Ben Vitale’s Triangle puzzle

I’m on vacation right now or I’d write something much more thorough longer, but I wanted to point out a really interesting puzzle about triangles posed by Ben Vitale on twitter yesterday:

Puzzle: Triangle, sides are radicals, area an integer number http://t.co/9lfSVl3VHd via @BenVitale #math

— Benjamin Vitale (@BenVitale) February 18, 2014

Since I never know if the link’s from twitter will be preserved in the embedding, here’s a direct link to his blog:

http://benvitalenum3ers.wordpress.com/2014/02/18/puzzle-triangle-sides-are-radicals-area-an-integer-number/

He gives a couple of examples and challenges the reader to find some more.  One of his examples solutions is the  triangle with sides , , and which has an area of 21.

Following the hint on Ben’s blog, I played around a little bit and found that the triangle with side lengths of , , and has an area of 3.   Here’s my work, where you’ll see the Fibonacci numbers hiding in the middle of the page!

I really like this problem, and as I mentioned to Ben via Twitter last night if we weren’t on vacation I’d be working through this problem straight away with my kids.  This problem shows a beautiful connection between geometry and arithmetic, and the Fibonacci numbers come up in my example from a neat connection to continued fractions.  What a wonderful problem.  Thanks Ben!