Cambridge O Level Mathematics: Coordinate Geometry (4024) – Revision Notes

Use this page in the last weeks before the exam. For full explanations and worked examples, read the study guide. The notes follow topic 3, Coordinate geometry (sections 3.1 to 3.7), of the Cambridge O Level Mathematics (Syllabus D) 4024 syllabus for 2025, 2026 and 2027. The content is the same for every candidate, and it can appear on the non-calculator paper and the calculator paper. Test yourself with the practice questions, see the course hub, tick off outcomes on the checklist and find weak spots with a free diagnostic.

Key terms

Term	Meaning
Cartesian coordinates	A point written (x, y): x across from the origin, y up
Gradient	How much y changes for each 1 unit increase in x
y-intercept	Where the line crosses the y-axis (x = 0)
x-intercept	Where the line crosses the x-axis (y = 0)
Line segment	The part of a line between two points
Midpoint	The point halfway along a line segment
Parallel	Same gradient, never meet
Perpendicular	Meet at 90°; gradients multiply to give −1
Perpendicular bisector	Cuts a segment in half at 90°

Formulas

Result	Rule
Gradient from two points	m = (y₂ − y₁) ÷ (x₂ − x₁)
Length of a segment	√((x₂ − x₁)² + (y₂ − y₁)²)
Midpoint	((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
Gradient-intercept form	y = mx + c
Line through (x₁, y₁) with gradient m	y − y₁ = m(x − x₁)
Parallel lines	m₁ = m₂
Perpendicular lines	m₁ × m₂ = −1, so the perpendicular gradient is −1/m
Horizontal line	y = k, gradient 0
Vertical line	x = k, gradient undefined

Section by section

3.1 Coordinates

• x first, then y. Positive x is right, positive y is up.
• On the x-axis y = 0. On the y-axis x = 0.
• Sketch any problem that gives several points. Shapes such as rectangles and parallelograms then become easy to read.

3.2 Drawing linear graphs

Two ways to draw a line:

1. Table of values for y = mx + c. Choose three x-values, work out y, plot and join.
2. Intercepts for ax + by = c. Set x = 0 to get the y-intercept, set y = 0 to get the x-intercept, then join.

Always extend the line across the whole grid and check with a third point. Use a table when the line passes through the origin.

Reminder: y = 6 − 3x gives the points (0, 6), (1, 3) and (2, 0). It falls as x rises because the coefficient of x is negative.

3.3 Gradient

Method in steps:

1. Write the two points: (x₁, y₁) and (x₂, y₂).
2. Subtract y-values on top and x-values below, in the same order.
3. Simplify. Keep the sign.

Reminder: through (0, 5) and (4, −3), the gradient is (−3 − 5) ÷ (4 − 0) = −8 ÷ 4 = −2.

On a graph, use a triangle with corners on grid crossings. Rising to the right is positive. Falling to the right is negative.

3.4 Length and midpoint

• Length: find the two differences, square, add, square root.
• Midpoint: add the coordinates and halve each.
• Surds: if a question asks for an exact value, give the answer as a surd. Otherwise the syllabus asks for non-exact answers to 3 significant figures unless the question says otherwise.
• End-point trick: if M is the midpoint of AB, then B = 2M − A, working in x and y separately.

Reminder: for (−1, −2) and (5, 6) the midpoint is (2, 2) and the length is √(36 + 64) = 10.

3.5 Equations of linear graphs

Method in steps (given two points, or a point and a gradient):

1. Find m.
2. Substitute one point into y − y₁ = m(x − x₁), or into y = mx + c to find c.
3. Simplify to the form asked.
4. Check with the other point.

To read m and c from ax + by = c, rearrange to y = mx + c. For 3x − 4y = 24: −4y = 24 − 3x, so y = (3/4)x − 6, gradient 3/4 and y-intercept −6. The x-intercept is where y = 0: x = 8.

Give a fully simplified equation. Do not leave brackets or a mix of forms unless asked.

3.6 Parallel lines

• Rearrange to y = mx + c to read the gradient.
• Keep the gradient, then use the new point to find the new c.

Reminder: parallel to y = 3x + 1 through (2, 5): 5 = 3(2) + c, so c = −1 and the line is y = 3x − 1.

3.7 Perpendicular lines

• Flip the fraction and change the sign.
• Whole-number gradients: 4 becomes −1/4; −5 becomes 1/5.
• For a perpendicular bisector: midpoint, then perpendicular gradient, then the equation.
• To prove a right angle, show that the two gradients multiply to −1 and write the conclusion.

Two combined reminders

Perpendicular bisector of P(2, −3) and Q(6, 5).

1. Midpoint = (4, 1).
2. Gradient of PQ = (5 − (−3)) ÷ (6 − 2) = 8 ÷ 4 = 2.
3. Perpendicular gradient = −1/2.
4. y − 1 = −(1/2)(x − 4), so y = −(1/2)x + 3, or x + 2y = 6.
5. Check: the midpoint gives 4 + 2 = 6.

