[Logo]   Langara College - Department of Mathematics and Statistics 

Internet Resources for Mathematics Students

 

Pythagoras' Theorem
Its most practical application is that it allows us to find the distances between distinct points in space. A simple example of this would be to find the distance between opposite corners of a room whose walls are of length a and b, since if a and b are given we can find c by taking the square root of a2 +b2 . (Do you see that if we put another triangle above the one in the picture - upside down and facing in the opposite direction - then we could make a rectangle with sides a and b and diagonal c?)
For example, if the wall lengths are b=3 and a=4, then c2 = 42 +32 =16+9=25, so c is the square root of 25, ie c=5.
But that's not all! We can apply the same idea to find the distance between any two points in the plane in terms of their Cartesian Coordinates, and if you are clever enough you may see how the rule generalizes to the case of three dimensions as well.

Of course, not all examples work out as nicely as the 3,4,5 example mentioned above. In fact, from a mathematical point of view what makes Pythagoras' Theorem almost magical in its implications is that, as a special case of the distance rule, it shows us the existence of lengths which cannot be related by whole number ratios. (An example of such an "irrational" length occurs if we take a and b of length 1 in the picture, for then c squared is 2 and this is not possible for any ratio of whole numbers. This was very upsetting to Pythagoras and his followers as their philosophy idealized such ratios. Ironically it is the theorem bearing Pythagoras' own name which was the downfall of much that he believed in.)

There are many ways of proving this important theorem and no-one's education can be said to be complete without understanding at least one.

Here are some links to proofs of the theorem that you can find on the Web:
 

This award winning presentation of an animated proof of Pythagoras' Theorem by UBC's Jim Morrey was an early example of the use of JAVA to help with the understanding of mathematical ideas.

Here's another presentation of the same proof (by the IES group in Japan)

(This latter site also illustrates a number of other proofs of the theorem, and allows you to see how some apparently different arguments are really just different aspects of the same basic principle.)

Review Contents of Math&Stats Dep't Website ....... Give Feedback ....... Return to Langara College Homepage