Thursday, November 4, 2010

Special Right Triangles

 You know the ones...



Convenient, nice to know going into trig, time-saving...especially for the SATs and GREs. It's a little hard to believe just how much the College Board hearts it some special right triangles.

But still, hard to motivate.

Enter...the humble dollar bill. It's a rectangle. What sort of rectangle? Not all rectangles are the same shape, of course. Some are squares! Their sides are in a 1:1 ratio. Some are square-like. Some are long and skinny. A long, skinny one's sides might be more like 10:1 or even more severe.

What about our paper currency in the US? All denominations are the same size and shape. This is not  true in every country. But ours are all this very familiar rectangle.


What ratio do you think its sides are in? If you use two bills to measure, to see how many short sides make up the long side, you can see that it's skinnier than 2:1 (If it were 2:1, it would make a square when you folded it in half, and it doesn't. Credit for that observation: Rachel B., class of '13), but not quite 5:2 (or 2.5:1, if you prefer). (Cue annoying kid shouting "TWO AND A THIRD! IT'S TWO AND A THIRD" employing a technique commonly known as "proof by intimidation.")

Hm.

Now we do a little folding. I handed out photocopies of dollar bills. They were black and white, and one-sided, so I don't think it was a felony.



It's an isosceles trapezoid!

It's a rhombus!

It's an equilateral triangle!
At this point, you can also gently unfold it and coax it into a tetrahedron.

Unfold it all the way, and behold...


Don't see it? Look at the creases.


What does this mean? In my classroom it means I can launch the derivation of side relationships in 30-60-90 triangle with a tiny amount of Davinci-code-type intrigue which is totally worth navigating the murky waters of incommensurate lengths. But in a nutshell, it means our familiar dollar bill is made of a rectangle in a supremely weird ratio of $4:\sqrt{3}$

and come on, that was no accident. I bet the Freemasons were involved.