Bouncy Face Barney

by Zack Kroll

There are three ways to investigate the problem of whether Barney will always return to the starting point. The first is to look at what happens when he starts at the midpoint of the side XY. He leaves the starting point and travels in a line that is parallel to side XZ. From there he bounces off of that wall at point B and moves to side XZ (parallel to XY). From point C he moves parallel to YZ and returns to the starting point, which is also point D.

Next, Barney can begin at one of the two vertices: X or Y. From there he will travel to the other two vertices (corners of the room) before returning to the starting point. This trip is essetially Barney walking along the edges of the room into a corner before he follows another wall to another corner.

The final option for Barney's travels is to start at any point except the midpoint or one of the two vertices: X or Y. From there Barney will bounce to a point on the segment YZ in a pathway that is parallel to XZ. From that point he will bounce to a point on XZ, followed by a different point than the starting one on XY, then back to another point on YZ, next back to XZ, before finally returning to the starting point.

Attached is the GSP file in which we can move the starting point along the side in order to see what happens to Barney's path.

As we can see in the images below, the number of walls that Barney comes into contact with entirely depend upon where his starting position is. If he is located in the middle he will reach two walls before returning to his starting position. However, if he begins anywhere between the vertices B and C then he will reach a wall five times before returning. Lastly, if he begins at one of the vertices then he will bounce to two corners (vertices) before he returns to his starting place.

After discovering how Barney always returns to the same point and exploring the different paths he can take, we want to determine whether the path he travels is always going to be the same distance. Using GSP we are able to determine that in fact he does travel the same distance regardless of where he begins his journey. We did this determining the perimeter of the triangle XYZ (Barney's room). Then we simply found the measure of each of the segment that represented Barney's travels. The total distance traveled is 54.64 cm. We have included the GSP file that will allow others to see how the measure of Barney's travels does not change regardless of where he begins.

What if Barney chose to begin and conduct his journey outside the triangle. We have developed another GSP file that can be used to investigate this peculiar situation.

Lastly, we have developed a file that allows us to show how how the path of triangles created by Barney are actually similiar to one another. The GSP file also shows the area of each of these triangles as well. We leave the reader to explore this and draw his/her own conclusions.