Bouncing Barney

by Kristy Hawkins

Barney lives in this triangular room ABC.

He walks from a point on AC parallel to BC. When he reaches AB, he turns and walks parallel to AC. When he reaches BC, he turns and walks parallel to AB back to AC. He continues this process until he reaches his starting point again. How many times will Barney reach a wall before this happens?

Well, when we construct this situation in GSP, it looks like Barney will return to his starting position after reaching the sixth wall. Move Barney around in this GSP sketch to see if you can find any other cases. These two look interesting to me.

What do you think? These might be cases when Barney is starting at the midpoint of AC or at a point that trisects AC. I think their might be congruent triangle as well! Let's see what we can prove!

Let's begin with the most simple case i.e. Barney begins at the midpoint of AC. Denote AC=b, AB=c, and BC=a.

Prove that Barney returns to his starting point after two bounces.

First let's prove that triangle AXY is similar to triangle ACB. This is pretty simple to do because we know that YX is parllel to BC from the problem. Since this is true,

by two parallel lines cut by a transversal.

Also, both of these triangles contain angle BAC, and therefore they are similar.

Since they are similar,

In the same way, we can show that BZ=.5a. Since this is true, when Barney travels from Z parallel to AB, he must land at X. Therefore he lands at the same place where he began after two bounces.

This conjecture is also pretty easy to prove wherever Barney begins on AC by similar triangles and parallel lines.

As I was investigating this situation, I thought about what would happen if Barney was traveling on lines outside of triangle ABC. What would that look like? Do you think Barney would still eventually end up at the same place?

He still ends up at the same point after only five bounces! Can you prove this?

Now let's look at the distance that Barney travels before he returns to his beginning point.

At first glance, it looks like the distance that Barney travels might be equal to the perimeter of the triangle. Would you agree with this? Let's measure the distances in GSP and see what we find.

It looks like this is true for all triangles. Pretty cool!