|Return to Session #927 at the NCTM 2002 Annual Meeting
Point P Perambulates - posted April 8, 2002I found the basis for this problem on the 1972 "Annual High School Contest of the Mathematical Association of America." While the actual question they asked is the bonus here, I thought a shorter version of the question would be a good geometry problem to use.
Bonus: Instead of stopping when the triangle returns to the
lower left-hand corner, continue until P, A, and B all return to their
original positions. Now what is the length of the path that P travels?
One good way to tackle a problem like this is to actually "model" what's happening - cut out a square and a triangle of the appropriate size, label them, and then do exactly what the problem says. Be sure to label P so that you can easily keep track of it. (You could even trace P as you rotate the triangle.)
The path P takes may be broken up into separate pieces, some of which are "long" and some of which are "short." What is the shape of the pieces of the path traveled by P?
To find the lengths of the pieces, try to figure out the angle of rotation each time P moves. For example, the first time we rotate the triangle, P moves to X. What is angle PBX?
Once you know the angle of rotation, what portion of a whole circle does that represent? How does that help us find the length of each piece of the path?
We found that when the triangle returned to the lower left-hand corner, P was in the corner, A was on AX (which now looks like PX), and B was the top of the triangle (floating in the square).
We noticed that the path P takes as it moves was composed of arcs of circles. We also found that there were 5 different arcs formed as the triangle is rotated once inside the square, 3 "long" and 2 "short."
We found that the total length of the path traveled by P is 14pi/3 inches.
What's the length of the path traveled by the center of the triangle? What if we rotated the triangle inside a different polygon with edgelength 4, like an equilateral triangle or a pentagon?
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