Cartesian Coordinates

Fix the position of a point on a flat surface with an ordered pair: read and plot points in all four quadrants, find lengths and areas on a grid, and move points.

Before you start

On a number line, positive numbers lie to the right of $0$ and negative numbers to the left: $-3$ is $3$ units to the left of $0$ .
The distance between two numbers on a number line is the larger number minus the smaller one. From $-2$ to $5$ is $5 - (-2) = 5 + 2 = 7$ units.
Area of a rectangle $=$ length $\times$ breadth. Area of a triangle $= \frac{1}{2} \times$ base $\times$ height.

The Cartesian Plane

A number line fixes a position along a line. To fix a position on a flat surface, such as a page or a map, we use two number lines that cross at right angles at their zeros. These two number lines are called the axes. This is the Cartesian coordinate system, named after the French mathematician René Descartes, and the flat surface with the two axes is called the Cartesian plane.

The Cartesian plane

The $x$ -axis is horizontal; its numbers increase from left to right. The $y$ -axis is vertical; its numbers increase from bottom to top. The axes meet at the origin $O$ , the point $(0, 0)$ .

Every point on the Cartesian plane is given by an ordered pair $(x, y)$ . The $x$ -coordinate says how far the point is to the right (positive) or to the left (negative) of the $y$ -axis. The $y$ -coordinate says how far it is above (positive) or below (negative) the $x$ -axis.

The $x$ -coordinate is always written first.

The two axes divide the plane into four parts called quadrants. They are numbered anticlockwise, starting from the top right: the first, second, third and fourth quadrants. The signs of a point's two coordinates tell you which quadrant it lies in.

Quadrant	Where it is	$x$ -coordinate	$y$ -coordinate	Example
First	top right	positive	positive	$A(3, 2)$
Second	top left	negative	positive	$B(-4, 3)$
Third	bottom left	negative	negative	$C(-2, -4)$
Fourth	bottom right	positive	negative	$D(4, -1)$

Points on the axes

A point on the $x$ -axis has $y$ -coordinate $0$ , such as $(4, 0)$ or $(-2, 0)$ . A point on the $y$ -axis has $x$ -coordinate $0$ , such as $(0, 3)$ or $(0, -5)$ .

A point on an axis does not lie in any quadrant. The origin $(0, 0)$ lies on both axes.

Example 1: Which quadrant, or which axis?

State the quadrant in which each point lies, or the axis on which it lies.
(a) $A(4, 7)$ (b) $B(-3, 5)$ (c) $C(-2, -6)$ (d) $D(5, -3)$ (e) $E(0, -8)$ (f) $F(6, 0)$

Solution

(a) $A(4, 7)$ : the $x$ -coordinate is positive and the $y$ -coordinate is positive, so $A$ is in the first quadrant.

(b) $B(-3, 5)$ : $x$ -coordinate negative, $y$ -coordinate positive, so $B$ is in the second quadrant.

(d) $D(5, -3)$ : $x$ -coordinate positive, $y$ -coordinate negative, so $D$ is in the fourth quadrant.

(e) $E(0, -8)$ : the $x$ -coordinate is $0$ , so $E$ is neither to the left nor to the right of the $y$ -axis. It lies on the $y$ -axis, $8$ units below the origin.

(f) $F(6, 0)$ : the $y$ -coordinate is $0$ , so $F$ lies on the $x$ -axis, $6$ units to the right of the origin.

$A$ is in the first quadrant, $B$ in the second, $C$ in the third and $D$ in the fourth. $E$ lies on the $y$ -axis and $F$ lies on the $x$ -axis.

Ordered Pairs and Plotting Points

The pair $(x, y)$ is called an ordered pair because the order of its two numbers matters. The first number is always the $x$ -coordinate and the second is always the $y$ -coordinate. One way to remember it: go along the corridor first, then up the stairs.

Example 2: The order matters

Plot the points $P(2, 5)$ and $Q(5, 2)$ on the same grid. Are they the same point?

Solution

$P(2, 5)$ : from $O$ , move $2$ units to the right, then $5$ units up.

$Q(5, 2)$ : from $O$ , move $5$ units to the right, then $2$ units up.

The dashed lines from each point to the two axes show its two coordinates.

