2004 AIME II Problems/Problem 7

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*}$ (Error compiling LaTeX. Unknown error_msg)

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*}$ (Error compiling LaTeX. Unknown error_msg)

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 $2(AB+BC)=2(25+\frac{70}{3})=2(\frac{145}{3})=$ \frac{290}{3} $=$ \boxed{293}$.

2004 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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