2009 AMC 10B Problems/Problem 13

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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))[1]; 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 $\boxed{C}$.

See Also

2009 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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

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