Parallel and perpendicular lines

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What this lesson covers

  • Parallel lines have equal gradients.
  • Perpendicular gradients are negative reciprocals.
  • A perpendicular bisector goes through the midpoint at right angles.
Key idea

The two gradient rules

  • Parallel: m₁ = m₂.
  • Perpendicular: m₁ × m₂ = −1, so flip the fraction and change the sign.
  • A gradient of 2 pairs with −1/2. A gradient of −3/4 pairs with 4/3.
Worked example

Lines related to y = 2x + 1

  • Find the parallel line through (3, 4), then the perpendicular line through (2, 5).
  1. parallel: y − 4 = 2(x − 3), so y = 2x − 2
  2. perpendicular gradient = −1/2
  3. y − 5 = −(1/2)(x − 2)
  4. y = −(1/2)x + 6
Worked example

Perpendicular bisector of A(2, 1) and B(6, 5)

  • It needs the midpoint and the perpendicular gradient.
  1. midpoint = (4, 3)
  2. gradient AB = 4 ÷ 4 = 1
  3. perpendicular gradient = −1
  4. y − 3 = −1(x − 4)
  5. y = −x + 7

Common mistakes

  • Taking the reciprocal and keeping the sign.
    Do both: the perpendicular of 2 is −1/2, not 1/2.
  • Using A or B as the point on a perpendicular bisector.
    The bisector goes through the midpoint, not through A or B.

Your turn

  • 1. Find the line parallel to y = −3x + 2 through (1, 1).
  • 2. Find the line perpendicular to y = −3x + 2 through (3, 0).
Show answer

1. y = −3x + 4 2. gradient 1/3, so y = (1/3)x − 1

Summary

What to take away

  • Same gradient means parallel.
  • Negative reciprocal means perpendicular.
  • A bisector always starts from the midpoint.

More in AS 91256

Distance, midpoint and gradientNotesWriting the equation of a lineNotesParallel and perpendicular linesNotesModelling problems on the coordinate planeNotes

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