$P$ and $Q$ are different points: $(2, 5)$ and $(5, 2)$ use the same two numbers in a different order.

To plot a point $(x, y)$

Step 1: Start at the origin $O$ .

Step 2: Move along the $x$ -axis by the number in $x$ , without its sign: to the right if $x$ is positive, to the left if it is negative. For $(-3, 2)$ , move $3$ units to the left.

Step 3: From there, move parallel to the $y$ -axis by the number in $y$ , without its sign: up if $y$ is positive, down if it is negative. For $(-3, 2)$ , move $2$ units up.

Step 4: Mark the point with a dot and write its letter beside it.

Example 3: Plotting points in all four quadrants

Plot the points $A(3, 4)$ , $B(-2, 5)$ , $C(-4, -3)$ , $D(2, -1)$ and $E(-3, 0)$ on one grid, and say where each point lies.

Solution

Choose the axes first. The $x$ -coordinates run from $-4$ to $3$ and the $y$ -coordinates from $-3$ to $5$ , so an $x$ -axis from $-5$ to $5$ and a $y$ -axis from $-5$ to $6$ hold every point with room to spare.

$A(3, 4)$ : $3$ units right, then $4$ units up.

$B(-2, 5)$ : $2$ units left, then $5$ units up.

$C(-4, -3)$ : $4$ units left, then $3$ units down.

$D(2, -1)$ : $2$ units right, then $1$ unit down.

$E(-3, 0)$ : $3$ units left, and no move up or down, so $E$ is on the $x$ -axis.

$A$ is in the first quadrant, $B$ in the second, $C$ in the third and $D$ in the fourth; $E$ lies on the $x$ -axis.

Reading the Coordinates of a Point

To read the coordinates of a point, go from the point straight up or down to the $x$ -axis and read the $x$ -coordinate there. Then go from the point straight across to the $y$ -axis and read the $y$ -coordinate. Write them as $(x, y)$ , with the $x$ -coordinate first.

From $R$ , straight down to $4$ on the $x$ -axis and straight across to $3$ on the $y$ -axis: $R$ is the point $(4, 3)$ .

Example 4: Reading coordinates from a grid

Write down the coordinates of the points $A$ to $F$ on the grid below. Which of them lie on an axis?

Solution

$A$ : straight down to $4$ on the $x$ -axis; straight across to $2$ on the $y$ -axis. So $A$ is $(4, 2)$ .

$B$ : straight down to $-3$ ; straight across to $5$ . So $B$ is $(-3, 5)$ .

$C$ : straight up to $-5$ ; straight across to $-2$ . So $C$ is $(-5, -2)$ .

$D$ : straight up to $3$ ; straight across to $-4$ . So $D$ is $(3, -4)$ .

$E$ is on the $y$ -axis, $3$ units below $O$ , so its $x$ -coordinate is $0$ : $E$ is $(0, -3)$ .

$F$ is on the $x$ -axis, $5$ units to the right of $O$ , so its $y$ -coordinate is $0$ : $F$ is $(5, 0)$ .

$A(4, 2)$ , $B(-3, 5)$ , $C(-5, -2)$ , $D(3, -4)$ , $E(0, -3)$ and $F(5, 0)$ . $E$ lies on the $y$ -axis and $F$ lies on the $x$ -axis.

Axes with Other Scales

One square of the grid does not always stand for $1$ unit. Before you read or plot a point, look at the numbers on each axis and work out what one square is worth. The two axes may have different scales.

What one square is worth

One square $=$ the gap between two numbered lines $\div$ the number of squares between them.

If there is a number at every grid line and the numbers go $0, 2, 4, 6, \ldots$ , the gap is $2$ with $1$ square between two numbers, so one square is $2 \div 1 = 2$ units.

If the numbers go $0, 1, 2, 3, \ldots$ but are written only at every second grid line, the gap is $1$ with $2$ squares between two numbers, so one square is $1 \div 2 = 0.5$ unit.

Example 5: Reading points when one square is not 1 unit

Write down the coordinates of the points $P$ , $Q$ , $R$ and $S$ on the grid below.

Solution

Read the scales first. The $x$ -axis has a number at every grid line, going up in $2$ s, so one square across is $2 \div 1 = 2$ units. The $y$ -axis has a number at every grid line, going up in $5$ s, so one square up or down is $5 \div 1 = 5$ units.

