Maths · 4 Coordinate geometry

4.1 Coordinates

Before you start

Negative numbers on a number line, and subtracting a negative number, e.g. $5 - (- 2) = 7$ .
Multiplying and dividing by decimals such as $0.5$ and $0.4$ .
The names and basic properties of a square, a rectangle and a parallelogram.
The area of a rectangle as length times width.

Syllabus objectives covered (1)

Use and interpret Cartesian coordinates in two dimensions.

Part 1 of 6

Two number lines and an ordered pair

A Cartesian grid is two number lines that cross at right angles. The horizontal one is the $x$ -axis and the vertical one is the $y$ -axis. They cross at the origin, the point $(0, 0)$ .
Every point on the grid is fixed by two numbers, written $(x, y)$ . The first number tells you how far to go across from the origin. The second tells you how far to go up or down.
The order matters, which is why $(x, y)$ is called an ordered pair. $(2, 5)$ is 2 across and 5 up. $(5, 2)$ is 5 across and 2 up. They are two different points.
A way to hold the order in your head: you walk along the corridor before you take the stairs. Across first ( $x$ ), then up or down ( $y$ ).
Write coordinates in round brackets with a comma between the two numbers: $(4, - 1)$ . The brackets and the comma show that the two numbers belong together as one point.
A point on an axis has a zero in it. On the $x$ -axis you have not moved up or down, so $y = 0$ , as in $(3, 0)$ . On the $y$ -axis you have not moved across, so $x = 0$ , as in $(0, 3)$ .

Part 2 of 6

Negative coordinates and the four quadrants

Each axis continues past the origin into negative numbers. A negative $x$ -coordinate means go left. A negative $y$ -coordinate means go down.
So $(- 3, 2)$ is 3 left and 2 up, and $(4, - 5)$ is 4 right and 5 down. The sign tells you the direction; the number tells you the distance.
The two axes divide the grid into four regions called quadrants. They are numbered anticlockwise, starting at the top right.
graph diagram — not drawn yet
A coordinate grid from -6 to 6 on both axes with the origin labelled O, the four quadrants labelled First, Second, Third and Fourth anticlockwise from the top right, and the points (2, 5), (5, 2), (-3, 2), (-6, -1) and (4, -5) plotted as small crosses with their coordinates written beside them.
(2, 5) and (5, 2) are different points, and the signs of the two coordinates decide the quadrant.
First quadrant (top right): $x > 0$ and $y > 0$ . Second quadrant (top left): $x < 0$ and $y > 0$ . Third quadrant (bottom left): $x < 0$ and $y < 0$ . Fourth quadrant (bottom right): $x > 0$ and $y < 0$ .
This means the signs alone tell you roughly where a point is before you plot it. $(- 6, - 1)$ has both coordinates negative, so it must be bottom left. Use this as a check on every point you plot or read.
A point on an axis, such as $(0, - 4)$ , is not in any quadrant. It sits on the boundary between two of them.

Part 3 of 6

Read the scale before you read the point

The grid lines are not always one unit apart. Before you plot or read anything, work out what one small square is worth on each axis.
Take two labels next to each other on the axis, find the difference between them, and divide by the number of small squares between them.
$one small square = \frac{difference between two labels}{number of small squares between them}$
Example: the $y$ -axis is labelled $0, 4, 8, 12$ with 10 small squares between each pair of labels. One small square is $4 \div 10 = 0.4$ units. A point 7 small squares above the $x$ -axis has $y = 7 \times 0.4 = 2.8$ .
The two axes can have different scales. Work out each scale separately. Never assume that the $y$ -axis uses the same scale as the $x$ -axis.
graph diagram — not drawn yet
First-quadrant axes: the x-axis labelled 0, 5, 10 with 10 small squares between labels, the y-axis labelled 0, 4, 8, 12 with 10 small squares between labels, and one point plotted 18 small squares right of the y-axis and 7 small squares above the x-axis, with dashed lines from the point down to the x-axis and across to the y-axis.
One small square is 0.5 across but 0.4 up, so the point is (9, 2.8), not (18, 7).
To plot a point, turn each value back into squares by dividing by the value of one square. With $0.4$ per square, $y = 6$ is $6 \div 0.4 = 15$ squares up.
Mark a plotted point with a small, neat cross centred exactly on its position, and write its letter beside it. A large blob covers more than one position, so it does not show which point you mean.

