Exploring a 13-14-15 triangle

My younger son was struggling with one of the challenge problems from Art of Problem Solving’s Introduction to Geometry book today. I hadn’t done a project with him this week, so I thought it would be fun to turn talk about this problem in detail.

Here’s the introduction to the problem and some of his initial thoughts:

He had a pretty good idea about how to proceed by the end of the last video. Here we started to work through the algebra. There is one simplification that helps get to the solution fairly quickly – it took him a little while to find it. Once he did, though, the solution came quickly.

Now all that was left was a little arithmetic and checking that the answer we found was correct.

At the end he gives a nice summary of the problem.

It turns out that I did a previous project on the 13-14-15 triangle with my older son:

Mr. Honner’s 13-14-15 triangle and a surprising unsolved problem

Amazing how useful this triangle is!