Cambridge IGCSE Maths (0580) · Revision notes · Topic 3 of 9

Cambridge IGCSE Maths 0580 Coordinate Geometry Notes

Revision notes for topic 3, Coordinate geometry, of the Cambridge IGCSE Mathematics (0580) syllabus for exams from 2025 to 2030 (Cambridge says the 2028–2030 syllabus has no significant changes that affect teaching). Each of the 7 sub-topics lists what the syllabus asks, marked Core or Extended, with a short explanation and a worked example. Cover the answer and try each example first.

7 sub-topics
5 Core and Extended
2 Extended only
1 with notes written for 0580

Core Core students (Papers 1 and 3) learn the Core statements. Extended Extended students (Papers 2 and 4) learn everything: the Core statements, the Extended additions and the Extended-only sub-topics. Papers 1 and 2 are non-calculator.

3.1 Coordinates

Core C3.1 Extended E3.1

Use and interpret Cartesian coordinates in two dimensions

Same mathematics as Edexcel 4MA1.

0580 practice questions

8 questions written for 0580 in the Coordinate geometry practice: 8 Core, 5 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Write down the coordinates of

(a) the point 4 units to the right of the origin and 2 units down

(b) the point on the $y$ -axis 3 units above the origin.

Show the answer and mark scheme

B1 (a) (4, −2)
B1 (b) (0, 3)

Answer: (a) $(4, - 2)$ (b) $(0, 3)$

Practice question (Core, non-calculator, 4 marks):

$(1, 2)$ and $(4, 6)$ are two adjacent vertices of a square. The other two vertices are to the left of the line through these two points.

Find the coordinates of the other two vertices.

Show the answer and mark scheme

M1 from (1, 2) to (4, 6) is 3 across and 4 up
M1 turns through 90°: 4 left and 3 up
A1 (−3, 5)
A1 (0, 9)

Answer: $(- 3, 5)$ and $(0, 9)$

0580 questions: Coordinates and Linear Graphs questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Linear Graphs
Practise (4MA1 questions, same mathematics): Straight Line Graphs questions

3.2 Drawing linear graphs

Core C3.2 Extended E3.2

Draw straight-line graphs from their equations

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

A linear graph is y = mx + c where m is the gradient (rise/run) and c is the y-intercept.

Example: Find the gradient of the line through (2, −3) and (5, 6).

Answer: m = (6 − (−3)) ÷ (5 − 2) = 9/3 = 3.

0580 practice questions

8 questions written for 0580 in the Coordinate geometry practice: 8 Core, 5 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Complete the table of values for $y = 2 x + 1$ .

$\begin{array}{cccccc} x & - 1 & 0 & 1 & 2 & 3 \\ y \end{array}$

Show the answer and mark scheme

B2 −1, 1, 3, 5, 7 (B1 for 3 or 4 correct)

Answer: $- 1, 1, 3, 5, 7$

Practice question (Core, non-calculator, 2 marks):

Complete the table of values for $y = 5 - x$ .

$\begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ y \end{array}$

Show the answer and mark scheme

B2 5, 4, 3, 2, 1 (B1 for 3 or 4 correct)

Answer: $5, 4, 3, 2, 1$

3.3 Gradient of linear graphs

Core C3.3 Extended E3.3

Find the gradient of a straight line from its graph

Extended also: Calculate the gradient from the coordinates of two points.

Same mathematics as Edexcel 4MA1.

0580 practice questions

8 questions written for 0580 in the Coordinate geometry practice: 7 Core and 1 Extended, 5 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

The graph of a straight line is drawn on a grid. The line passes through the grid points $(0, 1)$ and $(2, 7)$ . Use these points to find the gradient of the line.

Show the answer and mark scheme

M1 (7 − 1) ÷ (2 − 0)
A1 3

Answer: $3$

Practice question (Extended, non-calculator, 3 marks):

The points $A (2 k, 5)$ and $B (k + 4, - 1)$ are such that the gradient of $A B$ is 3.

Find the value of $k$ .

Show the answer and mark scheme

M1 (−1 − 5)/((k + 4) − 2k) = 3
M1 −6 = 3(4 − k)
A1 k = 6

Answer: $k = 6$

0580 questions: Gradient of a Straight Line questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Linear Graphs
Practise (4MA1 questions, same mathematics): Straight Line Graphs questions

3.4 Length and midpoint

Extended only E3.4

Extended: The length of a line segment and the coordinates of its midpoint from its end points.

