Coordinates

Coordinates are a way to locate points on a plane. If you think of a map or a graph, coordinates tell you exactly where something is. The plane we usually use for this in maths is called the Cartesian plane, named after the French mathematician René Descartes.

The main thing to remember is ‘you go along the corridor and up/down the stairs.’ This helps us to know how to write the coordinates. The x-axis is horizontal, and the y-axis is vertical. There are 4 quadrants (regions).

Components of Coordinates

Every coordinate has two main components: the x-value and the y-value. The x-value tells you how far along left or right the point is, and the y-value tells you how far up or down it is. Have a look below:

• The coordinate for point A is , first you move 3 points to the left (along the x-axis) then up 2 points (along the y-axis). Coordinates are always written in brackets.

• For point B, the coordinate is . This means that you move 4 points to the right (along the x-axis) then up 3 points (along the y-axis).

• For point C, the coordinate is . This means that you move 5 points to the left (along the x-axis) then 4 points down (along the y-axis).

• The coordinate for point D is , first you move 2 points to the right (along the x-axis) then down 2 points (along the y-axis).

Plotting Points on a Cartesian Plane

To plot these points on a Cartesian Plane, you start at the point where x and y are both zero. This is known as the origin, represented by the coordinate (0, 0).

From there, you move to the right if x is positive and to the left if x is negative. Then, you go up if y is positive and down if y is negative. For example, to plot , you’d move 3 units to the right and 2 units up from the origin.

The Four Quadrants

The Cartesian Plane is divided into four areas called quadrants:

• When both x and y are positive, the point will be in the first quadrant

• If x is negative while y is positive, the point will be located in the second quadrant

• In cases where both x and y are negative, the point will be located in the third quadrant

• If x is positive, but y is negative, the point will be in the fourth quadrant

Distance Between Two Points

Finding the distance between two points on a Cartesian plane is like drawing a straight line between them. Imagine you have points A and B .

If you draw a line connecting these points, you could measure its length to find the distance. Sometimes, this is done using a formula, but at this stage, visualising it can often be enough.

Midpoint of a Line Segment

The midpoint between two coordinates is just what it sounds like: the point exactly halfway between them.

To find the midpoint, you average the x-values and the y-values of the two endpoints. If you have coordinates and , the midpoint would be calculated as follows:

For the x-coordinate:

For the y-coordinate:

So, the midpoint is , which is point f.

Practice Questions

Question 1:

In which quadrant is the point located?

Solution

The x-value is negative and the y-value is positive, placing this point in the second quadrant.

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Question 2:

What’s the midpoint between the points and ?

Solution

Average the x-values, , and the y-values, , to get the midpoint at .