Triangles, Trig and Squares, oh my.

Over the weekend I happened accross this loevely puzzle on twitter. It was tweeted out by Diego Rattaggi (@diegorattaggi) and I saved it to my ever growing folder of puzzles to try on my phone. When i had a bit of time spare I thought I would give it a go.

Have you had a try? If not, you should. Its a nice one and I’d love to hear some different approaches.

Anyway, here is how I approached it. Initially, I wondered about finding an algebraic expression that might fall out with some nice numbers, but i got a nasty looking equation with z and y both raised to the power 4 so I thought I’d try a different tactic.

I did plenty of annotating on a diagram to visualise the information, and found that 180 = 180, so that wasn’t particularly useful.

Then I looked at a trigonometric approach. Which fell out nicely.

I has Tan 2b = 2, and I needed tan (90-2b), my insitinc was to use double angle formulae, but I quickly realised tan 90 was undefined so that wouldn’t work. I sketched a quick right angle triange, then realised I was being a tad silly.

While the triangle did give me the correct answer, it also made me remember that the definition of Cotangent is “The tangent of the complementary angle”, so cot 2b is quite literally tan (90-2b), so all i needed to do was take the reciprocal.

It was a nice little puzzle that incorporated a number of things, and it has me wondering how my year 13s would approach it. I’m gonna try it out with them later this week. I’d be intrested to hear how you approaced it, expecially if you soved it a different way.