Partly in Edexcel 4MA1: 4MA1 practises plotting points and Pythagoras separately; no lesson on the length and midpoint of a segment from coordinates.

Notes for 0580

For A(x₁, y₁) and B(x₂, y₂), the midpoint of AB is ((x₁ + x₂)/2, (y₁ + y₂)/2): the mean of the x-coordinates and the mean of the y-coordinates.

The length AB is √((x₂ − x₁)² + (y₂ − y₁)²): Pythagoras' theorem on the horizontal and vertical steps between the points. Take care with negative coordinates: the step from −2 to 4 is 6.

Example (Extended, non-calculator): A is (−2, 3) and B is (4, −5). Find the coordinates of the midpoint of AB and the length of AB.

Midpoint: ((−2 + 4)/2, (3 + (−5))/2) = (1, −1)
Steps: 4 − (−2) = 6 across and −5 − 3 = −8 up
AB = √(6² + (−8)²) = √(36 + 64) = √100 = 10

Answer: Midpoint (1, −1); AB = 10 units

Example (Extended, non-calculator): M(3, 1) is the midpoint of PQ, and P is (−1, 4). Find the coordinates of Q.

From P to M: x goes up by 3 − (−1) = 4 and y changes by 1 − 4 = −3.
Q is the same step on from M: (3 + 4, 1 − 3)

Answer: Q is (7, −2)

0580 practice questions

15 questions written for 0580 in the Coordinate geometry practice: 15 Extended, 12 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Extended, non-calculator, 2 marks):

$M$ is halfway along the line segment from $(- 4, 9)$ to $(6, - 3)$ . What are the coordinates of $M$ ?

Show the answer and mark scheme

M1 (−4 + 6)/2 or (9 − 3)/2
A1 (1, 3)

Answer: $(1, 3)$

Practice question (Extended, non-calculator, 4 marks):

$A$ is the point $(1, k)$ and $B$ is the point $(7, 2 k + 1)$ . The midpoint of $A B$ has $y$ -coordinate 8.

(a) Find the value of $k$ .

(b) Find the length of $A B$ . Give your answer as a surd in its simplest form.

Show the answer and mark scheme

M1 (a) (k + 2k + 1)/2 = 8
A1 (a) k = 5
M1 (b) A(1, 5), B(7, 11): 6² + 6² = 72
A1 (b) 6√2

Answer: (a) $k = 5$ (b) $6 \sqrt{2}$

0580 questions: Length and Midpoint of a Line Segment questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Pythagoras' Theorem
Practise the shared part (4MA1 questions): Straight Line Graphs questions · Pythagoras' Theorem questions

3.5 Equations of linear graphs

Core C3.5 Extended E3.5

Interpret and find the equation of a straight line in the forms y = mx + c and x = k

Extended also: Use lines in other forms, e.g. ax + by = c, and find the equation of a line through two points.

Same mathematics as Edexcel 4MA1.

0580 practice questions

15 questions written for 0580 in the Coordinate geometry practice: 11 Core and 4 Extended, 10 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

The line $y = 5 - 2 x$ is drawn on a grid.

(a) What is its gradient?

(b) At what value does it cross the $y$ -axis?

Show the answer and mark scheme

B1 (a) −2
B1 (b) 5

Answer: (a) $- 2$ (b) $5$

Practice question (Extended, non-calculator, 3 marks):

On a map, a straight road passes through the points $(- 3, 4)$ and $(6, - 2)$ . Find the equation of the road, written as $a x + b y = c$ where $a$ , $b$ and $c$ are whole numbers with no common factor.

Show the answer and mark scheme

M1 gradient −6/9 = −2/3
M1 y − 4 = −2/3 (x + 3) or 4 = −2/3 × (−3) + c
A1 2x + 3y = 6 oe

Answer: $2 x + 3 y = 6$

0580 questions: Equation of a Straight Line questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Linear Graphs
Practise (4MA1 questions, same mathematics): Straight Line Graphs questions

3.6 Parallel lines

Core C3.6 Extended E3.6

Find the gradient and equation of a line parallel to a given line

Same mathematics as Edexcel 4MA1.

0580 practice questions

8 questions written for 0580 in the Coordinate geometry practice: 8 Core, 5 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

(a) Which two of these lines are parallel? $y = 3 x + 1, y = 2 x + 3, y = 3 x - 4, y = - 3 x + 1$

(b) Write down the gradient of any line parallel to $y = 5 - 2 x$ .

