Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who need the distance, midpoint and gradient formulas at their fingertips — and who must spot parallel and perpendicular lines from gradients.
What query it owns: how to find distance, midpoint and gradient in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the revision-guide angle, while Tutopiya’s Distance, Midpoint and Gradient subtopic page owns the learning resource and the free Distance, Midpoint and Gradient quiz owns the practice.

Distance, midpoint and gradient are the building blocks of Coordinate Geometry in Cambridge IGCSE Mathematics (0580/0607). Every line question — from finding the equation of a line to proving two lines are perpendicular — starts here. This guide gives you the three core formulas, shows when to use them, and maps the command words examiners use.

Key takeaways

• Distance between (x₁, y₁) and (x₂, y₂): d = √[(x₂ − x₁)² + (y₂ − y₁)²].
• Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
• Gradient: m = (y₂ − y₁)/(x₂ − x₁) — rise over run.
• Parallel lines have equal gradients; perpendicular lines have gradients that multiply to −1.

What are distance, midpoint and gradient in Cambridge IGCSE Maths?

The distance formula measures how far apart two points are. The midpoint formula finds the point exactly halfway between them. The gradient (slope) measures how steep a line is — the change in y divided by the change in x. Together they underpin every equation-of-a-line question and many geometry proofs on the coordinate plane.

Read the full notes on Tutopiya’s Distance, Midpoint and Gradient subtopic page before you attempt questions.

The three formulas you must know

Quantity	Formula	Example: A(1, 2), B(4, 6)
Distance	√[(x₂ − x₁)² + (y₂ − y₁)²]	√[9 + 16] = 5
Midpoint	((x₁ + x₂)/2, (y₁ + y₂)/2)	(2.5, 4)
Gradient	(y₂ − y₁)/(x₂ − x₁)	4/3

How to find distance, midpoint and gradient — step by step

1. Write down the coordinates of both points clearly — label (x₁, y₁) and (x₂, y₂).
2. Substitute into the correct formula.
3. Simplify — for distance, evaluate the squares before square-rooting.
4. Check the answer makes sense (distance positive; midpoint between the two points).

Test yourself with the free Distance, Midpoint and Gradient quiz.

Parallel and perpendicular: gradient relationships

Relationship	Gradient condition	Example
Parallel	m₁ = m₂	m = 2 and m = 2
Perpendicular	m₁ × m₂ = −1	m = 2 and m = −½
Horizontal line	m = 0	y = 3
Vertical line	Undefined gradient	x = 4

Distance, midpoint and gradient in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Calculate the distance	Apply the distance formula	”Calculate the distance between A(3, 1) and B(7, 4).”
Find the midpoint	Halve each coordinate	”Find the midpoint of AB.”
Find the gradient	Rise over run	”Find the gradient of the line through (2, 5) and (6, −3).”
Show that	Prove a property with working	”Show that triangle ABC is isosceles.”
Write down	State a value	”Write down the gradient of a line perpendicular to y = 2x + 1.”

Worked exam-style stems (how to answer the wording)

1. “A is (1, −2) and B is (4, 2). Calculate the length of AB.” AB = √[(3)² + (4)²] = √25 = 5. Reward: correct substitution, simplified surd or integer.

2. “Find the midpoint of the line joining P(−3, 4) and Q(5, 0).” Midpoint = ((−3 + 5)/2, (4 + 0)/2) = (1, 2). Reward: both coordinates correct.

3. “The gradient of a line is 3. Write down the gradient of a line perpendicular to it.” Perpendicular gradient = −1/3 = −⅓. Reward: negative reciprocal stated.

When you can recognise the wording instantly, work the full set on the Coordinate Geometry topical past paper questions and the Distance, Midpoint and Gradient quiz.

How these formulas connect to Coordinate Geometry

Distance and gradient feed directly into Equation of a Line, and coordinate methods support Linear Programming. Use the Cambridge IGCSE Maths resource hub to move between subtopics.

Common mistakes students make

• Subtracting coordinates in the wrong order inconsistently (mixing up signs).
• Forgetting to square-root at the end of the distance formula.
• Using m₁ + m₂ = −1 for perpendicular lines instead of m₁ × m₂ = −1.
• Saying a vertical line has gradient zero instead of undefined.

When you need more support

If distance or gradient questions keep costing marks, work through the Equation of a Line quiz and the Coordinate Geometry topical past papers, then get help from a Cambridge IGCSE Maths tutor.

Frequently asked questions

What is the distance formula for IGCSE? d = √[(x₂ − x₁)² + (y₂ − y₁)²]. It comes from Pythagoras’ theorem applied to the horizontal and vertical separations.

How do I find the gradient of a line? Gradient m = (change in y)/(change in x) = (y₂ − y₁)/(x₂ − x₁) for any two points on the line.

What gradient is perpendicular to m = 4? The negative reciprocal: −¼. Their product must equal −1.

How should I revise distance, midpoint and gradient? Memorise the three formulas, work five mixed examples, then take the Distance, Midpoint and Gradient quiz.

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