{21}{2} \qquad\textbf{(C) } 12 \qquad\textbf{(D) } \frac{25}{2} \qquad\textbf{(E) } 15</math>
  
−
==Solution==
+
==Solution 1==
−
<center><asy>
 
−
unitsize(8mm);
 
−
defaultpen(linewidth(.8pt)+fontsize(10pt));
 
−
dotfactor=4;
 
−
 
 
−
pair A=(0,0), B=(5,0), C=(5,3), D=(0,3);
 
−
pair F=(1,3), G=(3,3);
 
−
pair E=(5/3,5);
 
−
 
 
−
draw(A--B--C--D--cycle);
 
−
draw(A--E);
 
−
draw(B--E);
 
−
 
 
−
pair[] ps={A,B,C,D,E,F,G}; dot(ps);
 
−
label("$A$",A,SW);
 
−
label("$B$",B,SE);
 
−
label("$C$",C,NE);
 
−
label("$D$",D,NW);
 
−
label("$E$",E,N);
 
−
label("$F$",F,SE);
 
−
label("$G$",G,SW);
 
−
label("$1$",midpoint(D--F),N);
 
−
label("$2$",midpoint(G--C),N);
 
−
label("$5$",midpoint(A--B),S);
 
−
label("$3$",midpoint(A--D),W);
 
−
</asy></center>
 
  
 
<math>\triangle EFG \sim \triangle EAB</math> because <math>FG \parallel AB.</math> The ratio of <math>\triangle EFG</math> to <math>\triangle EAB</math> is <math>2:5</math> since <math>AB=5</math> and <math>FG=2</math> from subtraction. If we let <math>h</math> be the height of <math>\triangle EAB,</math>
 
<math>\triangle EFG \sim \triangle EAB</math> because <math>FG \parallel AB.</math> The ratio of <math>\triangle EFG</math> to <math>\triangle EAB</math> is <math>2:5</math> since <math>AB=5</math> and <math>FG=2</math> from subtraction. If we let <math>h</math> be the height of <math>\triangle EAB,</math>
Line 67: Line 41:
  
 
The height is <math>5</math> so the area of <math>\triangle EAB</math> is <math>\frac{1}{2}(5)(5) = \boxed{\textbf{(D)}\ \frac{25}{2}}</math>.
 
