Gradients of parallel and perpendicular lines
The gradient of a line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). Two lines are parallel when their gradients are equal, and perpendicular when the product of their gradients is −1.
Key formula
Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = −1, i.e. m₂ = −1/m₁
Line: y = mx + c or y − y₁ = m(x − x₁)
Finding equations
To find a line through a given point that is parallel or perpendicular to another, first work out the required gradient, then use y − y₁ = m(x − x₁). A useful check is to rewrite any given line in the gradient-intercept form y = mx + c before comparing. For a line written as ax + by = c, the gradient is −a/b.
When two gradients multiply to give −1 the lines meet at a right angle, and this condition is often combined with the midpoint or distance formulas in longer coordinate-geometry questions. Remember that a line parallel to the x-axis has gradient 0, while a line parallel to the y-axis has an undefined gradient, and these two are perpendicular to each other. To write the final equation, substitute the required gradient and the given point into y − y₁ = m(x − x₁), then simplify to the form asked for.
Worked example
Line L₁ is y = 2x + 1, so m₁ = 2. A line through (3, 4) perpendicular to L₁ has gradient m₂ = −1/2. Its equation is y − 4 = −½(x − 3), giving 2y + x = 11. A line through (3, 4) parallel to L₁ has gradient 2: y − 4 = 2(x − 3), so y = 2x − 2.
Remember
- Equal gradients ⇒ parallel; product −1 ⇒ perpendicular.
- A horizontal line (m = 0) is perpendicular to a vertical line (m undefined).
- Rearrange to y = mx + c to read the gradient.