Barney is in the triangular room shown here. 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. How many times will Barney reach a wall before returning to his starting point?
Let’s construct Barney’s path using GSP. Then we can extend the path, look at differ shape triangles, and consider different starting points.
We picked a point on AC and then used GSP to construct a line through that point and parallel to AC.
What does parallel mean?
GSP has a recorded construction of a line through a given point that is parallel to a given line. How is this construction done?
Continue Barney’s journey:
This picture shows Barney has bounced from AC to AB, then to BC, then to AC, then to AB. What will happen next?
Barney continues with a bounce to BC and then back to his starting point on AC.
Will he always return to the starting point? Why?
Will he always bounce to each side twice?
Does it matter what shape the triangle is?
What if Barney started at the midpoint of AC?
What is a midpoint of a segment? How does one construct it?
Click here for a GSP file to explore.
Suppose AB, AC, and BC are lines and Barney starts at a point on AC that is outside the segment AC. This is, the triangle is defined by lines rather than segments. We have this:
Use GSP to construct Barney’s path. Describe what you find.
Click here for a GSP Sketch of Barney’s path.