Final Assignment: Bouncing Barney
Chelsea Henderson
Barney's Journey
Our beloved purple dinasaur seems to be stuck in a triangle.
Barney starts out hanging out on the side BC of the triangle, as seen below.
Barney then begins exploring the room. He decides to walk only along straight lines parallel to the opposite side of the room. He first walks from his original starting point, D, on a parallel path to side AC, ending up at a point E on AB. See below.
ED is parallel to AC
Barney continues his journey by walking along lines parallel to the sides of the triangle, changing paths when we reaches a side of the triangle.
Above is the path Barney takes along his walk. He's not finished yet, in fact, it looks like he will never be finished. He'll just keep walking until someone sets him free. Where will he go next? What does it look like?
Let's see where he ends up next.
Barney continued on to his original point. It looks like from here he will be on the same path forever and ever.
Will Barney always end up back at his original starting point?
Below are scenarios of Barney's path when he starts at different points D along side AB.
It seems from above that Barney always ends up back where he began. The final picture above shows when Barney begins at the midpoint of side BC. This is a special case which we will discuss further later on.
Click here for a GSP file showing Barney's Path.
Can we prove that Barney will always end up at his starting point?
Yes!
To prove that Barney will always end up at his starting point, I will start by calling the point where Barney returns to BC J.
Let's suppose Barney returns to his original side BC at a point J not equal to point D, his starting point, as in the picture below.
(Note, J above was not constructed using parallels- I just chose a point on BC not equal to D to demonstrate our assumption).
Proof that J must be the same point as D, thus proving that Barney returns to his original starting point.
Also,
Above we have proven that the point of Barney's return is the same as the point of his departure.
The Length of Barney's Journey
We know that Barney will travel around the same path forever and he will eventually wear himself out.
What if, however, he only walked around his path once? Will he be as tired as if he had just walked the perimeter of the room? Would there be a better starting place for him on side BC that would make his journey shorter? Or is his distance the same no matter where he starts?
Above, we see that the length of the path is equal to the perimeter of the triangle.
Below is a picture of a different starting point of D along BC. I include it here only to illustrate that the starting point of D does not affect the proof above.
Am I fooling you? Is the proof above always always valid? What about if Barney starts at the midpoint? Look at the picture of this case above, what do you think? We'll examine this special case further down.
A New Beginning
What if Barney did not begin his journey on side BC? What is he decided on his path while he was inside the triangle? What would his path look like then?
What would his path be if he started at the centroid?
Above, Barney starts at point D, the centroid of the triangle. His path is very similar to all the possible paths we have seen so far. Also, after his first completion of the path, the path Barney will take will always be the same. Barney will always pass through his starting point- the centroid. The case here is the same as before in that Barney will always return to his starting point, even though here the starting poing is not on a side of the triangle.
What would his path be if he started at the incenter?
Above, Barney starts at the point D, the incenter of the triangle. His path is similar to other paths we have seen, but is different than the path seen above when he began at the centroid. Unlike above, a middle triangle is formed. Still, we have Barney eventually returning to (or continuing through) his starting point.
The picture below confirms that the distance of the path Barney takes when he starts at the incenter is equal to the triangle's perimeter. Below it is a visual proof following what we did in the distance proof above.
A Special Case
It seems that no matter where Barney starts inside or along the triangle, his path has the same properties: distance of path equals the perimeter of the triangle and he will always return to his starting point. The later of these properties is always true, but the first property changes in just one case- when Barney starts his path at the midpoint of a triangle side.
The case where Barney begins at the midpoint of a triangle side is pictured below.
The question now becomes, what is the distance traveled by Barney when he begins his journey at the midpoint of a side of the triangle?
We may not have the exact same picture that we did above when looking at his path's distance in relationship to the perimter, but we can apply the same ideas when answering our current question.
Above, we can see that Barney's path is somewhat related to the perimeter of the triangle. What else can we say?
When Barney begins his journey at the midpoint of a side of the triangle, his path is no longer equal to the perimeter of the triangle, but rather is equal to half the length of the perimeter of the triangle.
Barney cuts his journey in half by starting at the midpoint!
Barney Breaks Out!
With his magical powers, Barney has escaped the triangle. Now, Barney is outside the triangle, but he enjoyed his path so much, he wants to continue bouncing.
What will his path look like if Barney begins his journey on a point on line BC outside of the triangle? Will he re-enter the triangle, only to have to use magic to escape again? Will he still end up back at his starting point?
Click here for a picture of with more detail of how the above picture was constructed.
Click here for a GSP file with D, Barney's starting point, outside of the triangle ABC.
What can we say about Barney's journey now?
No triangles are created, as we had before. Also, the distance of the path is now larger than the perimeter of triangle ABC.
Barney does still return to his original starting point. There are still 6 points of intersection with the lines of the triangle ABC.
Barney has broken out of the triangle, but his path will still remain the same forever until he gets over his fear of not walking in a line parallel to the wall.