Show that 2y − 6x = 5 is parallel to y = 3x + 8. Rearranging the first gives y = 3x + 5/2. Both gradients are 3, so the lines are parallel.

Must-know distinctions

These look similar	The difference
Parallel and perpendicular	Parallel: equal gradients. Perpendicular: product −1
Negative reciprocal and reciprocal	The perpendicular gradient needs both the flip and the sign change
x = 3 and y = 3	The line x = 3 runs up and down through (3, 0); the line y = 3 runs across through (0, 3)
Gradient and y-intercept	Gradient is the number multiplying x. Intercept is the value where x = 0
Midpoint and length	Midpoint gives a point (two numbers). Length gives one number
Line segment and line	A segment has two ends. A line goes on for ever

Quick self-test

Answer without looking, then check below.

1. Find the gradient of the line through (−2, 1) and (4, 10).
2. Find the midpoint of (−1, −2) and (5, 6), and the length of the segment.
3. Find the gradient and y-intercept of 3x − 4y = 24. Where does it cross the x-axis?
4. Write down the equation of the line parallel to y = 3x + 1 that passes through (2, 5).
5. Find the gradient of a line perpendicular to 6x + 3y = 7.
6. Find the length and midpoint of the segment from (3, −2) to (3, 9).
7. M(1, 3) is the midpoint of AB. A is (−4, 7). Find B.
8. A line with gradient −3 passes through (2, 7) and (k, 1). Find k.
9. Find the equation of the line perpendicular to y = −x/4 + 1 that passes through (0, −4).
10. A line crosses the x-axis at 8 and the y-axis at −6. Find its equation.
11. Write down the equation of the horizontal line through (5, −3).

Answers

1. (10 − 1) ÷ (4 − (−2)) = 9 ÷ 6 = 3/2.
2. Midpoint (2, 2). Length = √(6² + 8²) = 10.
3. y = (3/4)x − 6. Gradient 3/4, y-intercept −6, and the line crosses the x-axis at (8, 0).
4. Gradient 3. 5 = 3(2) + c gives c = −1. y = 3x − 1.
5. 3y = 7 − 6x, so y = −2x + 7/3. The gradient is −2, so the perpendicular gradient is 1/2.
6. Same x-value, so the length is 9 − (−2) = 11. Midpoint = (3, 3.5).
7. B = (2 × 1 − (−4), 2 × 3 − 7) = (6, −1).
8. (1 − 7) ÷ (k − 2) = −3, so k − 2 = 2 and k = 4.
9. The given gradient is −1/4, so the perpendicular gradient is 4. The y-intercept is −4, so y = 4x − 4.
10. The points (8, 0) and (0, −6) give gradient 6/8 = 3/4. The y-intercept is −6, so y = (3/4)x − 6, or 3x − 4y = 24.
11. y = −3.

Where marks are usually lost

• Reversing the order of subtraction in only one of the numerator and the denominator, which gives the wrong sign.
• Reading the gradient of 3x − 4y = 24 as 3 or −3 without rearranging.
• Writing the perpendicular gradient with the right sign but not inverted, or inverted with the wrong sign.
• Using the midpoint formula when the question asks for length, or the reverse.
• Mixing up x = k (vertical) and y = k (horizontal).
• Finding c by substituting the point into the wrong place, and not checking with a second point.
• Stopping at a decimal for a gradient that is a fraction, such as 0.333 for 1/3.
• Leaving the final equation unsimplified, or in a different form from the one requested.
• Claiming lines are parallel or perpendicular without stating the gradients and the conclusion.

Frequently asked questions

What is the formula for the gradient between two points? The gradient is m = (y₂ − y₁) ÷ (x₂ − x₁). Subtract the coordinates in the same order on the top and the bottom, or the sign of the gradient will be wrong.

How do you find the length and midpoint of a line segment? For the length, find the differences in x and in y, square them, add, and take the square root. For the midpoint, add the x-coordinates and halve, then add the y-coordinates and halve.

What is the difference between the lines x = 3 and y = 3? The line x = 3 is vertical and runs up and down through (3, 0). The line y = 3 is horizontal and runs across through (0, 3).

How do you find an end point if you know the midpoint? If M is the midpoint of AB, then B = 2M − A, worked out separately for the x- and y-coordinates. For example, a midpoint of (1, 3) and an end point of (−4, 7) give B = (6, −1).

Should coordinate geometry answers be exact or rounded in Cambridge O Level Mathematics 4024? Give a surd if the question asks for an exact value. Otherwise, the 4024 syllabus asks for non-exact answers to 3 significant figures unless the question says otherwise.

Official syllabus

Cambridge O Level Mathematics (Syllabus D) 4024 syllabus for examination in 2025, 2026 and 2027, topic 3 Coordinate geometry (Cambridge University Press & Assessment).