The height is <math>5</math> so the area of <math>\triangle EAB</math> is <math>\frac{1}{2}(5)(5) = \boxed{\textbf{(D)}\ \frac{25}{2}}</math>.
 +
 +
==Solution 2==
 +
 +
We can look at this diagram as if it were a coordinate plane with point <math>A</math> being <math>(0,0)</math>. This means that the equation of the line <math>AE</math> is <math>y=3x</math> and the equation of the line <math>EB</math> is <math>y=\frac{-3}{2}x+\frac{15}{2}</math>. From this we can set of the follow equation to find the <math>x</math> coordinate of point <math>E</math>:
 +
 +
<cmath>3x=\frac{-3}{2}x+\frac{15}{2}</cmath>
 +
<cmath>6x=-3x+15</cmath>
 +
<cmath>9x=15</cmath>
 +
<cmath>x=\frac{5}{3}</cmath>
 +
 +
We can plug this into one of our original equations to find that the <math>y</math> coordinate is <math>5</math>, meaning the area of <math>\triangle EAB</math> is <math>\frac{1}{2}(5)(5) = \boxed{\textbf{(D)}\ \frac{25}{2}}</math>
 +
 +
==Solution 3==
 +
At points <math>A</math> and <math>B</math>, segment <math>AE</math> is 5 units from segment <math>BE</math>. At points <math>F</math> and <math>G</math>, the segments are 2 units from each other. This means that collectively, the two lines closed the distance between them by 3 units over a height of 3 units. Therefore, to close the next two units of distance, they will have to travel a height of 2 units.
 +
 +
Then calculate the area of trapezoid <math>FGBA</math> and triangle <math>EGF</math> separately and add them. The area of the trapezoid is <math>\frac {2+5}{2}\cdot 3 = \frac {21}{2}</math> and the area of the triangle is <math>\frac{1}{2}\cdot 2 \cdot 2 = 2</math>. <math>\frac{21}{2}+2=\boxed{\textbf{(D)}\ \frac{25}{2}}</math>
 +
 +
==Solution 4==
 +
Since <math>\Delta{ABE}\sim{\Delta{FGE}}</math> then <math>[AFGB]\sim{[FXYG]}</math>, where <math>X</math> and <math>Y</math> are ponts on <math>EF</math> and <math>EG</math> respectivley which make the areas similar. This process can be done over and over again multiple times by the ratio of <math>\frac{FG}{AB}=\frac{2}{5}</math>, or something like this
 +
<cmath>[AEB]=[AFGB]+[FXYZ]+...</cmath><cmath>[AEB]=[AFGB]+\frac{2}{5}[AFGB]+(\frac{2}{5})^2[AFGB]+...</cmath>we have to find the ratio of the areas when the sides have shrunk by length <math>\frac{2}{5}l</math>
 +
 +
<asy>
 +
unitsize(0.6 cm);
 +
 +
pair A, B, C, D, E, F, G;
 +
 +
A = (0,0);
 +
B = (5,0);
 +
C = (5,3);
 +
D = (0,3);
 +
F = (1,3);
 +
G = (3,3);
 +
E = extension(A,F,B,G);
 +
 +
draw(A--B--C--D--cycle);
 +
draw(A--E--B);
 +
 +
label("$A$", A, SW);
 +
label("$B$", B, SE);
 +
label("$C$", C, NE);
 +
label("$D$", D, NW);
 +
label("$E$", E, N);
 +
label("$F$", F, SE);
 +
label("$G$", G, SW);
 +
label("$2/5$", (D + F)/2, N);
 +
label("$4/5$", (G + C)/2, N);
 +
label("$6/5$", (B + C)/2, dir(0));
 +
label("$6/5$", (A + D)/2, W);
 +
label("$2$", (A + B)/2, S);
 +
</asy>
 +
 +
Let <math>[AFGB]'</math> be the area of the shape whose length is <math>\frac{2}{5}l</math>
 +
<cmath>[AFGB]'=[ADCB]-[ADF]-[BCG]</cmath><cmath>[AFGB]'=12/5-6/25-12/25</cmath><cmath>[AFGB]'=42/25</cmath>Now comparing the ratios of <math>[AFGB]'</math> to <math>[AFGB]</math> we get
 +
<cmath>\frac{[AFGB]'}{[AFGB]}=\frac{42}{25}/\frac{21}{2}\implies \frac{[AFGB]'}{[AFGB]}=\frac{4}{25}</cmath>By applying an infinite summation
 +
<cmath>[AEB]=\sum_{n=0}^{\infty} \frac{21}{2}\cdot{(\frac{4}{25})^n}</cmath><cmath>S=\frac{a_1}{1-r}</cmath><cmath>\boxed{[AEB]=\frac{25}{2}}</cmath>
 +
 +
==Solution 5==
 +
Drop a perpendicular from <math>E</math> to <math>AB</math> and call the intersection point <math>X.</math> <math>\triangle EXA</math> and <math>\triangle DFA</math> are similar and so are <math>\triangle BCG</math> and <math>\triangle EXB</math> (You can prove this with <math>AA</math> by observing that <math>\angle DFA</math> is congruent to <math>\angle EAB</math> and so on). This means that <math>\dfrac{AX}{BX}=\dfrac{1}{2}</math> so <math>AX=\dfrac{5}{3}.</math> <math>\dfrac{EX}{AX}=3,</math> which means that <math>EX=5.</math> Then, <math>[AEB]=\dfrac{5\cdot 5}{2}=\boxed{\textbf{(D) } \dfrac{25}{2}}.</math>
  
 
==See Also==
 
==See Also==

Latest revision as of 16:26, 3 July 2023

The following problem is from both the 2003 AMC 12B #14 and 2003 AMC 10B #20, so both problems redirect to this page.

Problem

In rectangle $ABCD, AB=5$ and $BC=3$. Points $F$ and $G$ are on $\overline{CD}$ so that $DF=1$ and $GC=2$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.

[asy] unitsize(8mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=0; pair A=(0,0), B=(5,0), C=(5,3), D=(0,3); pair F=(1,3), G=(3,3); pair E=(5/3,5); draw(A--B--C--D--cycle); draw(A--E); draw(B--E); pair[] ps={A,B,C,D,E,F,G}; dot(ps); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",E,N); label("$F$",F,SE); label("$G$",G,SW); label("$1$",midpoint(D--F),N); label("$2$",midpoint(G--C),N); label("$5$",midpoint(A--B),S); label("$3$",midpoint(A--D),W); [/asy]

$\textbf{(A) } 10 \qquad\textbf{(B) } \frac{21}{2} \qquad\textbf{(C) } 12 \qquad\textbf{(D) } \frac{25}{2} \qquad\textbf{(E) } 15$

Solution 1

$\triangle EFG \sim \triangle EAB$ because $FG \parallel AB.$ The ratio of $\triangle EFG$ to $\triangle EAB$ is $2:5$ since $AB=5$ and $FG=2$ from subtraction. If we let $h$ be the height of $\triangle EAB,$

\[\frac{2}{5} = \frac{h-3}{h}\] \[2h = 5h-15\] \[3h = 15\] \[h = 5\]

The height is $5$ so the area of $\triangle EAB$ is $\frac{1}{2}(5)(5) = \boxed{\textbf{(D)}\ \frac{25}{2}}$.