Part 4 of 6

Points that line up: horizontal and vertical sides

If two points have the same $y$ -coordinate, they are the same height above the $x$ -axis, so the line joining them is horizontal. $(- 2, 3)$ and $(5, 3)$ lie on a horizontal line.
If two points have the same $x$ -coordinate, the line joining them is vertical. $(4, - 1)$ and $(4, 6)$ lie on a vertical line, and every point on that line has $x$ -coordinate $4$ .
The length of a horizontal line segment is the difference between the $x$ -coordinates of its ends. For a vertical segment it is the difference between the $y$ -coordinates. Subtract the smaller from the larger.
$5 - (- 2) = 7$
The trap is the negative sign. From $x = - 2$ to $x = 5$ you go 2 units to reach the origin and then 5 more, so 7 in total. $5 - 2 = 3$ is wrong because it treats $- 2$ as if it were on the same side of zero as $5$ .
A horizontal line and a vertical line always meet at a right angle. So if one side of a shape is horizontal and the next side is vertical, the angle between them is $90 °$ , and the coordinates are your evidence.

Part 5 of 6

Using coordinates to complete a shape

A question can give three vertices of a shape and ask for the fourth. The letters go round the shape in order, so $A B C D$ has sides $A B$ , $B C$ , $C D$ and $D A$ .
For a rectangle whose sides are parallel to the axes, the missing vertex borrows from its two neighbours. It shares its $x$ -coordinate with the neighbour joined to it by a vertical side, and its $y$ -coordinate with the neighbour joined to it by a horizontal side.
For a parallelogram, opposite sides are parallel and equal in length. That means the move along one side is exactly the same as the move along the opposite side.
Describe the move as so many units across and so many up or down. If the move from $B$ to $C$ is 2 right and 4 down, then the move from $A$ to $D$ is also 2 right and 4 down.
graph diagram — not drawn yet
A coordinate grid from -3 to 7 across and -3 to 4 up showing parallelogram PQRS with P(1, 2), Q(6, 3), R(4, -1) and S(-1, -2) labelled, with one arrow from Q to R and an identical parallel arrow from P to S, each marked '2 left, 4 down'.
Opposite sides of a parallelogram are the same move, so the move Q to R also takes P to S.
Check the order of the letters before you copy a move. In $A B C D$ , side $B C$ is opposite side $A D$ , so the move from $B$ to $C$ goes from $A$ to $D$ . Applied from the wrong vertex, the same move gives a different point. That point is not $D$ : either the four letters do not go round the shape in the order $A B C D$ , or the shape is not a parallelogram at all (applied from $C$ , the move puts the new point on the line through $B$ and $C$ ).
Make a rough sketch of the points before you answer. A quick sketch shows whether your fourth point really makes the shape you were asked for.

Part 6 of 6

What a coordinate means on a real graph

On a graph of real data, each axis measures a quantity, with a label and a unit, such as time in minutes across and temperature in °C up. The grid works exactly as before: the first coordinate belongs to the horizontal axis, the second to the vertical axis.
So the point $(9, - 2.8)$ on a graph of temperature against time means: after 9 minutes, the temperature is $- 2.8$ °C.
When a question asks you to Describe or Explain what a point shows, give both values, with their units, in a sentence about the situation. Writing $(9, - 2.8)$ again only repeats the coordinates; it does not interpret them.
A negative coordinate has a meaning in context too. Here it is a temperature below zero, read from the part of the graph below the horizontal axis.

Worked examples

Example 1. Without plotting them, state where each point lies: $A (- 5, 2)$ , $B (3, - 0.5)$ , $C (- 1, - 4)$ , $D (0, - 3)$ and $E (6, 0)$ .

Show the working

$A$ : $x$ is negative and $y$ is positive, so go left and up. That is the second quadrant (top left).
$B$ : $x$ is positive and $y$ is negative, so go right and down. That is the fourth quadrant (bottom right). The decimal does not change anything; only the signs decide the quadrant.
$C$ : both coordinates are negative, so go left and down. That is the third quadrant (bottom left).
$D$ : the $x$ -coordinate is $0$ , so you do not move across at all. The point is on the $y$ -axis, 3 units below the origin.
$E$ : the $y$ -coordinate is $0$ , so you do not move up or down. The point is on the $x$ -axis, 6 units to the right of the origin.

