I stumbled across this question in a book yesterday:
I know the problem, and in fact have solved it a few times before, but upon looking at it I couldn’t quite remember how I solved it, so I thought I’d have another stab at it. It didn’t go well…!
I knew I needed something like this diagram, because there’s just nothing else I can think of doing with the tiny amount of information I’m given.
More specifically, I just need this half (I think). But this is where I got into a pickle. I stared at the above diagram for a little while, convinced it was what I needed, but completely lost at how it helps me. Sure I have one side, and what I need is part of the triangle, but surely I need *more* information??
Now as I said, I have solved this problem a few times, and in fact, the first time it was shown to me I solved it in about a minute or two, but for whatever reason I couldn’t for the life of me get past this point yesterday. The reason why I couldn’t progress was simple – I wasn’t following my own advice on solving problems. Specifically, I’d recommend to students just finding as many things as you can until you find a way in. So after staring at that little triangle for far too long, I decided to just use Pythagoras anyway, despite “knowing” that I didn’t have enough information for it to be useful.
So what? I surely needed to replace R with some kind of expression using r and a number… but I don’t have that information, so I have a ‘useless’ two variable equation and nothing else to work with.
I re-read the question and realised I’d been hung up with the seemingly useless application of Pythag instead of focusing on the fact that the question concerns area. Finally, I had that moment of clarity.
The bit I want is the shaded bit, so I can effectively ignore
Weird how it took me so long to ‘see’ it this time, and I spotted it so quickly a couple of years ago. I probably shouldn’t overthink that…