Daughter's algebra homework takes me back to school

My daughter is smart but was understandably grumpy with some very tiresome algebra homework. The trouble is I love numbers and my enthusiasm for solving even the alien-looking display of letters, symbols, lots of brackets and superscript probably annoyed more than helped her. I can just imagine today's conversation at break:

"Did you do your maths homework?"
"No. Daddy did."
"OMG. What, without getting cross or reminding you how useful algebra will be?"
"Well, he didn't get cross but he did do the 'it'll be really useful' thing. He seemed to quite enjoy it, actually."
"Whatever."

It had been a while since I'd had to do battle with x² and y³ but I really do think her school could have come up with more pleasant exercises - or at least less cumbersome answers that made you think you simply must have done something wrong when the last line of 'simplification' looked a damn sight more complex than the expression we'd started with!

Anyway, for some reason that I can't really explain, but I'll blame her for it, I woke up this morning trying to figure out how to slice a triangle so that the area of the small triangle at the top is one-half of the original area.

So the pink triangle above has 1/2 of the area of the bigger one.

What I wanted to know was where to draw the dotted line. Now there'll be people reading this who can just shout out the answer but I had to work it out. At one point I even looked up sine, cosines and tans on Google as I'd forgotten which was which. The answer, which I think is right, is delightfully simple (and that's the sort of problem Royal Latin School should be giving my daughter). You divide the original height by the square root of 2.

Then I had another thought. What height would a triangle with just a third of the area be? Ah - divide the height by the square root of 3! Brilliant. Ooops, no, surely, that can't be a series developing, can it? Because the next number is 4, so to make a small triangle with just one quarter of the original area would, if my thing were correct, simply mean having one half the height as the square root of 4 which is a nice, friendly number, 2, rather than some weird one with piles of never ending decimal places.

(The fact that I could, in those instances, never actually precisely measure where to draw the ruddy line did disturb me but I decided to leave that, together with why I can't measure a third of an inch properly, to another day.)

A picture helped me convince myself that I wasn't being silly.

So, with a little bit of algebra (I gave up on the trigonometry) I've discovered that you can make a smaller triangle of whatever proportion to the original just by diving the original height by the square root of whatever the fraction is to be. Along the way fractals made an appearance too when I played with numbers like 16 and 25. That's another story I'll share in a while.

And if my daughter is still struggling, there's always the excellent Khan Academy!