Pythagoras lived in ancient Greece and used the relationship that the square of the longest side equals the squares of the 2 shorter sides added together.  The shorter sides are called legs and are colored red in the diagram.  The longest side is the hypotenuse and is colored blue in the diagram.

Solving a problem:  Suppose we know the two shortest sides are 3 ft and 4 ft, then we can find the longest side, the hypotenuse by doing the following:

  where I am using x to stand for the length of the hypotenuse that I am trying to find.

The square of that side is 25 because .

What number squared will give you 25?  Since the answer is 5, the length of the long side must be 5 ft.

Notice the mathematical operation for getting 5 from 25 is taking the square root or .  If the number under the radical doesn’t turn out to be a perfect square, then you will have to use your calculator and give a rounded off decimal for your answer.

Another example:

This time we know the long side is 15 cm and one of the short sides is 6 cm.  What is the length of the missing side.

If the square of the long side equals the sum of the squares of the two short sides, then it must be the case that I can find the missing short side by subtracting the square of the one I know from the square of the longest side.

Thus,

.  This time I used “a” to stand for the side I don’t know.  Do you see with a little re-arrangement this says the same thing as ?

Solving gives me 225 – 36 = 189 = a².  Since= 13.75, the missing side is 13.75 cm.