Coordinate geometry: straight lines

The standard form y = mx + c

• m = gradient (slope)
• c = y-intercept (where the line crosses the y-axis)

Calculating gradient between two points

m = (y₂ − y₁) / (x₂ − x₁) — change in y over change in x.

Example: line through (1, 2) and (4, 11). m = (11 − 2)/(4 − 1) = 9/3 = 3.

Finding the equation of a line

Given gradient m and one point (x₁, y₁): y − y₁ = m(x − x₁) — point-slope form. Then rearrange.

Example: gradient 3, through (1, 2). y − 2 = 3(x − 1) → y = 3x − 1.

Given two points: first calculate m, then use either point in point-slope.

Parallel lines

Have the same gradient. Different y-intercepts.

Example: y = 2x + 5 is parallel to y = 2x − 3.

Perpendicular lines (Higher tier)

Their gradients multiply to −1: m₁ × m₂ = −1.

To find the perpendicular gradient, take the negative reciprocal:

• gradient 3 → perpendicular gradient −1/3
• gradient −2/5 → perpendicular gradient 5/2

Finding the equation of a perpendicular line

1. Find the gradient of the original line.
2. Negative reciprocal → new gradient.
3. Use point-slope form with the given point.

Midpoint formula

Midpoint of (x₁, y₁) and (x₂, y₂) = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Length of a line segment (Pythagoras)

Distance = √[(x₂ − x₁)² + (y₂ − y₁)²].

Common CCEA exam tip

When the question says "in the form y = mx + c", make sure your final answer has y on its own (no fractions on the y-side). Score the A1 by full simplification.