Answer: $A$ second quadrant; $B$ fourth quadrant; $C$ third quadrant; $D$ on the $y$ -axis below the origin; $E$ on the $x$ -axis to the right of the origin.

Example 2. On a grid, the $x$ -axis is labelled $0, 5, 10, 15$ with 10 small squares between each pair of labels. The $y$ -axis is labelled $0, 20, 40, 60$ with 10 small squares between each pair of labels. (a) Point $P$ is 7 small squares to the right of the $y$ -axis and 13 small squares above the $x$ -axis. Find the coordinates of $P$ . (b) How many small squares across and how many up must you go to plot $Q (12, 50)$ ?

Show the working

Work out the scales first, one axis at a time. Across: $5 \div 10 = 0.5$ units per small square. Up: $20 \div 10 = 2$ units per small square.
(a) Turn squares into values by multiplying by the value of one square.
$x = 7 \times 0.5 = 3.5, y = 13 \times 2 = 26$
Sense-check: $3.5$ is between the labels $0$ and $5$ , and $26$ is between $20$ and $40$ , which matches 7 squares and 13 squares.
(b) Turn values into squares by dividing by the value of one square.
$12 \div 0.5 = 24, 50 \div 2 = 25$

Answer: (a) $P (3.5, 26)$ (b) 24 small squares to the right and 25 small squares up.

Example 3. $A (- 3, 1)$ , $B (5, 1)$ and $C (5, - 4)$ are three vertices of rectangle $A B C D$ . One unit on each axis represents 1 cm. Find the coordinates of $D$ and the area of the rectangle.

Show the working

$A$ and $B$ both have $y = 1$ , so $A B$ is horizontal. $B$ and $C$ both have $x = 5$ , so $B C$ is vertical. The rectangle has its sides parallel to the axes.
$D$ is joined to $A$ and to $C$ . Side $D A$ is opposite $B C$ , so it is vertical, and $D$ takes the $x$ -coordinate of $A$ , which is $- 3$ . Side $C D$ is opposite $A B$ , so it is horizontal, and $D$ takes the $y$ -coordinate of $C$ , which is $- 4$ .
Now the side lengths. $A B$ is horizontal, so subtract the $x$ -coordinates, keeping the sign.
$A B = 5 - (- 3) = 8$
$B C$ is vertical, so subtract the $y$ -coordinates.
$B C = 1 - (- 4) = 5$
Area of a rectangle is length times width. One unit is 1 cm, so the answer is in cm².
$8 \times 5 = 40$

Answer: $D (- 3, - 4)$ ; area $= 40$ cm².

Example 4. $P Q R S$ is a parallelogram with its vertices in that order. $P$ is $(1, 2)$ , $Q$ is $(6, 3)$ and $R$ is $(4, - 1)$ . Find the coordinates of $S$ .

Show the working

Use the letter order. Going round $P Q R S$ , side $Q R$ is opposite side $P S$ . Opposite sides of a parallelogram are parallel and equal, so the move from $Q$ to $R$ is the same as the move from $P$ to $S$ .
Find the move from $Q (6, 3)$ to $R (4, - 1)$ . Across: $6$ to $4$ is 2 left. Up or down: $3$ to $- 1$ is 4 down.
Apply the same move to $P (1, 2)$ .
$x = 1 - 2 = - 1, y = 2 - 4 = - 2$
Check with the other pair of opposite sides. The move from $P (1, 2)$ to $Q (6, 3)$ is 5 right and 1 up. The move from $S (- 1, - 2)$ to $R (4, - 1)$ is also 5 right and 1 up, so $P Q$ and $S R$ match.

Answer: $S (- 1, - 2)$

Where people go wrong

Mistake: Writing the coordinates the wrong way round, e.g. giving the point 2 across and 5 up as $(5, 2)$ .

Instead: Always $x$ first, then $y$ : $(2, 5)$ .

The pair is ordered. $(5, 2)$ is a different point, so it is a wrong answer, not just a slip in notation.

Mistake: Counting every small square as one unit without looking at the labels.

