Perpendicular Lines

3.7 Perpendicular Lines – Cheat Sheet

1. Key Facts About Perpendicular Lines

• If two lines are perpendicular:
  m₁ × m₂ = −1 (product of gradients = −1)
• Gradient of perpendicular line = negative reciprocal of the original gradient.
  • Example: If m = 3, perpendicular m = −1/3

2. Finding Gradient of a Perpendicular Line

Example: Gradient of line perpendicular to 2y = 3x + 1

• Rewrite: y = (3/2)x + 1/2 → m₁ = 3/2
• Perpendicular gradient m₂ = −2/3

3. Equation of a Perpendicular Line Through a Point

Steps:

1. Find gradient m₁ of given line.
2. Perpendicular gradient m₂ = −1/m₁.
3. Use point–slope form: y − y₁ = m₂(x − x₁).
4. Rearrange to y = mx + c form.

Example: Find equation of line perpendicular to y = 4x − 1 through (2, 3)

• m₁ = 4 → m₂ = −1/4
• y − 3 = (−1/4)(x − 2)
• y − 3 = (−x/4) + 1/2
• y = (−x/4) + 7/2

4. Perpendicular Bisector of a Line Segment

Steps:

1. Find midpoint of the segment: ((x₁ + x₂)/2, (y₁ + y₂)/2).
2. Find gradient of segment: m₁ = (y₂ − y₁) / (x₂ − x₁).
3. Perpendicular gradient m₂ = −1/m₁.
4. Use midpoint and m₂ in point–slope form to get equation.

Example: Perpendicular bisector of line joining (−3, 8) and (9, −2)

• Midpoint = ((−3 + 9)/2, (8 − 2)/2) = (3, 3)
• m₁ = (−2 − 8) / (9 − (−3)) = (−10) / 12 = −5/6
• m₂ = 6/5
• Equation: y − 3 = (6/5)(x − 3)
• y = (6/5)x − 18/5 + 15/5
• y = (6/5)x − 3/5

5. Quick Reference Table