Solution 2

We can look at this diagram as if it were a coordinate plane with point $A$ being $(0,0)$. This means that the equation of the line $AE$ is $y=3x$ and the equation of the line $EB$ is $y=\frac{-3}{2}x+\frac{15}{2}$. From this we can set of the follow equation to find the $x$ coordinate of point $E$:

\[3x=\frac{-3}{2}x+\frac{15}{2}\] \[6x=-3x+15\] \[9x=15\] \[x=\frac{5}{3}\]

We can plug this into one of our original equations to find that the $y$ coordinate is $5$, meaning the area of $\triangle EAB$ is $\frac{1}{2}(5)(5) = \boxed{\textbf{(D)}\ \frac{25}{2}}$

Solution 3

At points $A$ and $B$, segment $AE$ is 5 units from segment $BE$. At points $F$ and $G$, the segments are 2 units from each other. This means that collectively, the two lines closed the distance between them by 3 units over a height of 3 units. Therefore, to close the next two units of distance, they will have to travel a height of 2 units.

Then calculate the area of trapezoid $FGBA$ and triangle $EGF$ separately and add them. The area of the trapezoid is $\frac {2+5}{2}\cdot 3 = \frac {21}{2}$ and the area of the triangle is $\frac{1}{2}\cdot 2 \cdot 2 = 2$. $\frac{21}{2}+2=\boxed{\textbf{(D)}\ \frac{25}{2}}$

Solution 4

Since $\Delta{ABE}\sim{\Delta{FGE}}$ then $[AFGB]\sim{[FXYG]}$, where $X$ and $Y$ are ponts on $EF$ and $EG$ respectivley which make the areas similar. This process can be done over and over again multiple times by the ratio of $\frac{FG}{AB}=\frac{2}{5}$, or something like this \[[AEB]=[AFGB]+[FXYZ]+...\]\[[AEB]=[AFGB]+\frac{2}{5}[AFGB]+(\frac{2}{5})^2[AFGB]+...\]we have to find the ratio of the areas when the sides have shrunk by length $\frac{2}{5}l$

[asy] unitsize(0.6 cm); pair A, B, C, D, E, F, G; A = (0,0); B = (5,0); C = (5,3); D = (0,3); F = (1,3); G = (3,3); E = extension(A,F,B,G); draw(A--B--C--D--cycle); draw(A--E--B); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, NE); label("$D$", D, NW); label("$E$", E, N); label("$F$", F, SE); label("$G$", G, SW); label("$2/5$", (D + F)/2, N); label("$4/5$", (G + C)/2, N); label("$6/5$", (B + C)/2, dir(0)); label("$6/5$", (A + D)/2, W); label("$2$", (A + B)/2, S); [/asy]

Let $[AFGB]'$ be the area of the shape whose length is $\frac{2}{5}l$ \[[AFGB]'=[ADCB]-[ADF]-[BCG]\]\[[AFGB]'=12/5-6/25-12/25\]\[[AFGB]'=42/25\]Now comparing the ratios of $[AFGB]'$ to $[AFGB]$ we get \[\frac{[AFGB]'}{[AFGB]}=\frac{42}{25}/\frac{21}{2}\implies \frac{[AFGB]'}{[AFGB]}=\frac{4}{25}\]By applying an infinite summation \[[AEB]=\sum_{n=0}^{\infty} \frac{21}{2}\cdot{(\frac{4}{25})^n}\]\[S=\frac{a_1}{1-r}\]\[\boxed{[AEB]=\frac{25}{2}}\]

Solution 5

Drop a perpendicular from $E$ to $AB$ and call the intersection point $X.$ $\triangle EXA$ and $\triangle DFA$ are similar and so are $\triangle BCG$ and $\triangle EXB$ (You can prove this with $AA$ by observing that $\angle DFA$ is congruent to $\angle EAB$ and so on). This means that $\dfrac{AX}{BX}=\dfrac{1}{2}$ so $AX=\dfrac{5}{3}.$ $\dfrac{EX}{AX}=3,$ which means that $EX=5.$ Then, $[AEB]=\dfrac{5\cdot 5}{2}=\boxed{\textbf{(D) } \dfrac{25}{2}}.$

See Also

2003 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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 (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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

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=2003_AMC_10B_Problems/Problem_20&oldid=195223"