Pythagoras’ Theorem

Pythagoras’ Theorem is one of the oldest and well known theorems known today. The reason for this is probably how simple, yet extremely useful it is. For those of you who need your memory jogging, the theorem is question is:

$a^2 + b^2 = c^2$

Where a, b and c are the sides of a right angled triangle, with c being the hypotenuse, (the longest side, opposite the right angle).

Did you know? – Pythagoras is said to have had a golden thigh, which – for some unknown reason – he exhibited in the Olympic games.

As the diagram above hopefully helps illustrate, you can see that if you add the square of the two shorter sides together, this is equal to the square of the hypotenuse.

Example 1 – A ship sails 10km east, and then 7km North. Find the straight line distance from it’s starting point to 2dp.

First thing we need to do with these types of problems is to draw a diagram.

You can see from the diagram that the path traveled by the ship and the distance from it’s starting point form a right angled triangle.

Using Pythagoras’s Theorem, we can deduce that $10^2 + 7^2 = d^2$, a little rearranging gives $d^2 = 149$, so $d = \sqrt{12.206556}$. So to two decimal places, the distance traveled by the ship is 12.21km.

Pythagoras’ Theorem in 3D

Pythagoras’ Theorem can also be extended for use in 3 dimensional problems.

Example – Consider then cuboid with sides of length 12km, 7km and 5km.

Let’s say we have a plane that travels from point A to point B. That is, it travels 12km forwards, 7km left and 5km upwards. What is the total distance traveled?

The three dimensional version of Pythagoras’ Theorem works in just the same way. To find the distance traveled we take the square the 3 distances, and then take the square root of the sum. So in our example:

$\sqrt{12^2 + 7^2 + 5^2} = \sqrt{218} = 14.76$ to 2 decimal places.

So the distance traveled by our plane is approximately 14.76km.