Amberly Roberts

Final Assignment: Bouncing Barney

Barney is in the triangular room shown here. He walks from a point on BC parallel to AC. When he reaches AB, he turns and walks parallel to BC. When he reaches AC, he turns and walks parallel to AB. Discuss and prove any mathematical conjectures you find in the situation.

Part 1: Prove that Barney will eventually return to his starting point.

Part 2: How many times will Barney reach a wall before returning to his starting point?

Part 3: Explore and discuss for various starting points on line BC, including points exterior to segment BC.

Before completing any of the parts labeled above, I completed a simulation of what would happen if Barney started from an arbitrary point on segment BC. Here's what happened.

 Observations: 1. Barney returned to his starting point. 2. Barney touched each segment of the triangle twice. (He traveled along 6 different segments)

GSP offers the option of moving the starting point any where along the segment on which it was constructed. When I moved the starting point along segment BC, I noticed several interesting cases. Click here to open the GSP file so that you may observe for yourself.

Below are two cases in which observation #2 in the table above is broken. Observation #1 seems to hold true in all cases.

If Barney starts on the vertex of the triangle, his path will be the perimeter of the triangle. He will travel along 3 distinct paths. If Barney starts on what appears to be the midpoint of segment BC, he will only touch each side once and travel along 3 distinct segments.

Also, it's very interesting to note that the animation suggests that Barney travels the same distance no matter what path he selects! Pretty cool. I will use this observations to try and create of proof for Part 1.

Now, I think we're ready for Part 1: Prove that Barney will eventually return to his starting point.

Part 2: How many times will Barney reach a wall before returning to his starting point?

Our work suggests that in all cases except for the two mentioned above, Barney will reach the walls a total of five times before returning to his starting point. When he starts at the midpoint of the wall, he will reach the walls twice before returning to his starting point.

When he begins at a vertex, he will travel along the walls. (I am not sure how to discuss the idea of reaching a wall if we are traveling along the wall....hmmm...is this case relevant?)

Part 3: Explore and discuss for various starting points on line BC, including points exterior to segment BC.

The purple dotted segments represent the path of Barney if he travels along paths parallel to the sides of the triangle. The concept of Barney bouncing OFF of walls is now irrelevant. In this case, he is bouncing THROUGH walls. Each placement of the starting point somewhere along the exterior of segment BC results in a similar construction. It appears that Barney reaches a wall five times before returning to his starting location. Don't miss that! It's interesting to note that he still returns to his starting point.