IGCSE Mathematics: Coordinate Geometry (Cambridge 0580)

This guide covers Topic 3 Coordinate geometry, for Cambridge IGCSE Mathematics 0580, 2025–2027 series. The Core subtopics (C3.1, C3.2, C3.3, C3.5, C3.6) are examined at all entry levels; the Extended-only subtopics (E3.4, E3.7, plus additions within E3.2, E3.3, E3.5) are required only for the Extended tier. Extended candidates are eligible for grades A* to E and Core candidates for grades C to G, so grades A* to B need the Extended tier (the syllabus advises Extended for anyone aiming at grades A* to C).

Where this fits in 0580

Coordinate geometry is the third of nine topics in 0580, and it sits directly after Algebra and graphs, which it leans on heavily: every technique in this topic is really algebra applied to points and lines on a grid. A line’s equation, its gradient, and the length or midpoint between two points are all calculated using the algebraic manipulation skills from Topic 2, so a shaky grasp of substitution and rearrangement resurfaces here as errors in an otherwise well-understood coordinate geometry method. This topic also feeds forward into later work on graphs of functions and, at Extended tier, differentiation, where finding a gradient at a point is a direct extension of the straight-line gradient skill taught here.

Syllabus coverage

CAMBRIDGE IGCSE MATHEMATICS 0580 — TOPIC 3 COORDINATE GEOMETRY

Core

• C3.1 Coordinates — using and interpreting Cartesian coordinates in two dimensions
• C3.2 Drawing linear graphs — drawing straight-line graphs for linear equations given in the form y = mx + c (for example, y = –2x + 5), unless a table of values is given
• C3.3 Gradient of linear graphs — finding the gradient of a straight line from a grid only
• C3.5 Equations of linear graphs — interpreting and obtaining the equation of a straight-line graph in the form y = mx + c, including finding the equation when the graph is given, and finding the gradient or y-intercept from an equation; equations must be given in a fully simplified form
• C3.6 Parallel lines — finding the gradient and equation of a straight line parallel to a given line

Note that C3.4 and C3.7 exist only as Extended-tier subtopics (E3.4 and E3.7 below) — there is no Core content at those numbers.

Extended only (in addition to the Core content above)

• E3.2 Drawing linear graphs (Extended) — as C3.2, plus equations given in forms such as y = 7 – 4x or 3x + 2y = 5
• E3.3 Gradient of linear graphs (Extended) — as C3.3, plus calculating the gradient of a straight line from the coordinates of two points on it
• E3.4 Length and midpoint — calculating the length of a line segment, and finding the coordinates of the midpoint of a line segment
• E3.5 Equations of linear graphs (Extended) — as C3.5, plus interpreting and obtaining the equation of a straight-line graph in different forms (for example, ax + by = c), including finding the gradient or y-intercept from an equation such as 5x + 4y = 8
• E3.7 Perpendicular lines — finding the gradient and equation of a straight line perpendicular to a given line, including finding the equation of a perpendicular bisector of the line joining two given points

How to approach it

Build the four core formulas — gradient, length, midpoint, and y = mx + c — as one connected toolkit rather than four separate recipes, since most exam questions combine at least two of them in sequence (for example, finding a gradient first, then using it to find an equation). Always subtract coordinates in the same order in both the numerator and denominator of the gradient formula, since swapping which point is (x₁, y₁) and which is (x₂, y₂) partway through a calculation is the most common source of sign errors in this topic. For length, remember Pythagoras’ theorem is doing the work: the formula is the horizontal and vertical distances between two points combined and square-rooted, so an answer left as the sum of two squares (without the final square root) is incomplete. For parallel and perpendicular lines specifically, learn the two rules as a contrasting pair — parallel lines share the same gradient, while perpendicular lines have gradients that are negative reciprocals of each other — since exam questions often test both within the same multi-part question to check the distinction is secure.

Worked example: finding a perpendicular bisector (Extended)

Find the equation of the perpendicular bisector of the line joining (–3, 8) and (9, –2).

Step 1 (gradient of the original line):
  m = (-2 - 8) / (9 - (-3)) = -10 / 12 = -5/6

Step 2 (perpendicular gradient):
  -5/6 x m = -1  ->  m = 6/5

Step 3 (midpoint, which lies on the bisector):
  midpoint = ((-3 + 9)/2, (8 + (-2))/2) = (3, 3)

Step 4 (substitute into y = mx + c using the midpoint):
  3 = (6/5)(3) + c  ->  c = 3 - 18/5 = -3/5

  y = (6/5)x - 3/5

This worked example draws on nearly every subtopic in this guide at once — gradient, the perpendicular rule, midpoint, and the equation of a line — which is exactly the kind of multi-step question the Extended paper favours over testing each skill in isolation.

Common mistakes

Dividing the change in x by the change in y when finding a gradient, rather than change in y over change in x. Leaving a length answer as the sum of two squares, having forgotten the final square root. Adding the coordinates for a midpoint but forgetting to divide by 2. Confusing the parallel rule (same gradient) with the perpendicular rule (negative reciprocal gradient), especially under exam time pressure. Substituting a point into y = mx + c to solve for c, then accidentally writing the point’s coordinates back into the final answer instead of the calculated values of m and c.

Quick revision checklist

• Know the gradient, length and midpoint formulas from memory, and practise applying all three to the same pair of points.
• Be able to find the equation of a line from two points, or from one point and a given gradient.
• Distinguish parallel (same gradient) from perpendicular (negative reciprocal gradient) confidently under time pressure.
• Practise multi-step questions that chain several of these techniques together, since this is how Extended-tier questions typically combine the subtopics.

Official syllabus

Cambridge International, Cambridge IGCSE Mathematics (0580) syllabus for examination in 2025, 2026 and 2027: official syllabus PDF, Subject content, section 3 “Coordinate geometry”. Verified 2026-09-06.