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Coordinates

What you'll learn

  • How to read and plot points on a coordinate grid.
  • How to use negative coordinates correctly.
  • How to find the midpoint of a line segment.
  • How to draw simple lines such as x=−1x=-1x=−1 and y=6y=6y=6.

The coordinate grid

A coordinate grid has two number lines crossing each other. You use it to describe exact positions of points.

A coordinate grid with the x-axis, y-axis and origin clearly labelled.

Definition

Coordinate grid

  • The x-axis is the horizontal line going left and right.

  • The y-axis is the vertical line going up and down.

  • The origin is the point where the axes meet. It is at (0,0)(0,0)(0,0).

A coordinate grid showing the horizontal x-axis, vertical y-axis, and origin at (0,0).

  • A coordinate pair tells you a point’s position and is written as (x,y)(x,y)(x,y).
Key Idea

Across first, then up or down

Coordinates are always read in this order: x first, then y.

Reading coordinates

To read a point from a grid, start at the origin.

  • The first number tells you how far left or right the point is.
  • The second number tells you how far up or down the point is.
  • Positive x means right.
  • Negative x means left.
  • Positive y means up.
  • Negative y means down.
Example

Writing down the coordinates of a marked point

Point A is 4 squares right of the origin and 2 squares below the x-axis, so its coordinates are (4,-2).

  1. Suppose point A is 4 squares to the right of the origin.

  2. That means the x-coordinate is 4.

  3. Point A is also 2 squares below the x-axis.

  4. That means the y-coordinate is -2.

  5. So the coordinates of point A are (4,−2)(4,-2)(4,−2).

Point A is 4 squares right of the origin and 2 squares below the x-axis, so it is at (4,-2).

Common Mistake

Swapping the coordinates

A very common mistake is writing the y-coordinate first. Remember: coordinates are always written as across, then up/down.

Plotting a point

To plot a point means to put it in the correct place on the grid.

Always start at the origin, then count across first, then up or down.

Example

Plotting point B at (−1,3)

To plot B(-1,3), move 1 square left from the origin and then 3 squares up.

  1. Start at the origin, (0,0)(0,0)(0,0).

  2. The x-coordinate is -1, so move 1 square left.

  3. The y-coordinate is 3, so move 3 squares up.

  4. Put a cross at that position.

  5. Label the point B.

To plot B(-1,3), move 1 square left from the origin and then 3 squares up.

Tip

Negative directions

Negative x goes left. Negative y goes down. If both are negative, you move left and down.

Midpoints

Sometimes you are given two points joined by a line segment. A line segment is a straight line between two endpoints.

Definition

Midpoint

The midpoint is the point exactly halfway between two endpoints.

To find a midpoint, you find the middle of the x-coordinates and the middle of the y-coordinates.

Another way to say this is: add the two x-coordinates and divide by 2, then add the two y-coordinates and divide by 2.

Key Idea

Midpoint rule

Find the average of the x-coordinates and the average of the y-coordinates.

Example

Finding the midpoint of B and C

The midpoint of B(-3,3) and C(3,0) lies halfway along the line segment at (0,1.5).

  1. Let B be at (−3,3)(-3,3)(−3,3) and C be at (3,0)(3,0)(3,0).

  2. Find the middle of the x-coordinates. The x-coordinates are -3 and 3.

  3. Add them and divide by 2:

    (−3+3)÷2=0(-3+3)\div 2=0(−3+3)÷2=0

  4. Find the middle of the y-coordinates. The y-coordinates are 3 and 0.

  5. Add them and divide by 2:

    (3+0)÷2=1.5(3+0)\div 2=1.5(3+0)÷2=1.5

  6. So the midpoint is (0,1.5)(0,1.5)(0,1.5).

The midpoint of B(-3,3) and C(3,0) lies halfway along the line segment between them at (0,1.5).

Common Mistake

Only finding one coordinate

A midpoint needs two coordinates. Make sure you find both the x-value and the y-value.

Drawing lines like x=−1x=-1x=−1 and y=6y=6y=6

An equation is a mathematical statement with an equals sign.

For example, x=−1x=-1x=−1 tells you that every point on the line has an x-coordinate of -1.

Definition

Equation of a line

An equation of a line tells you which points are on that line.

Lines where x is fixed

A line such as x=3x=3x=3 is vertical. Every point on the line has x-coordinate 3.

A line with equation x=3 is vertical because every point on it has the same x-coordinate.

Example

Drawing the line x=−2

  1. Find -2 on the x-axis.

  2. Draw a straight vertical line through -2.

  3. The line goes up and down through the grid.

  4. Check with a point on the line, such as (−2,4)(-2,4)(−2,4). Its x-coordinate is -2.

The line x=-2 is vertical because every point on it has x-coordinate -2.

Lines where y is fixed

A line such as y=6y=6y=6 is horizontal. Every point on the line has y-coordinate 6.

A line with equation y=4 is horizontal because every point on it has the same y-coordinate.

Example

Drawing the line y=4

  1. Find 4 on the y-axis.

  2. Draw a straight horizontal line through 4.

  3. The line goes left and right through the grid.

  4. Check with a point on the line, such as (1,4)(1,4)(1,4). Its y-coordinate is 4.

The line y=4 is horizontal because every point on it has y-coordinate 4.

Tip

Remember vertical and horizontal

Lines written as x=numberx=\text{number}x=number are vertical. Lines written as y=numbery=\text{number}y=number are horizontal.

Completing a square on a grid

A square is a shape with 4 equal sides and 4 right angles.

If you are given three corners of a square, you can find the missing corner by matching the movement from one side.

For a square called ABCD, the letters usually go around the shape in order.

Example

Finding the missing corner D

  1. Suppose A is at (−1,3)(-1,3)(−1,3), B is at (3,2)(3,2)(3,2), and C is at (2,−2)(2,-2)(2,−2).

  2. Look at the move from B to C.

  3. From B to C, you move 1 square left and 4 squares down.

  4. To find D, repeat that same move from A.

  5. From A, move 1 square left and 4 squares down.

  6. So D is at (−2,−1)(-2,-1)(−2,−1).

Repeating the move from B to C starting at A gives the missing corner D(-2,-1) of the square.

Tip

Use the matching side

In a square ABCD, the side AD matches the side BC. So the movement from B to C is repeated from A to D.

Exam technique

In the exam

  1. Always read coordinates in the order x first, then y.

  2. When plotting, start at the origin and count carefully.

  3. For a midpoint, average the x-coordinates and average the y-coordinates separately.

  4. For x=numberx=\text{number}x=number draw a vertical line; for y=numbery=\text{number}y=number draw a horizontal line.

Self review

Check yourself

  • Can you plot the point (−3,2)(-3,2)(−3,2) on a grid?

  • What is the midpoint of (4,−2)(4,-2)(4,−2) and (4,6)(4,6)(4,6)?

  • Is the line x=2x=2x=2 horizontal or vertical?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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Coordinates Revision Guide

  1. GCSE
  2. /Maths
  3. /Coordinates