Coordinates and Plotting Points

Read and plot points in all four quadrants, and find the missing corner of a rectangle on a grid.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 30 real past-paper questions below

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Across, Then Up

A coordinate is a pair of numbers that fixes one point on a grid. The first number is x: how far across. The second is y: how far up or down.

Start at the origin $(0, 0)$ , where the two axes cross. Minus means left or down.

Number	Tells you
First (x)	right if plus, left if minus
Second (y)	up if plus, down if minus

Memory trick

Along the corridor, then up the stairs: across first, then up.

Key idea

“Cartesian coordinates” are named after the French thinker René Descartes, who published the idea in 1637. Pierre de Fermat found it on his own at about the same time. The neat x and y axes you use were developed later by those who translated Descartes' book.

Writing Down Coordinates

The 1-mark question: “Write down the coordinates of A.” Read across first, then up.

Worked example

1
$x = - 3$ — Start at the origin and look along the x-axis. A is 3 squares left, and left means minus.
2
$y = 2$ — Now look up or down. A is 2 squares above the x-axis, and up means plus.
3
$A = (- 3, 2)$ — Write the across number first, then the up number, in brackets with a comma. Everyone reads coordinates in that order.

Common mistake

Writing (2, −3). The order matters: (2, −3) is a different point, in another corner of the grid.

Key idea

A point on an axis has a 0 in it. A point on the y-axis 4 up is $(0, 4)$ , not $(4, 0)$ .

Marking a Point

The 1-mark question: “Mark with a cross the point $(4, - 3)$ . Label it T.”

Worked example

1
$(4, - 3)$ — Read the pair. The first number is across, the second is up or down.
2
$4 right$ — Start at the origin. 4 is plus, so go 4 squares right along the x-axis.
3
$3 down$ — From there, the minus in −3 means go 3 squares down.
4
$\times T$ — Draw a small cross exactly where the grid lines meet, then write T next to it. The centre of the cross is the point, and the label shows which point it is.

Common mistake

Using the numbers the wrong way round. Going 3 left and 4 up plots $(- 3, 4)$ , a different point. The first number is always across.

The Missing Corner

The 1-mark question: “PQRS is a rectangle. Find the coordinates of S.” A rectangle's sides here run along the grid lines.

Worked example

1
$P (- 2, 3), Q (3, 3), R (3, - 1)$ — The three corners you have. Sketch them with the letters in order: P, Q, R, then S joins back to P.
2
$S is below P : x = - 2$ — PS is an upright side, like QR. Points straight above or below each other have the same across number, so S takes P's x.
3
$S is level with R : y = - 1$ — RS is a flat side, like PQ. Points level with each other have the same up number, so S takes R's y.
4
$S = (- 2, - 1)$ — Put x and y together. Check: S is below P and level with R, so the four sides close the rectangle.

Key idea

Sketch it first. Your fourth corner must close the shape with the letters in order round it.

Corner of a Parallelogram

Some questions use a parallelogram or a rhombus, so the sides are slanted. Use this when copying an x and a y does not work.

Rule

Opposite sides of a parallelogram are parallel and the same length. So they are the same step: the same number across and the same number up.

Worked example

1
$A (- 3, - 1), B (2, - 1), C (4, 3)$ — ABCD is a parallelogram. Sketch the three corners: the letters go round in order, so D joins on to A and to C.
2
$B \to C : + 2 across, + 4 up$ — Count the step from B to C: 4 − 2 = 2 across and 3 − (−1) = 4 up.
3
$A \to D is the same step$ — AD is the side opposite BC. Opposite sides match, so going from A to D is the same step.
4
$x : - 3 + 2 = - 1$ — Add the across step to A's x.
5
$y : - 1 + 4 = 3$ — Add the up step to A's y.
6
$D = (- 1, 3)$ — Check: D to C is 5 across and 0 up, the same step as A to B. So both pairs of opposite sides match.

Common mistake

Adding the step to C. That gives (6, 7), which does not close the shape. D sits next to A, so start from A.

Key idea

A rhombus is a parallelogram with four equal sides, so its missing corner is found the same way.

Which Method? Read the Question

The command word tells you which way to go.

You see	You do
“Write down the coordinates”	read across, then up
“Mark with a cross”	go across, then up, then label
“Find the fourth corner”	sketch the shape, copy an x and a y
Parallelogram or rhombus corner	copy the step of the opposite side
Gradients, diagonals or a kite	Coordinate Geometry of Shapes
The middle of two points	Length of a Line Segment

Key idea

Brackets and a comma every time: (x, y). The mark scheme wants both numbers in the right order.

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