From arithmetic to algebra to geometry

We found a really neat question about triangles in our Introduction to Geometry book. The question gives you two sides of a triangle and asks about the possible values of the third side that would make the triangle obtuse.

It is pretty interesting question on several levels. One of those levels is that you can solve the problem with just basic arithmetic. In the case we study below, I restrict the values of the third side to integers.

 

At the end of the discussion we have an answer to the problem, but no pictures. Pretty surprising for a geometry problem! We went to Wolfram Alpha to take a look at some of the triangles.

 

So, a fun little project which serves as a nice illustration of the connection between arithmetic, algebra, and geometry. Pretty amazing what you can learn about a triangle just from some simple observations about the side lengths.

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