$P$ is $3$ squares to the right of the $y$ -axis, so its $x$ -coordinate is $3 \times 2 = 6$ . It is $3$ squares above the $x$ -axis, so its $y$ -coordinate is $3 \times 5 = 15$ . So $P$ is $(6, 15)$ . (Counting squares alone would give $(3, 3)$ , which is wrong.)

$Q$ is $2$ squares left, so $x = -4$ , and $2$ squares up, so $y = 10$ . So $Q$ is $(-4, 10)$ .

$R$ is $4$ squares left, so $x = -8$ , and $4$ squares down, so $y = -20$ . So $R$ is $(-8, -20)$ .

$S$ lies halfway between the grid lines for $2$ and $4$ , so $x = 3$ . It is $2$ squares down, so $y = -10$ . So $S$ is $(3, -10)$ .

$P(6, 15)$ , $Q(-4, 10)$ , $R(-8, -20)$ and $S(3, -10)$ .

Points on a Horizontal or Vertical Line

Points on one line

Points with the same $y$ -coordinate are all the same distance above (or below) the $x$ -axis, so they lie on one horizontal line, parallel to the $x$ -axis. For example, $(-1, 2)$ , $(3, 2)$ and $(5, 2)$ .

Points with the same $x$ -coordinate are all the same distance to the right (or left) of the $y$ -axis, so they lie on one vertical line, parallel to the $y$ -axis. For example, $(4, -3)$ , $(4, 0)$ and $(4, 6)$ .

Example 6: Which points lie on one line?

Here are seven points: $A(-2, 5)$ , $B(2, -3)$ , $C(-2, 1)$ , $D(5, 4)$ , $E(6, -3)$ , $F(-2, 3)$ and $G(4, -3)$ .
(a) Which three points lie on a vertical line?
(b) Which three points lie on a horizontal line?
(c) Which point lies on neither of these lines?

Solution

(a) Look for a repeated $x$ -coordinate. $A$ , $C$ and $F$ all have $x$ -coordinate $-2$ , so they lie on a vertical line.

(b) Look for a repeated $y$ -coordinate. $B$ , $E$ and $G$ all have $y$ -coordinate $-3$ , so they lie on a horizontal line.

(a) $A$ , $C$ and $F$ . (b) $B$ , $E$ and $G$ . (c) $D$ .

Shapes, Lengths and Areas on a Grid

Plot the vertices of a shape and join them in order to see the shape. A side that is horizontal or vertical runs along a grid line, so its length can be counted in squares, or found by subtracting coordinates.

Lengths along grid lines

Two points with the same $y$ -coordinate are joined by a horizontal line segment. Its length $=$ larger $x$ -coordinate $-$ smaller $x$ -coordinate.

Two points with the same $x$ -coordinate are joined by a vertical line segment. Its length $=$ larger $y$ -coordinate $-$ smaller $y$ -coordinate.

For example, from $(-3, 2)$ to $(4, 2)$ the length is $4 - (-3) = 7$ units: $3$ units to reach the $y$ -axis and $4$ units more beyond it.

This works only for horizontal and vertical line segments.

Example 7: The missing vertex of a rectangle

The points $A(1, 2)$ , $B(5, 2)$ and $C(5, 5)$ are three vertices of the rectangle $ABCD$ .
(a) Find the coordinates of $D$ .
(b) Find the length $AB$ and the width $BC$ .
(c) Find the perimeter and the area of the rectangle.

Solution

(a) In rectangle $ABCD$ the vertices go round in order, so $D$ is joined to both $C$ and $A$ . $AB$ is horizontal (both $y$ -coordinates are $2$ ) and $BC$ is vertical (both $x$ -coordinates are $5$ ).

So $DC$ is horizontal: $D$ has the same $y$ -coordinate as $C$ , which is $5$ . And $AD$ is vertical: $D$ has the same $x$ -coordinate as $A$ , which is $1$ . So $D$ is $(1, 5)$ .

(b) $AB$ is horizontal: $AB = 5 - 1 = 4$ units. $BC$ is vertical: $BC = 5 - 2 = 3$ units.

Area $= 4 \times 3 = 12$ square units.

