Tiffany N. KeysÕ Final Assignment:

Bouncing
Barney

Barney is in a triangular room. 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. Prove that Barney
will eventually return to his starting point. How many times will Barney reach
a wall before returning to his starting point? Explore and discuss for various
starting points on line BC, including points exterior to segment BC. Discuss
and prove any mathematical conjectures you find in the situation.

OBSERVATIONS
AND IVESTIGATIONS:

á
It
is observed that Barney bounces off a wall five times before he returning to
his starting point.

á
This
raises the question of will he always touch the walls five times before
returning to his starting point.

á
To
investigate one of the relationships between Barney's path and the triangular
room, I used GSP to find the perimeter of the triangle ABC.

á
Then,
I found the length of each path.

á
The
sum of the length of Barney's paths was equal to the perimeter of the triangle.

á
By
tracing the intersections of Barney's paths, the medians of triangle ABC were

á
The
point of intersection of the three medians is called the centroid.

á
In
order for Barney's paths to have one common intersection, Barney must begin on
the point approximately one third of the segment AC from point A.

á
This
is the only way for the paths to all intersect at the centroid of the triangle.