We looked at a few solutions to the Find the Missing Length problem.

First we solved the problem using similar triangles, in a manner that evokes a standard proof of the Pythagorean Theorem.  With some extraction, rotation, and/or reflection, the two interior triangles can be seen to be similar to the original triangle:

Simple proportions will then do the trick, both for our problem and proving the Pythagorean Theorem.

Mark showed how this principle can be used in non-right triangles, and he also showed us Polya’s favorite proof of the Pythagorean Theorem, which is related to Proof #7 here.

We also looked at a straight-forward algebraic solution to our original problem:  declare some variables, and use the Pythagorean Theorem to set up a system of equations.

Lastly, we looked at the elegant solution.  Since the area of the triangle is the same regardless of how you look at, just look at it a couple of different ways!

After changing rooms, we looked at Curry’s Puzzle and discussed some of the ways you could play around with it.  Larry suggested having students make their own “paradox”, and Timon emailed me a link to this similar idea (scroll down to see Sam Loyd’s  rectangle puzzle).

We took a look at a NYML contest and worked out a few of the problems.  The NYML, aka the Math League,  has contests for all levels, and they have books of contests available on their website.  When I taught 9th grade algebra, I used to love closing off a unit by having a little NYML team competition:  very accessible problems, and everyone always got into it.

At the end, we looked at some folding, as many of us were inspired by Erik Demaine’s recent MoMath talk.   We talked abut the one-cut problem, as well as how to use folding to investigate concurrency of angle bisectors, medians, and perpendicular bisectors.

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