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

Last week Patrick Honner wrote about an old New York State Regents exam question involving a 13-14-15 triangle:

Yesterday I was pleasantly surprised to see that the Art of Problem Solving Introduction to Geometry book led off the section on Heron’s formula by looking at a 13-14-15 triangle.  Fun little coincidence:

Finding the area of the 13-14-15 triangle is a nice exercise in both algebra and geometry.  We start in on the algebra first:

Having set up two equations relating our two variables, we now tackle the algebra of solving those equations.  I like using examples like this to build up a little algebra sense.  The first instinct is often to multiply out all of the terms and see what happens, but here we have a few other algebraic techniques which provide an interesting (and informative) alternate path.

At the end of this video we discuss the surprising geometric solution to this problem, too.

We just found out that the 13-14-15 triangle can break into two right triangles with integer side lengths.  It turns out that constructing a box with integer lengths for the sides and for the diagonals is a famous (and unsolved) problem.  It seemed like talking about that problem was a natural extension of the discussion we had on the 13-14-15 triangle, so we took a short little diversion and looked at the “Euler brick.”

Finally, we went to the kitchen table to discuss the surprising property of the 13-14-15 triangle that Patrick Honner pointed out in his blog post.   The problem from an old NY State Regents exam asked students to find the angle between the sides of length 14 and 15 in the 13-14-15 triangle.  The question is designed  to test a student’s knowledge of the law of cosines, but, as Mr. Honner points out in his blog post, there are geometric solutions that do not require the law of cosines to answer this multiple choice question.

One of the geometric solution is so clever that I had to try it out for myself.   First I used a rule and compass to construct the triangles that Mr. Honner uses in his blog post:

Compass Triangles

An amazingly close fit – just as the picture in the blog suggested!

Following this construction, I used our 3D printer to make physical copies of these triangles so you could see how they fit together.  We used these objects to walk through Mr. Honner’s solution:

So, a fun project with the 13-14-15 triangle and an unsolved problem.  Love finding a neat blog post online just as the same concepts are coming up in one of the books we are following.  Can’t wait for the next time it happens!