Show the answer and mark scheme

B1 (a) y = 3x + 1 and y = 3x − 4
B1 (b) −2

Answer: (a) $y = 3 x + 1$ and $y = 3 x - 4$ (b) $- 2$

Practice question (Core, non-calculator, 4 marks):

Here are the equations of four lines.

A: $y = 3 x - 2$   B: $2 y = 6 x + 7$   C: $3 y = x + 4$   D: $y + 3 x = 1$

(a) Which two lines are parallel?

(b) Find the equation of the line parallel to line C that passes through $(6, 0)$ .

Show the answer and mark scheme

M1 gradients 3, 3, 1/3 and −3
A1 (a) A and B
M1 (b) gradient 1/3 and 0 = (1/3) × 6 + c
A1 y = (1/3)x − 2

Answer: (a) A and B (b) $y = \frac{1}{3} x - 2$

0580 questions: Parallel and Perpendicular Lines questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Linear Graphs
Practise (4MA1 questions, same mathematics): Straight Line Graphs questions

3.7 Perpendicular lines

Extended only E3.7

Extended: Find the gradient and equation of a line perpendicular to a given line, including perpendicular bisectors.

Same mathematics as Edexcel 4MA1.

0580 practice questions

15 questions written for 0580 in the Coordinate geometry practice: 15 Extended, all non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Extended, non-calculator, 1 mark):

Line $A$ has equation $y = - \frac{2}{5} x + 1$ . Line $B$ crosses line $A$ at right angles. What is the gradient of line $B$ ?

Show the answer and mark scheme

B1 5/2 oe

Answer: $\frac{5}{2}$

Practice question (Extended, non-calculator, 3 marks):

$L_{1}$ is the line $3 x - 4 y = 12$ . A second line, $L_{2}$ , meets $L_{1}$ at right angles and goes through the point $(6, 0)$ . Where does $L_{2}$ cut the $y$ -axis?

Show the answer and mark scheme

M1 gradient of L₁ is 3/4
M1 L₂: y = −4/3 (x − 6)
A1 (0, 8)

Answer: $(0, 8)$

Examiner Insights: common mistakes in IGCSE Maths 0580, with fixes (our analysis of public sources) →

Practise coordinate geometry

77 practice questions are written for 0580 on every sub-topic of this topic. For the 6 sub-topics that are the same mathematics in Edexcel 4MA1, the practice also has our 4MA1 Higher questions, which are Extended level. Core students: use the Core filter on the practice page; every sub-topic with Core content has questions written for 0580 Core. Practice is part of the IGCSE plan, with a free preview; these notes are free.

Coordinate geometry practice for 0580: the 77 questions written for 0580, tagged Core or Extended and non-calculator or calculator, with the Edexcel 4MA1 Higher questions for the sub-topics that are the same mathematics
Student hub: the Edexcel 4MA1 units (Unit 4, Unit 5), each question marked
Cambridge IGCSE Maths 0580 past papers: the 2025 specimen papers and past papers, with a timer
Formulas for coordinate geometry on the one-page 0580 formula sheet (free PDF), showing which are on the paper's formula list
Cambridge 0580 coordinate geometry questions by topic, Core and Extended, with worked answers

More on this topic

Knowledge organiser: Coordinate geometry on one page (printable)

Knowledge organiser: Coordinate geometry on one page (key facts, formulas and mistakes to avoid; printable PDF)

Questions about coordinate geometry

What is in Cambridge IGCSE 0580 coordinate geometry?

Topic 3 has 7 sub-topics: 3.1 Coordinates; 3.2 Drawing linear graphs; 3.3 Gradient of linear graphs; 3.4 Length and midpoint; 3.5 Equations of linear graphs; 3.6 Parallel lines; 3.7 Perpendicular lines.

Which coordinate geometry sub-topics are Extended only?

3.4 Length and midpoint; 3.7 Perpendicular lines. Core students (Papers 1 and 3) do not need them.

Are these notes free?

Yes. Every note and worked example on these pages is free to read, with no sign-in.

Join free: practise coordinate geometry with marking →

Cambridge IGCSE Maths 0580 Coordinate Geometry Notes

3.1 Coordinates

0580 practice questions

3.2 Drawing linear graphs

Revise it

0580 practice questions

3.3 Gradient of linear graphs

0580 practice questions

3.4 Length and midpoint

Notes for 0580

0580 practice questions

3.5 Equations of linear graphs

0580 practice questions

3.6 Parallel lines

0580 practice questions

3.7 Perpendicular lines

0580 practice questions

Practise coordinate geometry

Try it now

Questions about coordinate geometry