$D$ is $(1, 5)$ . $AB = 4$ units and $BC = 3$ units. The perimeter is $14$ units and the area is $12$ square units.

Example 8: A right-angled triangle

Plot the points $P(-4, 1)$ , $Q(3, 1)$ and $R(3, 5)$ and join them to make triangle $PQR$ .
(a) What kind of triangle is $PQR$ ?
(b) Find the lengths $PQ$ and $QR$ .
(c) Find the area of triangle $PQR$ .

Solution

(a) $P$ and $Q$ have the same $y$ -coordinate, $1$ , so $PQ$ is horizontal. $Q$ and $R$ have the same $x$ -coordinate, $3$ , so $QR$ is vertical. A horizontal line and a vertical line meet at a right angle, so the angle at $Q$ is $90^\circ$ .

(b) $PQ = 3 - (-4) = 3 + 4 = 7$ units. $QR = 5 - 1 = 4$ units.

Area $= \frac{1}{2} \times 7 \times 4 = 14$ square units.

$PQR$ is a right-angled triangle, with the right angle at $Q$ . $PQ = 7$ units and $QR = 4$ units. Its area is $14$ square units.

Example 9: The missing vertex of a parallelogram

The points $K(1, 1)$ , $L(6, 1)$ and $M(8, 4)$ are three vertices of the parallelogram $KLMN$ . Find the coordinates of $N$ .

Solution

In a parallelogram, opposite sides are parallel and equal in length. In $KLMN$ , the side $NM$ is opposite the side $KL$ .

From $K$ to $L$ is $6 - 1 = 5$ units to the right, with no move up or down (both $y$ -coordinates are $1$ ).

So from $N$ to $M$ is also $5$ units to the right. Working back from $M$ , $N$ is $5$ units to the left of $M$ : $N = (8 - 5, 4) = (3, 4)$ .

Check the other pair of sides. From $K(1, 1)$ to $N(3, 4)$ is $2$ units right and $3$ units up. From $L(6, 1)$ to $M(8, 4)$ is also $2$ units right and $3$ units up. $\checkmark$

$N$ is $(3, 4)$ .

Moving a Point

Moving a point

Moving right: add the number of units to the $x$ -coordinate.

Moving left: subtract the number of units from the $x$ -coordinate.

Moving up: add the number of units to the $y$ -coordinate.

Moving down: subtract the number of units from the $y$ -coordinate.

To find where a point started, make the opposite moves from where it ended.

Move from $(2, 3)$	Working	New point
$4$ units right	$(2 + 4, 3)$	$(6, 3)$
$4$ units left	$(2 - 4, 3)$	$(-2, 3)$
$4$ units up	$(2, 3 + 4)$	$(2, 7)$
$4$ units down	$(2, 3 - 4)$	$(2, -1)$

Example 10: Moving a point

The point $A(-2, 7)$ is moved $6$ units to the right and then $10$ units down, to the point $B$ . Find the coordinates of $B$ .

Solution

Moving $6$ units to the right changes only the $x$ -coordinate: $-2 + 6 = 4$ . The point is now at $(4, 7)$ .

Moving $10$ units down changes only the $y$ -coordinate: $7 - 10 = -3$ .

In the drawing, the dashed path shows the two moves.

$B$ is $(4, -3)$ .

Example 11: Working back to the starting point

A point $P$ is moved $5$ units to the left and then $3$ units up. It ends at $Q(-1, 4)$ . Find the coordinates of $P$ .

Solution

Work back from $Q$ by making the opposite moves: $5$ units to the right and $3$ units down.

$x$ -coordinate: $-1 + 5 = 4$ . $y$ -coordinate: $4 - 3 = 1$ . So $P$ is $(4, 1)$ .

Check: from $P(4, 1)$ , moving $5$ units left gives $4 - 5 = -1$ , and moving $3$ units up gives $1 + 3 = 4$ . That is $Q(-1, 4)$ . $\checkmark$

$P$ is $(4, 1)$ .

Coordinates in Everyday Life

Maps, seating plans and board games use the same idea: a position is given by two numbers, one across and one up. On a map, north is usually up the page, so east is to the right.