Instead: Work out the value of one small square from two neighbouring labels, separately on each axis, before you read or plot.

If 10 squares represent 4 units, a point 7 squares up is at $2.8$ , not $7$ .

Mistake: Finding the distance from $x = - 2$ to $x = 5$ as $5 - 2 = 3$ .

Instead: Subtract the coordinates with their signs: $5 - (- 2) = 7$ .

The two points are on opposite sides of zero, so the distance is the two parts added together, $2 + 5$ .

Mistake: Plotting $(0, 4)$ on the $x$ -axis at $4$ .

Instead: $(0, 4)$ is on the $y$ -axis: 0 across, then 4 up.

The zero is the $x$ -coordinate, so you do not move across at all. The point on the $x$ -axis at $4$ is $(4, 0)$ .

Mistake: Applying the move for a missing vertex from the wrong vertex.

Instead: Follow the letter order: in $A B C D$ , the move from $B$ to $C$ is the same as the move from $A$ to $D$ .

The move applied from the wrong vertex gives a different point, which either makes the letters go round in the wrong order or does not make a parallelogram at all, so it is not the point asked for.

Check yourself

In which quadrant does the point $(- 7, 3)$ lie?
Answer
The second quadrant (top left), because $x$ is negative and $y$ is positive.
Write down the coordinates of the point that is 4 units to the left of the origin and 6 units below it.
Answer
$(- 4, - 6)$
A point lies on the $y$ -axis, 2.5 units below the origin. Write down its coordinates.
Answer
$(0, - 2.5)$ . It is on the $y$ -axis, so $x = 0$ .
On the $y$ -axis, 5 small squares represent 1 unit. A point is 8 small squares above the $x$ -axis. What is its $y$ -coordinate?
Answer
One small square is $1 \div 5 = 0.2$ , so $y = 8 \times 0.2 = 1.6$ .
Find the distance between $(- 6, 3)$ and $(2, 3)$ , and the distance between $(1, - 7)$ and $(1, 4)$ .
Answer
Same $y$ -coordinate, so horizontal: $2 - (- 6) = 8$ units. Same $x$ -coordinate, so vertical: $4 - (- 7) = 11$ units.
Three vertices of a square are $(2, 1)$ , $(5, 1)$ and $(5, 4)$ . Find the fourth vertex.
Answer
$(2, 4)$ . It takes its $x$ -coordinate from $(2, 1)$ and its $y$ -coordinate from $(5, 4)$ . Each side is 3 units long.
Explain why $(3, 5)$ and $(5, 3)$ are not the same point.
Answer
The first number is the $x$ -coordinate and the second is the $y$ -coordinate. $(3, 5)$ is 3 across and 5 up; $(5, 3)$ is 5 across and 3 up.
A graph shows the height of a plant in cm (vertical axis) against time in weeks (horizontal axis). Describe what the point $(6, 14)$ tells you.
Answer
After 6 weeks, the plant is 14 cm tall.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. Points $A (- 4, 3)$ , $B (2, 3)$ and $C (2, - 5)$ are plotted on a coordinate grid. One unit on each axis represents 1 cm.

(a) Write down the coordinates of the point where the line $B C$ crosses the $x$ -axis. 1 mark No calculator

Mark scheme

B1 (2, 0) cao

(b) Show that angle $A B C$ is $90 °$ . 2 marks No calculator

Mark scheme

B1 A and B have the same y-coordinate (3), so AB is horizontal oe
B1 B and C have the same x-coordinate (2), so BC is vertical, and a horizontal and a vertical line are perpendicular oe

(c) Calculate the length of $A B$ and the length of $B C$ . 2 marks No calculator

Mark scheme

B1 AB = 6 [cm]
B1 BC = 8 [cm]

(d) $A B C D$ is a rectangle. Write down the coordinates of $D$ . 1 mark No calculator

Mark scheme

B1 (-4, -5) cao

(e) Calculate the area of rectangle $A B C D$ . 2 marks No calculator

Mark scheme

M1 6 × 8, FT their lengths from (c)
A1 48 [cm²]

Question 2. A graph shows the temperature, $T$ °C, inside a freezer $t$ minutes after it is switched on. Time is on the horizontal axis and temperature is on the vertical axis, with $T = 0$ along the horizontal axis. On the time axis, 10 small squares represent 5 minutes. On the temperature axis, 10 small squares represent 4 °C. Point $P$ on the graph is 18 small squares to the right of the vertical axis and 7 small squares below the horizontal axis.

