Knowledge Organiser: Parallel and Perpendicular Lines
Key Terms
- Parallel lines: Lines that never meet; they have the same gradient
- Perpendicular lines: Lines that meet at exactly 90°
- Negative reciprocal: For gradient m, the perpendicular gradient is −1 ÷ m
- Gradient product: Two perpendicular lines satisfy m₁ × m₂ = −1
Must-Know Facts
- Parallel lines have identical gradients but different y-intercepts
- Perpendicular gradients multiply to give −1 (they are negative reciprocals)
- To find a parallel line through a point: use same m, substitute point to find c
- To find a perpendicular line through a point: use m₂ = −1 ÷ m₁, then find c
- Horizontal (m = 0) and vertical (undefined m) lines are always perpendicular to each other
- If m = 2, the perpendicular gradient is −½; if m = −3, it is ⅓
Key Formulas
- Parallel condition: m₁ = m₂
- Perpendicular condition: m₁ × m₂ = −1
- Perpendicular gradient: m₂ = −1 ÷ m₁
- Equation of new line: substitute m and a given point into y = mx + c to find c
Common Mistakes
- Perpendicular gradient = negative reciprocal: If m = 3, perpendicular gradient is −⅓ (flip and negate), not just −3
- Parallel lines have the SAME gradient: A parallel line through a different point has the same m but a different c
- Not finding c: After finding the gradient, always substitute the given point to find the y-intercept c
- m₁ × m₂ = −1 check: Verify perpendicularity by multiplying the two gradients: result must be −1 exactly