Example 12: A map of a park

The map shows part of a park. The fountain is at the origin $O$ , each square stands for $50$ m, and north is up the page.
(a) What is at $(-2, -3)$ ?
(b) Write down the coordinates of the playground.
(c) Siti walks in a straight line from the car park to the café. How far does she walk?
(d) Ali starts at the toilets. He walks $4$ units east (to the right) and then $3$ units north (up). Where does he end up?

Solution

(a) From $O$ , go $2$ units left and $3$ units down: this is the café.

(b) The playground is $3$ units to the right of the $y$ -axis and $2$ units above the $x$ -axis, so it is at $(3, 2)$ .

(c) The car park is at $(4, -3)$ and the café is at $(-2, -3)$ . They have the same $y$ -coordinate, so the straight path between them is horizontal. Its length is $4 - (-2) = 6$ units, and $6 \times 50 = 300$ m.

(d) The toilets are at $(-4, 1)$ . $4$ units east: $-4 + 4 = 0$ . $3$ units north: $1 + 3 = 4$ . Ali ends at $(0, 4)$ , which is the pond.

(a) The café. (b) $(3, 2)$ . (c) $300$ m. (d) At the pond, $(0, 4)$ .

Common Mistakes

Writing the $y$ -coordinate first. The point $2$ units right and $5$ units up is $(2, 5)$ , not $(5, 2)$ : along first, then up.
Counting squares when one square is not $1$ unit. Read the numbers on each axis first: if one square is $5$ units, $3$ squares up is $15$ , not $3$ .
Subtracting wrongly across an axis. From $(-3, 2)$ to $(4, 2)$ is $4 - (-3) = 7$ units, not $4 - 3 = 1$ unit.
Mixing up the axes for a point with a $0$ . $(0, -3)$ lies on the $y$ -axis, because it is $0$ units across; $(-3, 0)$ lies on the $x$ -axis.
Changing the wrong coordinate when moving a point. Moving up or down changes only the $y$ -coordinate; moving left or right changes only the $x$ -coordinate.

Practice Questions

Try each question on your own, then open its solution to check your working. Use squared paper for the questions that ask you to plot points.

Question 1

(a) State the coordinates of the origin.
(b) In which quadrant does the point $P(-7, -2)$ lie?
(c) A point $Q$ lies in the fourth quadrant. What can you say about its coordinates?
(d) On which axis does the point $R(0, 6)$ lie?

ShowHide solution

(a) The origin is where the axes meet, $0$ units across and $0$ units up.

(b) Both coordinates of $P$ are negative, so $P$ is in the third quadrant.

(d) The $x$ -coordinate of $R$ is $0$ , so $R$ is on the $y$ -axis, $6$ units above the origin.

(a) $(0, 0)$ (b) The third quadrant. (c) Its $x$ -coordinate is positive and its $y$ -coordinate is negative. (d) The $y$ -axis.

Question 2

Write down the coordinates of the points $A$ to $F$ on the grid below. Which of them lie on an axis, and on which axis?

ShowHide solution

$A$ : down to $2$ on the $x$ -axis, across to $3$ on the $y$ -axis: $(2, 3)$ .

$B$ : down to $-4$ , across to $1$ : $(-4, 1)$ .

$C$ : up to $-2$ , across to $-4$ : $(-2, -4)$ .

$D$ : up to $5$ , across to $-2$ : $(5, -2)$ .

$E$ is on the $x$ -axis, $3$ units left of $O$ : $(-3, 0)$ .

$F$ is on the $y$ -axis, $4$ units above $O$ : $(0, 4)$ .

$A(2, 3)$ , $B(-4, 1)$ , $C(-2, -4)$ , $D(5, -2)$ , $E(-3, 0)$ , $F(0, 4)$ . $E$ lies on the $x$ -axis and $F$ lies on the $y$ -axis.

Question 3

Draw a pair of axes and plot the points $A(-4, 2)$ , $B(3, -5)$ , $C(0, 4)$ , $D(-2, -3)$ and $E(5, 0)$ . Choose the range of each axis yourself. State the quadrant or the axis on which each point lies.

ShowHide solution

The $x$ -coordinates run from $-4$ to $5$ and the $y$ -coordinates from $-5$ to $4$ . An $x$ -axis from $-5$ to $6$ and a $y$ -axis from $-6$ to $5$ hold them all.

