Playing around with Ed Southall’s tweet about triangles with angles near 45 degrees

I saw a neat tweet from Ed Southall when I got up this morning:

The (137903, 137904, 195025) Pythagorean triple triangle has an angle ~44.99979 degrees. It's trying *very* hard to be isosceles.

— Ed Southall (@edsouthall) July 18, 2020

I thought trying to find other triangles with near 45 degree angles would make for a great project, so I introduced the idea to the boys and asked them how they thought we could find other triangles with this property:

My younger son went first – here we explore a triangle with side lengths 99, 100, and :

My older son noticed that Ed’s triangle was a right triangle with sides whose legs different in length by 1 unit. We were going to search for other right triangles like that (with integer sides), but he noticed that a 3-4-5 triangle had that property. So we looked to see how close the angles in that triangle were to 45 degrees:

Finally, I showed them how you could used continued fractions to find triangles with angles that are really close to 45 degrees. They were surprised that we could find a triangle that was smaller than the one we looked at in the 2nd video with angles that were much closer to 45 degrees:

This was a really fun exercise – I think it is a great way to review some basic ideas from geometry and trigonometry with kids.