Coordinate geometry is a method of analysing geometric shapes.

To start off with, let us consider one of the simplest geometric problems: describing the position of points on a 2-dimensional surface (such as a piece of paper).

The position of points will usually be referred to in one of these three ways:

1. Fixed point: for example, (2,6).
2.  
Generic fixed points: for example, (x₁, y₁), (x₂, y₂).
3. General points: for example, (x, y). This is telling us that the point can be anywhere along a shape.

Using coordinates means that we have to define where we start measuring from. When we are defining points on a graph, our point of reference is nearly always the origin or (0, 0).

The next thing to be happy with is how to calculate the distance between any 2 points.

We are interested in finding the distance from A to B. A is at (x₁, y₁) and B is at (x₂, y₂). Let's take a look at this using the diagram below:

We want to find the distance between A and B. If you look at the diagram above, we have added an imaginary point L, so that the triangle ABL is a right-angled triangle.

Using Pythagoras' Theorem we can now express the length of the line AB in terms of A, B and L:

AB² = AL² + BL²

Remember that the position of A is (x₁, y₁) and the position of B is (x₂, y₂). So the distance between A and L is x₂ - x₁ and the distance between B and L is y₂ - y₁ (see diagram above).

Hence, we can re-write the above equation as:

AB² = (x₂ - x₁)² + (y₂ - y₁)²

Solving this for AB gives us the equation we are trying to prove, telling us that the distance between A and B can be expressed by:

So now you can find the distance between any two points (x₁, y₁) and (x₂, y₂), and also understand how the formula was derived!

Here's a quick example of how this would work with numbers.

The distance between (2,6) and (-3,9) is:

The next step is to learn how to calculate the midpoint, M, between any two points.

Consider this diagram:

M is the midpoint of A and B.

The x coordinate of the midpoint M is found by taking the average of the x coordinate of the points A and B (this is half way between x₁ and x₂ on the diagram above).

The y coordinate of the midpoint M is also found by taking the average of the y coordinates of the points A and B (this is half way between y₁ and y₂ on the diagram).

Therefore, we can say that:

The x coordinate of M equals

The y coordinate of M equals

So using this technique you can find the midpoint of any line defined by two points. e.g. the midpoint of (2,6) and (-3,9) is (-0.5, 7.5)

The gradient of a straight line measures its slope in relation to the x-axis. In other words, the amount that y increases for every unit increase in x.

The gradient of a line can be positive or negative.

A positive gradient has is an 'uphill' slope in relation to the positive x-axis, and this means that a positive increase in the x direction is accompanied by a positive increase in the y direction.

A negative gradient has a 'downhill' slope in relation to the positive x-axis. This means that a positive increase in the x direction is accompanied by a decrease in the y direction.

How to calculate the gradient of a line

The gradient of a line is the increase in the y coordinate divided by the increase in the x coordinate between any 2 points on a line.

Looking at the graph we can say that:

Note: This method can only be used to find the gradient of a straight line. If you need to find the gradient of a point on a curve, you need to use differentation (see the Differentiation Learn-It. There is a whole S-cool! topic dedicated to differentiation.)