2009 AMC 10B Problems/Problem 13
Problem
As shown below, convex pentagon has sides , , , , and . The pentagon is originally positioned in the plane with vertex at the origin and vertex on the positive -axis. The pentagon is then rolled clockwise to the right along the -axis. Which side will touch the point on the -axis?
Solution
The perimeter of the polygon is . Hence as we roll the polygon to the right, every units the side will be the bottom side.
We have . Thus at some point in time we will get the situation when and is the bottom side. Obviously, at this moment .
After that, the polygon rotates around until point hits the axis at .
And finally, the polygon rotates around until point hits the axis at . At this point the side touches the point . So the answer is
Solution 2: Mod Arithmetic
The perimeter is and mod , so it will end up on side + a total of 8 more units. , but , so it ends on side for an answer of .
See Also
2009 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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