My high attaining Year 9 class didn’t quite get this on their own yesterday but they enjoyed the challenge and were able to follow the explanation.
The crux of the problem is getting a right angle triangle with sides 1, (1+x) and (2-x). It is the (2-x) side which is hardest to spot. There were groans when I finally showed them.
Then it requires some algebra – namely expanding (1+x)² and (2-x)² which Year 9 hadn’t had much practice in, so it was good to show them why (1+x)²≠1²+x²
I gave them the problem printed out, here they are 2 to a page.
Here is link to the Geogebra file that I created this on.
I had a few Twitter responses to this including @ProfSmudge who kindly set us an extension question:
It’s an example of an Apollonian gasket, apparently (thanks to @mathforge for pointing that out!). That gets properly hard, involving Cosine Rule. Certainly not something I’d give to my students, but I’ve got a few teachers working on it!
