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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 in 2025, 2026 and 2027. 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.

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

6 questions written for 0580 in the Coordinate geometry practice: 6 Core, 3 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, 2 marks):

(a) Write down the coordinates of the origin.

(b) What is the \(x\)-coordinate of every point on the \(y\)-axis?

Show the answer and mark scheme
  • B1 (a) (0, 0)
  • B1 (b) 0

Answer: (a) \((0, 0)\) (b) \(0\)

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

6 questions written for 0580 in the Coordinate geometry practice: 6 Core, 3 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 \(y = 2x + 1\).

\[\begin{array}{c|ccccc} x & -1 & 0 & 1 & 2 & 3 \\ \hline y & & & & & \end{array}\]

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 \(y = 5 - x\).

\[\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \\ \hline y & & & & & \end{array}\]

Show the answer and mark scheme
  • B2 5, 4, 3, 2, 1 (B1 for 3 or 4 correct)

Answer: \(5, 4, 3, 2, 1\)

Edexcel 4MA1 notes: Linear Graphs
Practise (4MA1 questions, same mathematics): Straight Line Graphs questions

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

6 questions written for 0580 in the Coordinate geometry practice: 6 Core, 3 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 (Core, non-calculator, 2 marks):

The graph of a straight line is drawn on a grid. The line passes through the grid points \((1, 5)\) and \((3, 1)\). Use these points to find the gradient of the line.

Show the answer and mark scheme
  • M1 (1 − 5) ÷ (3 − 1)
  • A1 −2

Answer: \(-2\)

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.

  1. Midpoint: ((−2 + 4)/2, (3 + (−5))/2) = (1, −1)
  2. Steps: 4 − (−2) = 6 across and −5 − 3 = −8 up
  3. 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.

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

Answer: Q is (7, −2)

0580 practice questions

12 questions written for 0580 in the Coordinate geometry practice: 12 Extended, 10 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):

Find the coordinates of the midpoint of the line joining \((2, 7)\) and \((8, -3)\).

Show the answer and mark scheme
  • M1 (2 + 8)/2 or (7 + (−3))/2
  • A1 (5, 2)

Answer: \((5, 2)\)

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

\(P\) is the point \((k, 3)\) and \(Q\) is the point \((5, 9)\). The length of \(PQ\) is 10. Find the two possible values of \(k\).

Show the answer and mark scheme
  • M1 (5 − k)² + 6² = 10²
  • M1 (5 − k)² = 64 or 5 − k = ±8
  • A1 −3
  • A1 13

Answer: \(k = -3\) or \(k = 13\)

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

6 questions written for 0580 in the Coordinate geometry practice: 6 Core, 3 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 gradient and the \(y\)-intercept of the line \(y = 4x - 7\).

Show the answer and mark scheme
  • B1 gradient 4
  • B1 y-intercept −7

Answer: gradient \(4\), \(y\)-intercept \(-7\)

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

Write down the equation of

(a) the vertical line through \((3, -1)\)

(b) the horizontal line through \((3, -1)\).

Show the answer and mark scheme
  • B1 (a) x = 3
  • B1 (b) y = −1

Answer: (a) \(x = 3\) (b) \(y = -1\)

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

6 questions written for 0580 in the Coordinate geometry practice: 6 Core, 3 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? \(\quad y = 3x + 1, \quad y = 2x + 3, \quad y = 3x - 4, \quad y = -3x + 1\)

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

Show the answer and mark scheme
  • B1 (a) y = 3x + 1 and y = 3x − 4
  • B1 (b) −2

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

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

Find the equation of the line parallel to \(y = 4x - 1\) that passes through \((0, 6)\).

Show the answer and mark scheme
  • M1 gradient 4
  • A1 y = 4x + 6

Answer: \(y = 4x + 6\)

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.

Edexcel 4MA1 notes: Linear Graphs
Practise (4MA1 questions, same mathematics): Straight Line Graphs questions

Practise coordinate geometry

42 practice questions are written for 0580 on 3.1, 3.2, 3.3, 3.4, 3.5, 3.6. 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.