2009 AMC 10B Problem 13

Question

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?

\overline{AB}

\overline{BC}

\overline{CD}

\overline{DE}

\overline{EA}

The perimeter is

23,

and

2009=23\cdot87+8

After

87

full turns, track where vertices

A, B, C, D

land

The pentagon has perimeter

3+4+6+3+7=23.

One full roll advances the contact point by

23,

and

2009=23\cdot87+8.

After

87

rolls, vertex

A

sits at

x=23\cdot87=2001

and

B

at

2004.

Rolling further,

C

touches at

2004+4=2008

and

D

at

2008+6=2014.

Since

2009

lies between

2008

and

2014,

side

\overline{CD}

touches that point.

Thus, the correct answer is C.

Problem 13 in other years