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Difference between revisions of "2004 AIME II Problems/Problem 7"

m (→‎Solution 1 (synthetic))
m (→‎Solution 2 (analytic))
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=== Solution 2 (analytic) ===
 
=== Solution 2 (analytic) ===
 
Let <math>A = (0,0), B=(0,25)</math>, so <math>E = (0,8)</math> and <math>F = (l,22)</math>, and let <math>l = AD</math> be the length of the rectangle. The [[slope]] of <math>EF</math> is <math>\frac{14}{l}</math> and so the equation of <math>EF</math> is <math>y -8 = \frac{14}{l}x</math>. We know that <math>EF</math> is perpendicular to and bisects <math>BB'</math>. The slope of <math>BB'</math> is thus <math>\frac{-l}{14}</math>, and so the equation of <math>BB'</math> is <math>y -25 = \frac{-l}{14}x</math>. Let the point of intersection of <math>EF, BB'</math> be <math>G</math>. Then the y-coordinate of <math>G</math> is <math>\frac{25}{2}</math>, so
 
Let <math>A = (0,0), B=(0,25)</math>, so <math>E = (0,8)</math> and <math>F = (l,22)</math>, and let <math>l = AD</math> be the length of the rectangle. The [[slope]] of <math>EF</math> is <math>\frac{14}{l}</math> and so the equation of <math>EF</math> is <math>y -8 = \frac{14}{l}x</math>. We know that <math>EF</math> is perpendicular to and bisects <math>BB'</math>. The slope of <math>BB'</math> is thus <math>\frac{-l}{14}</math>, and so the equation of <math>BB'</math> is <math>y -25 = \frac{-l}{14}x</math>. Let the point of intersection of <math>EF, BB'</math> be <math>G</math>. Then the y-coordinate of <math>G</math> is <math>\frac{25}{2}</math>, so
−
<center> <math>
+
<cmath>
 
\begin{align*}
 
\begin{align*}
 
\frac{14}{l}x &= y-8 = \frac{9}{2}\
 
\frac{14}{l}x &= y-8 = \frac{9}{2}\
 
\frac{-l}{14}x &= y-25 = -\frac{25}{2}\
 
\frac{-l}{14}x &= y-25 = -\frac{25}{2}\
 
\end{align*}
 
\end{align*}
−
</math> </center>
+
</cmath>
 
Dividing the two equations yields
 
Dividing the two equations yields
 
<center><math>l^2 = \frac{25 \cdot 14^2}{9} \Longrightarrow l = \frac{70}{3}</math></center>
 
<center><math>l^2 = \frac{25 \cdot 14^2}{9} \Longrightarrow l = \frac{70}{3}</math></center>

Revision as of 19:56, 13 March 2015

Problem

$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Solution 1 (synthetic)

[asy] pointpen = black; pathpen = black +linewidth(0.7); pair A=(0,0),B=(0,25),C=(70/3,25),D=(70/3,0),E=(0,8),F=(70/3,22),G=(15,0); D(MP("A",A)--MP("B",B,N)--MP("C",C,N)--MP("D",D)--cycle); D(MP("E",E,W)--MP("F",F,(1,0))); D(B--G); D(E--MP("B'",G)--F--B,dashed); MP("8",(A+E)/2,W);MP("17",(B+E)/2,W);MP("22",(D+F)/2,(1,0)); [/asy]

Since $EF$ is the perpendicular bisector of $\overline{BB'}$, it follows that $BE = B'E$ (by SAS). By the Pythagorean Theorem, we have $AB' = 15$. Similarly, from $BF = B'F$, we have \begin{align*} BC^2 + CF^2 = B'D^2 + DF^2 &\Longrightarrow BC^2 + 9 = (BC - 15)^2 + 484 \\ BC &= \frac{70}{3} \end{align*} Thus the perimeter of $ABCD$ is $2\left(25 + \frac{70}{3}\right) = \frac{290}{3}$, and the answer is $m+n=\boxed{293}$.

Solution 2 (analytic)

Let $A = (0,0), B=(0,25)$, so $E = (0,8)$ and $F = (l,22)$, and let $l = AD$ be the length of the rectangle. The slope of $EF$ is $\frac{14}{l}$ and so the equation of $EF$ is $y -8 = \frac{14}{l}x$. We know that $EF$ is perpendicular to and bisects $BB'$. The slope of $BB'$ is thus $\frac{-l}{14}$, and so the equation of $BB'$ is $y -25 = \frac{-l}{14}x$. Let the point of intersection of $EF, BB'$ be $G$. Then the y-coordinate of $G$ is $\frac{25}{2}$, so \begin{align*} \frac{14}{l}x &= y-8 = \frac{9}{2}\\ \frac{-l}{14}x &= y-25 = -\frac{25}{2}\\ \end{align*} Dividing the two equations yields

$l^2 = \frac{25 \cdot 14^2}{9} \Longrightarrow l = \frac{70}{3}$

The answer is $\boxed{293}$ as above.

Solution 3 (Coordinate Bashing)

Firstly, observe that if we are given that $AE=8$ and $BE=17$, the length of the triangle is given and the height depends solely on the length of $CF$. Let Point $A = (0,0)$. Since $AE=8$, point E is at (8,0). Next, point $B$ is at $(25,0)$ since $BE=17$ and point $B'$ is at $(0,-15)$ since $BE=AE$ by symmetry. Draw line segment $BB'$. Notice that this is perpendicular to $EF$ by symmetry. Next, find the slope of EB, which is $\frac{15}{25}=\frac{3}{5}$. Then, the slope of $EF$ is -$\frac{5}{3}$.

Line EF can be written as y=$-\frac{5}{3}x+b$. Plug in the point $(8,0)$, and we get the equation of EF to be y=$_\frac{5}{3}x+\frac{40}{3}$. Since the length of $AB$=25, a point on line $EF$ lies on $DC$ when $x=25-3=22$. Plug in $x=22$ into our equation to get $y=-\frac{70}{3}$. $|y|=BC=\frac{70}{3}$. Therefore, our answer is $2(AB+BC)=2\left(25+\frac{70}{3}\right)=2\left(\frac{145}{3}\right)=\frac{290}{3}= \boxed{293}$.

See also

2004 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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