By Courtney Cody
In this assignment, we will investigate the situation of Bouncing Barney, who is in a triangular room ABC. 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. We will explore and discuss BarneyŐs movement for various starting points and we will prove that Barney will eventually return to his starting point, including how many times will Barney reach a wall before returning to his starting point.
LetŐs begin by sketching a general diagram of BarneyŐs movement for some arbitrary starting point D on BC.
By studying the figure above, BarneyŐs path was D-E-F-G-H-I-D and we see that every 3 steps Barney hits each wall of the triangular room. Also, it appears as though Barney will return to the starting point, for which he traveled six paths in the diagram above.
Now let's explore BarneyŐs path for a different starting point. LetŐs have him start where D is the midpoint of segment BC.
By studying this figure, we see that when D is the midpoint of BC, then BarneyŐs path is the medial triangle of the room. While the diagram shows us that he returns to his starting point again, this time it appears as though it only takes him three paths to get back to his starting point. While it is not immediately obvious, upon closer investigation of the figure it appears as though the length of BarneyŐs complete path is twice the length perimeter of the medial triangle and that Barney returns to his starting point twice.
Now letŐs have Barney start where D is one-third the length of segment BC.
By having Barney start one-third along BC we see that his path trisects each side of the triangular room.
Now suppose Barney starts at one of the vertices of the triangle, namely B. BarneyŐs path is as follows:
From the figure, we see that Barney once again returns to his starting point and that it takes him three paths to do so. In addition, BarneyŐs path is the same as the length of the perimeter of the triangular room ABC.
Click HERE to explore further BarneyŐs path for any starting point on BC!
Now that we have explored BarneyŐs path for various starting points, it is significant to determine whether the length of the path Barney has traveled is always constant. Click on the figure below for a GSP construction to move BarneyŐs starting point D around to verify with measurements that the length of his path is constant no matter where his starting point is on BC.
Now letŐs prove that Barney will return to his starting point. Suppose D is BarneyŐs starting point and J is his ending point.
As stated in the problem for Bouncing Barney, DE is parallel to AC. Then by the Angle-Angle similarity axiom we know that, triangle BED is similar to triangle BAC. Therefore, we can set up the proportion.
Similarly, since EF is parallel to BC as stated in the problem, then we know that triangle AEF is similar to triangle ABC. Hence, we can set up the proportion . Using these first two proportions, we know that .
In keeping with this pattern of similar triangles, we set up the following proportions:
This implies that , or .
Since we are trying to prove that the next point J is equal to D in order to verify that the ending point is the same as the starting point, we can set up another proportion to get since BarneyŐs next direction must head parallel to AB and intersect BC and we are letting this end point be J. Therefore, and so BarneyŐs path will always begin and end at the same point and thus it is a closed circuit, as desired.
How does BarneyŐs path change if his starting point is outside of triangle ABC? Is the length of BarneyŐs path preserved when he starts outside of the triangle? LetŐs investigate!
If Barney starts on a point not on a point on the line through AC but not on the segment AC and moves parallel to BC, he must go towards AB if he wishes to change his direction again. If he does this, he will see six different point, as in the usual case. The proof of this fact is similar to the proof above.
To see if length is preserved when the starting point is outside of triangle ABC, letŐs make some measurements. First, letŐs recall how when D is inside of triangle ABC, BarneyŐs path is always the length of the perimeter of triangle ABC. Now, letŐs compare this fact to the case when D lies outside of ABC.
As we can see by comparing the two diagrams above, the length of BarneyŐs path changes depending on how far outside of triangle ABC he starts at (point D). Therefore, the length Barney travels is not preserved when he starts outside of the triangle. Click HERE to see the effect of changing BarneyŐs starting point.