This module covers working with points and straight lines on a coordinate grid: finding a gradient, a midpoint and a length, writing the equation of a line, deciding whether lines are parallel or perpendicular, and testing whether points are collinear. Questions in this area often chain two or three of these skills in one problem.
Check the current Cambridge IGCSE Mathematics 0580 syllabus for the exact wording of each content point in your exam year. The habits taught here stay the same across versions. Our Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to substitute into an expression, rearrange a simple equation, simplify fractions and work with negative numbers. You should also know Pythagoras’ theorem and how to read a point such as (3, −2) on a grid. If graphs and transformations felt comfortable, you are ready.
An orienting example
A is the point (1, 2) and B is the point (7, 10). Find the gradient of AB, its midpoint, its length and its equation.
Gradient: (10 − 2)/(7 − 1) = 8/6 = 4/3.
Midpoint: ((1 + 7)/2, (2 + 10)/2) = (4, 6).
Length: the differences are 6 and 8, so AB = √(36 + 64) = √100 = 10.
Equation: y − 2 = 4/3(x − 1). Multiply by 3: 3y − 6 = 4x − 4, so 3y = 4x + 2.
Check: substitute B: 3(10) = 30 and 4(7) + 2 = 30 ✓. Substitute A: 3(2) = 6 and 4(1) + 2 = 6 ✓.
Four different results came from the same two points. Each lesson below takes one of them and builds it carefully.
In which order should you study it?
- Calculate a gradient from two points: the base skill, and the place where most sign errors start.
- Find a midpoint and a segment length: averaging and Pythagoras from the same coordinates.
- Write the equation of a line through a point: uses the gradient from lesson one.
- Identify parallel and perpendicular lines: compares gradients to build new lines.
- Check whether three points are collinear: applies gradients as a proof.
Then work through the mixed practice set. A steady pace is one lesson per study session and the practice set once all five are done.
Which traps catch most students here?
- Subtracting a negative coordinate wrongly, for example 5 − (−3) written as 2.
- Swapping the order in only the numerator or only the denominator of the gradient fraction.
- Adding the differences instead of squaring, adding and square-rooting for length.
- Forgetting to halve when finding a midpoint.
- Flipping a perpendicular gradient without changing the sign.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt every question on paper before opening the answer, and set out the working as you would in an exam. Sketch the points first, because a quick sketch catches sign errors before they grow. The non-calculator working trainer lets you check fraction steps without a calculator.
When you get something wrong, read the routing notes at the end of the practice set and go back to the lesson it names. Fix the lesson, then retry a fresh question a few days later. If you want a teacher to go through your working with you, see our online one-to-one Mathematics tuition.