imc

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 $\boxed{\overline{CD}}$