$A(-4, 2)$ : $4$ left, $2$ up; second quadrant.

$B(3, -5)$ : $3$ right, $5$ down; fourth quadrant.

$C(0, 4)$ : no move across, $4$ up; on the $y$ -axis.

$D(-2, -3)$ : $2$ left, $3$ down; third quadrant.

$E(5, 0)$ : $5$ right, no move up or down; on the $x$ -axis.

$A$ : second quadrant; $B$ : fourth quadrant; $C$ : on the $y$ -axis; $D$ : third quadrant; $E$ : on the $x$ -axis.

Question 4

On the grid below, the axes are numbered at every second grid line. Write down the coordinates of $A$ , $B$ , $C$ and $D$ .

ShowHide solution

On each axis the numbered lines are $1$ unit apart, with $2$ squares between them, so one square is $1 \div 2 = 0.5$ unit.

$A$ : $3$ squares right, $3 \times 0.5 = 1.5$ ; $4$ squares up, $4 \times 0.5 = 2$ . So $A$ is $(1.5, 2)$ .

$B$ : $5$ squares left, $-2.5$ ; $2$ squares up, $1$ . So $B$ is $(-2.5, 1)$ .

$C$ : $2$ squares left, $-1$ ; $3$ squares down, $-1.5$ . So $C$ is $(-1, -1.5)$ .

$D$ : $4$ squares right, $2$ ; $5$ squares down, $-2.5$ . So $D$ is $(2, -2.5)$ .

$A(1.5, 2)$ , $B(-2.5, 1)$ , $C(-1, -1.5)$ , $D(2, -2.5)$ .

Question 5

(a) The points $(-2, 5)$ , $(-2, -1)$ and $(k, 3)$ lie on one vertical line. Find $k$ .
(b) Write down the coordinates of two other points on the horizontal line through $(4, -3)$ .
(c) Which two of the points $(1, 6)$ , $(6, 1)$ , $(-3, 6)$ and $(6, -3)$ lie on a horizontal line?

ShowHide solution

(a) All points on one vertical line have the same $x$ -coordinate. $(-2, 5)$ and $(-2, -1)$ have $x$ -coordinate $-2$ , so $k = -2$ .

(b) Every point on the horizontal line through $(4, -3)$ has $y$ -coordinate $-3$ . Any $x$ -coordinate will do, for example $(0, -3)$ and $(-5, -3)$ .

(c) $(1, 6)$ and $(-3, 6)$ both have $y$ -coordinate $6$ , so they lie on a horizontal line. ( $(6, 1)$ and $(6, -3)$ share their $x$ -coordinate, so they lie on a vertical line instead.)

(a) $k = -2$ (b) For example, $(0, -3)$ and $(-5, -3)$ . (c) $(1, 6)$ and $(-3, 6)$ .

Question 6

The points $A(-3, -1)$ , $B(3, -1)$ and $C(3, 5)$ are three vertices of the square $ABCD$ .
(a) Find the coordinates of $D$ .
(b) Find the length of a side of the square.
(c) Find the perimeter and the area of the square.

ShowHide solution

(a) $DC$ is horizontal, so $D$ has the same $y$ -coordinate as $C$ : $5$ . $AD$ is vertical, so $D$ has the same $x$ -coordinate as $A$ : $-3$ . So $D$ is $(-3, 5)$ .

(b) $AB = 3 - (-3) = 6$ units. Check: $BC = 5 - (-1) = 6$ units, as a square needs.

(a) $D(-3, 5)$ (b) $6$ units (c) Perimeter $24$ units, area $36$ square units.

Question 7

Plot the points $X(-4, -5)$ , $Y(1, -5)$ and $Z(1, -2)$ and join them to make triangle $XYZ$ .
(a) Name the kind of triangle and say where its right angle is.
(b) Find the area of triangle $XYZ$ .

ShowHide solution

(a) $X$ and $Y$ have the same $y$ -coordinate, so $XY$ is horizontal. $Y$ and $Z$ have the same $x$ -coordinate, so $YZ$ is vertical. They meet at a right angle at $Y$ .

(b) $XY = 1 - (-4) = 5$ units and $YZ = -2 - (-5) = -2 + 5 = 3$ units.

