Every kid knows "Pythagoras' Theorem"...most can parrot it decades later "The square on the hypotenuse equals the sum of the squares on the other two sides".
What the heck does it mean?
It means that if you have a triangle with a right angle in one corner - then if you know the lengths of two sides of the triangle, you can always find the length of the third side.
This was invented by a Greek guy named "Pythagoras" about 2,500 years ago. But it's amazing how many people have it memorised - but have no clue what it's for.
This is a true story:
We all know that advancements in technology can cost people their jobs. However, in the case of the building industry in Texas, the effect of introducing new technology can often be somewhat delayed.
Back in 1997, my new house was in the slow process of changing from plans on paper into bricks on concrete. One of the tasks that has to be done early on is to lay out the shape of the house accurately onto the land. My builder uses a sub-contractor to do that - and I had occasion to watch him work. He arrived in a beat up old pickup truck with four 'migrant workers' sitting in the back. In order to lay out the initial 'bounding rectangle' of the building, they follow this algorithm:
Well, I watched this with some amusement - and asked why they didn't just calculate the length of the diagonal. The boss guy said that you couldn't do that - "It's impossible". I told him about Pythagoras' theorem. With the aid of a calculator (he didn't know what that funny 'square-root' key was for), I was able to show him how easy it is to calculate the length of the diagonal and do away with all the ugly 'jiggling'.
"Wow!" he said. Then he thought for a moment - "Now I'll only need three guys to hold the string!"...and fired one of them on the spot! I thought he was kidding - but the next day when they were measuring out the place for the garage, there was one less guy holding the string.
So, a 2,500 year old technological advance cost some poor guy his job.
Of course what I told him doesn't sound anything like "The square on the hypotenuse is equal to the sum of the squares on the other two sides" - but what Pythagoras was getting at was this:
The "hypotenuse" is just the longest side of a triangle that has a right angle. The "square on the hypotenuse" is that red square - and the green and purple squares are "the squares on the other two sides.
So the area of the green square plus the area of the purple square equals the area of the red square. That's all Pythagoras has been trying to tell us for the past 2,500 years.
For my house, 'a' and 'b' were two sides of the rectangle and 'c' (the hypotenuse) was the diagonal. So by multiplying the length of the house by itself, I was calculating the area of the purple square, multiplying the width by itself got me the area of the green square...and adding them together got me the area of the red square.
The "square root" button figures out what the length of the red square must be - and that's the length of the diagonal line across my house.
With all three sides of the triangle measured out in string, pulled tight and staked down - we have a perfect right angle. Then they just had to do it again with the other side - and they'd have a perfect rectangle.
Pythagoras is handy any time you need to compute a diagonal distance. You could use it to figure out the length of the roof timbers for a house too - if you know how wide the house is - and how high you want the peak of the roof - then pythagoras' theorem will tell you how long your timbers need to be.
Some people want to know how Pythagoras figured this out - to be honest, we're not absolutely sure - but probably he did something this:
You can take the bottom square and chop it into four pieces - then assemble the four pieces with the yellow square from the shorter side to make a square on the hypotenuse.
It's interesting to know that Pythagoras didn't have to pick squares. The sum of the teapot on the hypotenuse is equal to the sum of teapots on the other two sides! Any shape will do - it's just easier to use squares because calculators don't have a "teapot area" button.