Overview
The geometry of straight lines, plus turning curved relationships into linear ones.
Key facts
- Equation: y = mx + c (gradient m, y-intercept c).
- Gradient between two points = (y₂ − y₁) ÷ (x₂ − x₁).
- Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Length = √[(x₂ − x₁)² + (y₂ − y₁)²].
- Parallel lines have the same gradient.
- Perpendicular lines: m₁ × m₂ = −1.
Reducing to linear form
A relationship like y = a xⁿ becomes linear by taking logs: log y = n log x + log a — plot log y against log x; gradient = n, intercept = log a.
Worked example
Line through (1, 2) and (3, 8): gradient = (8 − 2)/(3 − 1) = 3, so y = 3x − 1.
Common mistakes
- Using m₁ = m₂ for perpendicular lines — that's parallel; perpendicular is the
negative reciprocal.
Exam tips
- "Show the relationship is y = a xⁿ" → plot log–log and read off the line.