Pythagoras' Theorem
There are several proofs of Pythagoras' Theorem one of which can be found elsewhere on this site. Here we will look at how to use the theorem to solve problems. Before using Pythagoras' Theorem you must be comfortable rounding numbers and using powers and roots.
Pythagoras, a Greek mathematician, discovered a peculiar property of right angled triangles, namely that if you add together the lengths of the shorter sides squared you will get the length of the longest side squared. So, in this triangle:
If we add together the lengths of the shorter sides squared we get:
32 + 42 = 9 + 16 = 25
If we square the length of the longest side we also get 25. (52 = 25). The longest side of a right angled triangle is also called the hypotenuse.
The theorem works for any right angled triangle, not just the one with the above dimensions, so it is common to state the theorem using the letters a, b, and c to represent whatever the lengths of the sides are. c is the length of the longest side (the hypotenuse) and a and b represent the lengths of the shorter two sides.
Pythagoras' Theorem: a2 + b2 = c2
Using Pythagoras' Theorem
It is unlikely that you will come accross a question that asks you to find c2. It is more common to be asked to find the length of one of the sides, so, unless you are good at rearranging equations, rather than learn the above formula, it is better to use these two formulas:
To find one of the shorter sides a = √(c2 - b2)
To find the longest side c = √(a2 + b2)
The longest side of a right angled triangle is often called the hypotenuse. These formulas can be seen in action here.