0580

Linear Graphs

Coordinate Geometry and Graphs

Negative reciprocal gradients

  • Two lines are perpendicular when they cross at a right angle
  • Their gradients multiply to give −1, so each is the negative reciprocal of the other
    • 4 pairs with −0.25, and ⅔ pairs with −1.5
  • To turn one gradient into the other, flip the fraction over and change the sign
  • Rearrange both equations to y = mx + c before comparing, otherwise the gradients cannot be read
  • The special case is a horizontal and a vertical line: y = c and x = k are perpendicular, and this is the one pair whose gradients cannot be multiplied

Building the equation

  • Rearrange the given line and read its gradient
  • Take the negative reciprocal to get the gradient of the new line
  • Substitute the point you are given to find c, exactly as for a parallel line
Worked example

Perpendicular through a given point

Find the equation of the line perpendicular to y = ⅔x + 4 that passes through (6, 1). Give your answer in the form y = mx + c.

Solution:

  • The given gradient is ⅔
  • Flip it and change the sign: the new gradient is −1.5
  • Check: ⅔ × (−1.5) = −1
  • Start from y = −1.5x + c
  • Substitute x = 6 and y = 1: 1 = −9 + c
  • Add 9: c = 10
  • The equation is y = −1.5x + 10

The perpendicular bisector

  • Bisecting a segment at right angles means passing through its midpoint at 90°, and the line that does so is its perpendicular bisector
  • It needs both skills together: the midpoint gives the point it passes through, and the negative reciprocal gives its gradient
  • Find the midpoint, find the gradient of the segment, flip and negate that gradient, then substitute the midpoint to find c
Exam tip

Rearrange to y = mx + c before reading any gradient — leaving 4y on the left and calling the gradient −3 is the usual slip. Write the given gradient, then the perpendicular one, then the substitution that finds c, each on its own line: mark schemes credit all three and each follows through from your own earlier value. These papers never say "y-intercept"; they say "the point where the graph crosses the y-axis".

Worked example

Perpendicular bisector of a segment

Find the equation of the perpendicular bisector of the line segment joining P(2, −1) and Q(10, 5).

Solution:

  • Midpoint: ((2 + 10) ÷ 2, (−1 + 5) ÷ 2) = (6, 2)
  • Gradient of PQ: (5 − (−1)) ÷ (10 − 2) = 6 ÷ 8 = 0.75
  • Negative reciprocal: −4 ÷ 3
  • Start from y = (−4 ÷ 3)x + c
  • Substitute (6, 2): 2 = −8 + c, so c = 10
  • The perpendicular bisector is y = (−4 ÷ 3)x + 10