Fun with right triangles
Last year my son enjoyed learning the pythagorean theorem. He was fascinated by a picture containing a proof and being able to calculate the length of one of the sides from the other two.
On vacation a couple weeks ago, we found a fun (well, for geeks anyway) driving game, which we called “Iron Chef Triangles”. The game is very simple: given a number, construct a pythagorean triple containing it – that is, find a right triangle whose sides are whole numbers one of which matches the given number. For example, if I say “5”, you might respond “3, 4, 5” (because 3 squared plus 4 squared equals 5 squared, so you can make a right triangle with sides of length 3, 4 and 5) or you might say “5, 12, 13” (because 5 squared plus 12 squared is 13 squared). If I say 8 you might say “6, 8, 10” or you might say “8, 15, 17”.
Try a few: how about 11? How about 20? See any patterns?
Once you’ve figured out how to easily do these in your head, here are two harder problems.
Hard: given right triangle whose sides aren’t necessarily integers, can you make a right triangle whose sides are integers with approximately the same angles? How close can you get?
Observation: the product of the three parts of a pythagorean triple is always a multiple of 60.
Unfairly hard problem: can you find two different pythagorean triples whose product is the same?