Understanding Pythagorean principles

From simple things like a thread of a screw or the construction of a ramp to more complex issues such as the calculation of distance and angle for the layout of a building; all these things rely heavily on Pythagoras' Theorem. In complex scientific language Pythagoras states:

"In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle)."

In Mathematical Formula: a2+b2 = c2

In plain English, what Pythagoras is trying to say is that the length of the longest side multiplied by itself is the same as the sum of each of the remaining sides when they are multiplied by themselves. In essence it is a simple statement, but the ramifications of that statement are profound.

There are three different types of triangles: scalene, isosceles and equilateral. So what is a "right" triangle? A "right" triangle as used by Pythagoras is a triangle in which one of the three angles is a "right" angle; an angle of 90o. To demonstrate an angle of 90o is simple; a loose-leaf sheet of plain paper is sliced at 90o and may be used as a crude measure of constructing one.

The above figure is that of a right triangle, that is labeled mathematically correctly. The right angle is denoted by the capital letter C, opposite to the longest side the hypotenuse (labeled common letter c). It is the convention to label the triangle with the sides being denoted by the common letter a, b and c and the angles denoted by the capital letters A, B, and C. In keeping with this convention, the angle labeled with a capital letter is usually opposite to the side labeled with the corresponding common letter (as was seen earlier with the right angle and the hypotenuse). The descriptions "Opposite" and "Adjacent" hold true for the angle A in this case. The Adjacent line to angle A is side b, and the Opposite line to angle A is side a (following the labeling convention). These are used to determine angular ratios: sine, cosine and tangent.

Now you are probably wondering what use is all this in doing anything in the real world. The answer is quite a lot. If you look at the right triangle above, you may see a resemblance between it and the cross section of an inclined plane. The inclined plane is one of the simplest of tools but construction of it and tools that have sprung from it, are wholly due to Pythagoras' work. Distance measurements with co-ordinates rely on the same principle, as do the construction of screws and anything that has a thread in it. A thread on a screw is simply an inclined plane wrapped around the screw to allow it to hold properly with as little effort applied to get it in place as is possible.

So, it can be seen how the Pythagorean principle works and we demonstrated how the right triangle is used in conjunction with Pythagoras' theorem to relate distances and angles together. We have shown how Pythagoras related to everyday life and structures and even how it is responsible for things we had no idea about.