Pythagoras’s Theorem

In this section we’re going to be exploring Pythagoras’s theorem. You will understand how to use Pythagoras’s theorem and be able to find the distance between 2 points in a coordinate grid.
Pythagoras was a Greek mathematician about 2699 years ago who discovered the method of calculating the length of the hypotenuse of a right angled triangle without drawing it.
What he did was took a triangle and drew squares on each side of the triangle as shown below.

Pythagoras looked and analysed the areas of the squares drawn at each end of the triangle. And concluded that;
The area of the square on the hypotenuse is equal to the sum of the squares on the shortest two sides of the triangle.

The areas of the two small squares drawn on the triangle sum up to give the area of the largest square area, the hypotenuse.

The following shows an example of finding the hypotenuse at the longest side the hypotenuse.

Example

Find the length of the hypotenuse of the following marked x.

Here we imagine that there are three squares drawn in the sides of the above right angled triangle as shown below.

The area of the smallest square A is 64;

And the area of the other smaller side B is 100;

The area of the largest square C is the sum of the areas A and B.

If the area of the largest square C is 100 that must mean that;

We could have done this quick and simple;

Shortest length

Above we have found the largest length the hypotenuse. Suppose we knew the hypotenuse and had to find one of the other smaller lengths. Here is an example:

We know that;

So we have to make x² the subject to work it out. Move 8² to the other side and you get;

We have managed to find the smaller side of the right angled triangle using Pythagoras theorem.

Finding distance between 2 points

Suppose we wanted to find the distance between two points on the grid. Here is a grid.

A has the coordinate of (3, 3) and B has coordinate (9, 7), suppose we wanted to find the length AB.
We can take advantage of Pythagoras here. Looking at the grid we can draw a right angled triangle as shown below.

We need to find the length of the triangle, to do that we look at the coordinates as shown on the graph. Now we can use Pythagoras.

There we have managed to find the length of AB by drawing the triangle on the grid in connection with the given coordinates and then used Pythagoras to solve the unknown lengths.