Area $= \frac{1}{2} \times 5 \times 3 = 7.5$ square units.

(a) A right-angled triangle, with the right angle at $Y$ . (b) $7.5$ square units.

Question 8

The points $P(-2, -1)$ , $Q(3, -1)$ and $R(5, 3)$ are three vertices of the parallelogram $PQRS$ . Find the coordinates of $S$ .

ShowHide solution

In the parallelogram $PQRS$ , the side $SR$ is opposite the side $PQ$ , so it is parallel to $PQ$ and equal in length.

From $P$ to $Q$ is $3 - (-2) = 5$ units to the right, with no move up or down (both $y$ -coordinates are $-1$ ).

So from $S$ to $R$ is also $5$ units to the right. Working back from $R$ , $S$ is $5$ units to the left of $R$ : $S = (5 - 5, 3) = (0, 3)$ .

Check the other pair of sides. From $P(-2, -1)$ to $S(0, 3)$ is $2$ units right and $4$ units up. From $Q(3, -1)$ to $R(5, 3)$ is also $2$ units right and $4$ units up. $\checkmark$

$S$ is $(0, 3)$ , on the $y$ -axis.

Question 9

(a) The point $A(-3, 2)$ is moved $7$ units to the right and then $5$ units down. Find its new position.
(b) A point $B$ is moved $4$ units to the right and then $6$ units down. It ends at $(1, -2)$ . Find the coordinates of $B$ .

ShowHide solution

(a) $7$ units right: $(-3 + 7, 2) = (4, 2)$ . Then $5$ units down: $(4, 2 - 5) = (4, -3)$ .

(b) Work back with the opposite moves: $4$ units left and $6$ units up. $x$ -coordinate: $1 - 4 = -3$ . $y$ -coordinate: $-2 + 6 = 4$ . So $B$ is $(-3, 4)$ .

Check: $-3 + 4 = 1$ and $4 - 6 = -2$ , giving $(1, -2)$ . $\checkmark$

(a) $(4, -3)$ (b) $B(-3, 4)$

Question 10 (Setting: Getting Around the Neighbourhood)

The map shows Wei Ming's neighbourhood. The MRT station is at the origin $O$ , each square stands for $200$ m, and north is up the page. Streets run along the grid lines.
(a) What is at $(-4, -2)$ ?
(b) Write down the coordinates of the school.
(c) Wei Ming walks from his home to the MRT station, first due east along one street and then due south along another. How far does he walk?
(d) After school, he walks $1000$ m due south and then $600$ m due west to the library. Find the coordinates of the library.

ShowHide solution

(a) $4$ units left and $2$ units down from $O$ is the hawker centre.

(b) The school is $4$ units right and $3$ units up: $(4, 3)$ .

(c) Home is at $(-3, 2)$ . Due east to the $y$ -axis is $0 - (-3) = 3$ units; then due south to $O$ is $2 - 0 = 2$ units. Total $= 3 + 2 = 5$ units $= 5 \times 200 = 1000$ m.

(d) $1000$ m south is $1000 \div 200 = 5$ units down: $3 - 5 = -2$ . $600$ m west is $600 \div 200 = 3$ units left: $4 - 3 = 1$ .

(a) The hawker centre. (b) $(4, 3)$ (c) $1000$ m, or $1$ km. (d) The library is at $(1, -2)$ .

Question 11 (Setting: A Classroom Seating Plan)

Mr Tan writes each seat in his classroom as (column, row). Columns are counted from the left and rows from the front, so row $1$ is nearest the board. On the plan, row $1$ is at the bottom. Priya sits at $(4, 2)$ and Ali sits at $(2, 4)$ .
(a) Ali says they sit in the same seat, because both seats use the numbers $2$ and $4$ . Explain why he is wrong.
(b) Siti sits $3$ seats to the right of Ali, in the same row. Write down her seat.
(c) Wei Ming sits at $(5, 1)$ . How many rows in front of Siti is he?
(d) Mei Ling sits $1$ column to the left of Priya and $2$ rows behind her. Write down Mei Ling's seat.

ShowHide solution

(a) The first number is the column and the second is the row. Priya is in column $4$ , row $2$ ; Ali is in column $2$ , row $4$ . These are different seats.

