# 2009 AMC 10B Problems/Problem 13

## Problem

As shown below, convex pentagon $ABCDE$ has sides $AB=3$, $BC=4$, $CD=6$, $DE=3$, and $EA=7$. The pentagon is originally positioned in the plane with vertex $A$ at the origin and vertex $B$ on the positive $x$-axis. The pentagon is then rolled clockwise to the right along the $x$-axis. Which side will touch the point $x=2009$ on the $x$-axis? $[asy] unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,0), Ep=7*dir(105), B=3*dir(0); pair D=Ep+B; pair C=intersectionpoints(Circle(D,6),Circle(B,4)); pair[] ds={A,B,C,D,Ep}; dot(ds); draw(B--C--D--Ep--A); draw((6,6)..(8,4)..(8,3),EndArrow(3)); xaxis("x",-8,14,EndArrow(3)); label("E",Ep,NW); label("D",D,NE); label("C",C,E); label("B",B,SE); label("(0,0)=A",A,SW); label("3",midpoint(A--B),N); label("4",midpoint(B--C),NW); label("6",midpoint(C--D),NE); label("3",midpoint(D--Ep),S); label("7",midpoint(Ep--A),W); [/asy]$ $\text{(A) } \overline{AB} \qquad \text{(B) } \overline{BC} \qquad \text{(C) } \overline{CD} \qquad \text{(D) } \overline{DE} \qquad \text{(E) } \overline{EA}$

## Solution

The perimeter of the polygon is $3+4+6+3+7 = 23$. Hence as we roll the polygon to the right, every $23$ units the side $\overline{AB}$ will be the bottom side.

We have $2009 = 23 \times 87 + 8$. Thus at some point in time we will get the situation when $A=(2001,0)$ and $\overline{AB}$ is the bottom side. Obviously, at this moment $B=(2004,0)$.

After that, the polygon rotates around $B$ until point $C$ hits the $x$ axis at $(2008,0)$.

And finally, the polygon rotates around $C$ until point $D$ hits the $x$ axis at $(2014,0)$. At this point the side $\boxed{\overline{CD}}$ touches the point $(2009,0)$. So the answer is $\boxed{C}$

## Solution 2: Mod Arithmetic

The perimeter is $23$ and $2009\equiv8($mod $23)$, so it will end up on side $AB$ + a total of 8 more units. $4<8$, but $4+6=10>8$, so it ends on side $CD$ for an answer of $\boxed{C}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 