Barney is in the
triangular room shown here. 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.
The following figure shows Barney's path
around a random triangular room ABC. Notice that in 5 turns
(6 lengths) he arrives back at his original starting point.
Is this true for all starting
points on segment BC, or does the figure above merely represent a
special case? Notice in the next figure, that as the starting
point moves along BC, this observation still holds.
However, when the starting
point lies coincident with vertex B, it only takes Barney 2 turns to
arrive at his original starting point.
The same is true when the
starting point is the midpoint of segment BC. It only takes
Barney 2 turns (3 lengths) to arrive back at his original starting
When Barney walks on a path parallel to the sides of the room, he will
always return to his starting point. It also appears that as
Barney walks his path creates a similar triangle at each vertex.
Let's look at ways to prove our conjecture holds true. One way is
by examining the parallelograms created by Barney's path.
The perimeter of these parallelograms, when the segments that are part
of the triangle are excluded, are equivalent to the perimeter of the
triangle. Barney will always return to his starting point since
his path is the same measure as the triangle.