An irrational triangle

The sad news about Don Steward last week prompted many people to share blog posts and tweets about how he has helped and inspired them over the years. One of the people who blogged about him was Jo Morgan (@mathsjem) who wrote this piece. Jo had the pleasure of knowing Don personally and it was really nice to read the things she shared. If you missed it, do give it a read.Today I want to look at the problem she shared at the end of the post, which she says is her favourite Don Steward problem:It’s not one I remember seeing before, although I have spent so long on Don’s fantastic median website it is highly probably I have seen it. But I thought I’d have a go. Usually when working out an are of a triangle with 3 sides I will apply Heron’s formula, but I looked at this one and realised I would end up with a massive expansion of surds and quite possibly nested radicals and that would take a long time to work through by hand, so I considered other options. I thought a locigal first attempt would be to use trigonometry. I drew a sketch:

Labelled one if the angles x. I can then work out cos (x) fairly easily:

I did think I might get stuck at this point, with the limitation in the question prohibiting calculators, but actually it worked out really nicely. If I know cos (x) I can use a right angled triangle to work out sin (x):

Here I only have to square 1 (1) and the root of 26 (26) do a subtraction and take the square root of 25 (5) to complete my triangle using Pythagoras’s Theorem. All easily done without a calculator.Now I have sin x I can work out the area using area = absin(c)/2

When I started playing with the surds I wasn’t quite expecting it to fall out so nicely. Who’d have thought a triangle with 3 irrational side lengths would have such a nice integer area? I think I will explore this more when I get time. See how easy this type of triangle is to generate and if there is a rule to triangle of this sort occurring.

I said earlier that I didn’t think I’d seen this one before. But now I think about it, I think I may have. I think I remember giving it to a year 12 class maybe last year and some of them cheating and using the calculator but not getting the right answer as it had rounded the cosine and sine values, I will need to check this later on a calculator too. If this is the case, then I think this is a great discussion point when looking at rounding and why we shouldn’t round too early. I’m also left wondering how easy/difficult it would have been to get to the nice answer using Heron’s Formula, so I will give that a go in due course too.