• The gradient m of a line measures its slope: m = (y₂−y₁)/(x₂−x₁)
• Three ways to write a line's equation: y = mx + c (gradient-intercept), y − y₁ = m(x − x₁) (point-gradient), and ax + by + d = 0 (general form)
• The midpoint of two points is just the average of their x's and the average of their y's
• The length of a line segment comes from Pythagoras: d = √[(x₁−x₂)² + (y₁−y₂)²]
• Parallel lines have the exact same gradient (m₁ = m₂)
• Perpendicular lines have gradients that multiply to −1 (m₁ × m₂ = −1) — i.e. they're negative reciprocals of each other

What "gradient" actually means

Think of the gradient as the answer to the question: "For every step I take to the right, how much does the line go up (or down)?" It's a single number that completely describes how steep and in which direction a line tilts.

A gradient of 1 means: move 1 unit right → the line rises 1 unit. A gradient of −2 means: move 1 unit right → the line falls 2 units (the negative sign tells you it's going downhill as you move right). The bigger the number (ignoring the sign), the steeper the line. A gradient of 0 means the line is perfectly flat (horizontal).

  y = mx + c — Gradient-Intercept Form

This is the most useful form because you can read the gradient and y-intercept straight off it — no rearranging needed. Here, m is the gradient, and c is the y-intercept, meaning the line crosses the y-axis at the point (0, c).

  y − y₁ = m(x − x₁) — Point-Gradient Form

This form is your best friend when you're building an equation from scratch, because it only needs a gradient and a single point on the line — you plug them straight in. Once you've written it, you can rearrange into y = mx + c or ax + by + d = 0 if the question asks for a specific format.

  ax + by + d = 0 — General Form

This is the "tidy, no-fractions" form examiners often ask for explicitly, especially when a, b and d must be integers. Its main superpower: you can instantly read off both intercepts without rearranging into y = mx + c first.

Finding an equation from scratch — the full method

Just an average of the coordinates

The midpoint is exactly what it sounds like: the point sitting exactly halfway between two endpoints, the same distance from each. There's no clever trick here beyond averaging — average the x-coordinates to get the midpoint's x, and average the y-coordinates to get the midpoint's y.

It's really just Pythagoras in disguise

Picture the line segment as the hypotenuse of a right-angled triangle, where one leg is the horizontal distance between the points (the difference in x) and the other leg is the vertical distance (the difference in y). Pythagoras' theorem (a² = b² + c²) then gives you the length directly.

Same steepness, never meet

Parallel lines are always the same distance apart, no matter how far you follow them — which means they never intersect. The reason this works geometrically is simple: if two lines tilt at exactly the same angle, they can never converge or diverge. That "same angle" condition translates mathematically into "same gradient."

How to check: rearrange each line into y = mx + c form, then compare the coefficients of x. If they match, the lines are parallel.

Meeting at a perfect right angle

Perpendicular lines cross each other at exactly 90°. The relationship between their gradients is less obvious than for parallel lines, but it's a rule worth just memorising: multiply the two gradients together and you always get −1. Equivalently, one gradient is the negative reciprocal of the other — flip the fraction upside down, and flip the sign.

Mistakes students actually make

• Subtracting coordinates in the wrong order — if you do (y₂−y₁) on top, you MUST do (x₂−x₁) on the bottom, using the same point order both times, or your gradient's sign flips.
• Forgetting to flip AND negate for perpendicular gradients — going from gradient 2 to gradient 2 (forgot to flip) or to gradient −2 (forgot to reciprocate) instead of the correct −1/2.
• Leaving fractions in "integer coefficient" answers — if a question asks for ax + by + d = 0 with integers, always multiply through to clear any fractions before giving your final answer.
• Rounding too early in distance calculations — always keep the exact square root (or square) until the very last step.
• Assuming lines with similar-looking equations are parallel without actually rearranging both into y = mx + c form first to properly compare gradients.
• Missing hidden geometric clues — not recognising that "the tangent to the circle" implies perpendicularity to the radius, or that a rectangle's sides imply both parallel and perpendicular relationships.

What examiners are looking for

• Clear, methodical working — showing the gradient calculation as a separate step before substituting into point-gradient form.
• Final answers given in the exact format requested in the question (check for "in the form y = mx + c" vs "in the form ax + by + c = 0" instructions).
• Integer coefficients when the general form is requested — don't leave halves or thirds floating around.
• A quick sketch as a sanity check on multi-step geometry questions — it costs no marks and catches sign errors instantly.