(b) Moving right along a row changes only the column: $2 + 3 = 5$ . Siti sits at $(5, 4)$ .

(d) $1$ column to the left: $4 - 1 = 3$ . $2$ rows behind, further from the board: $2 + 2 = 4$ . Mei Ling sits at $(3, 4)$ .

(a) $(4, 2)$ is column $4$ , row $2$ , but $(2, 4)$ is column $2$ , row $4$ : the order matters. (b) $(5, 4)$ (c) $3$ rows (d) $(3, 4)$

Question 12 (Setting: Sports Day)

For Sports Day, Mdm Lim marks out a rectangle $ABCD$ on the school field. The grid below is in metres, with the flagpole at the origin $O$ .
(a) Write down the coordinates of $A$ , $B$ , $C$ and $D$ .
(b) Find the length $AB$ and the width $BC$ of the rectangle.
(c) She puts a rope all the way round the rectangle. How long is the rope?
(d) Find the area of the rectangle.

ShowHide solution

(a) On each axis the numbered lines are $20$ m apart, with $2$ squares between them, so one square is $20 \div 2 = 10$ m. $A$ is $3$ squares left and $1$ square down: $(-30, -10)$ . In the same way, $B(50, -10)$ , $C(50, 30)$ and $D(-30, 30)$ .

(b) $AB$ is horizontal: $AB = 50 - (-30) = 80$ m. $BC$ is vertical: $BC = 30 - (-10) = 40$ m.

(d) Area $= 80 \times 40 = 3200 \text{ m}^2$ .

(a) $A(-30, -10)$ , $B(50, -10)$ , $C(50, 30)$ , $D(-30, 30)$ (b) $AB = 80$ m, $BC = 40$ m (c) $240$ m (d) $3200 \text{ m}^2$

Key Points

The $x$ -axis is horizontal and the $y$ -axis is vertical; they meet at the origin $O(0, 0)$ .
A point is given by an ordered pair $(x, y)$ , the $x$ -coordinate first; $(2, 5)$ and $(5, 2)$ are different points.
To plot $(x, y)$ : from $O$ , move across (right if $x$ is positive, left if it is negative), then up or down (up if $y$ is positive, down if it is negative), by the sizes of $x$ and $y$ without their signs. For $(-3, 2)$ : $3$ units left, then $2$ units up.
First quadrant $(+, +)$ , second $(-, +)$ , third $(-, -)$ , fourth $(+, -)$ . A point on an axis lies in no quadrant.
A point on the $x$ -axis has $y$ -coordinate $0$ ; a point on the $y$ -axis has $x$ -coordinate $0$ .
Read the scale of each axis before you read or plot a point: one square $=$ the gap between two numbered lines $\div$ the number of squares between them.
Points with the same $y$ -coordinate lie on a horizontal line; points with the same $x$ -coordinate lie on a vertical line.
A horizontal length is the difference of the $x$ -coordinates; a vertical length is the difference of the $y$ -coordinates.
Moving right or left changes only the $x$ -coordinate; moving up or down changes only the $y$ -coordinate. To work back, make the opposite moves.

Before you start

The Cartesian Plane

The Cartesian plane

Points on the axes

Example 1: Which quadrant, or which axis?

Ordered Pairs and Plotting Points

Example 2: The order matters

To plot a point (x,y)(x, y)(x,y)

Example 3: Plotting points in all four quadrants

Reading the Coordinates of a Point

Example 4: Reading coordinates from a grid

Axes with Other Scales

What one square is worth

Example 5: Reading points when one square is not 1 unit

Points on a Horizontal or Vertical Line

Points on one line

Example 6: Which points lie on one line?

Shapes, Lengths and Areas on a Grid

Lengths along grid lines

Example 7: The missing vertex of a rectangle

Example 8: A right-angled triangle

Example 9: The missing vertex of a parallelogram

Moving a Point

Moving a point

Example 10: Moving a point

Example 11: Working back to the starting point

Coordinates in Everyday Life

Example 12: A map of a park

Common Mistakes

Practice Questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10 (Setting: Getting Around the Neighbourhood)

Question 11 (Setting: A Classroom Seating Plan)

Question 12 (Setting: Sports Day)

Key Points

To plot a point $(x, y)$