(a) Work out the coordinates of $P$ . 2 marks No calculator

Mark scheme

B1 t-coordinate 9
B1 T-coordinate -2.8, negative sign required

(b) Describe what the coordinates of $P$ tell you about the freezer. 1 mark No calculator

Mark scheme

B1 9 minutes after it is switched on, the temperature inside the freezer is -2.8 °C oe; FT their (a)

(c) The point $Q (15, - 6)$ is also on the graph. Work out how many small squares to the right of the vertical axis, and how many small squares below the horizontal axis, $Q$ is plotted. 2 marks No calculator

Mark scheme

B1 30 [small squares to the right]
B1 15 [small squares below]

Question 3. $P Q R S$ is a parallelogram with its vertices in that order. $P$ is $(- 2, 1)$ , $Q$ is $(3, 2)$ and $R$ is $(5, - 2)$ .

(a) Find the coordinates of $S$ . 2 marks No calculator

Mark scheme

M1 the move from Q to R (2 right, 4 down) applied to P, or the move from Q to P (5 left, 1 down) applied to R oe
A1 (0, -3) cao

(b) Explain how you can tell from its coordinates that $S$ lies on the $y$ -axis. 1 mark No calculator

Mark scheme

B1 the x-coordinate of S is 0, so S is 0 units across from the origin oe

How this is examined

Side lengths and area of a shape whose sides are horizontal and vertical: A Calculate part after the vertices are given as coordinates: the lengths of two sides, then the area, about 2 marks each Each length as the difference of the coordinates that change, with the sign kept: $2 - (- 4) = 6$ , not $2 - 4$ . For the area, the product of your two lengths ( $6 \times 8$ ) is the method; the value with the right unit (48 cm² when one unit is 1 cm) is the answer.
Reading or plotting a point when one small square is not one unit: A Work out or Write down part on axes with a stated scale, e.g. 10 small squares to 4 units, worth 1 mark per coordinate Each coordinate correct after working out one square's value on that axis ( $4 \div 10 = 0.4$ ), with the sign right for a point below or left of an axis: $(9, - 2.8)$ , not $(18, 7)$ .
Finding the missing vertex of a rectangle or parallelogram: A 2-mark Find the coordinates of the fourth vertex, with three vertices given and the letter order stated The move along one side written as so many across and so many up or down (e.g. '2 right, 4 down') and applied from the correct vertex shows the method; the final coordinates in the form $(x, y)$ are the answer.
Showing that an angle is a right angle from the coordinates: A 2-mark Show (that) about one angle of a triangle or quadrilateral, with its vertices given Two statements, one for each side: 'A and B have the same $y$ -coordinate, so AB is horizontal' and 'B and C have the same $x$ -coordinate, so BC is vertical', then that a horizontal and a vertical line meet at $90 °$ . Naming the equal coordinate is the evidence.
Writing down coordinates the right way round, including on an axis: A 1-mark Write down the coordinates of a marked point or of the point where a line crosses an axis $x$ first, then $y$ , in brackets with a comma, with every sign correct. A point on an axis keeps its zero: $(2, 0)$ , not just $2$ .
Saying what a point means on a real-life graph: A 1-mark Describe or Explain what a given point tells you about the situation A sentence that gives both values with their units in the context: 'After 9 minutes the temperature is $- 2.8$ °C.' Repeating $(9, - 2.8)$ does not interpret it.

Words to know

Cartesian coordinates: A pair of numbers $(x, y)$ that fixes the position of a point on a grid made of two perpendicular number lines.
Origin: The point $(0, 0)$ , where the $x$ -axis and the $y$ -axis cross.
Ordered pair: Two numbers written in a fixed order. In $(x, y)$ the first is the $x$ -coordinate (across) and the second is the $y$ -coordinate (up or down).
Quadrant: One of the four regions the axes divide the grid into, numbered anticlockwise from the top right.
Scale: How much one small square represents on an axis. The two axes can have different scales.
Vertex: A corner of a shape (plural vertices).

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