Distance, midpoint and gradient

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Worksheet with answersDownload PDF

What this lesson covers

  • Gradient is the rise divided by the run.
  • Distance comes from Pythagoras on the rise and run.
  • The midpoint averages the x values and averages the y values.
Key idea

One triangle, three formulas

  • Take any two points and draw the right-angled triangle between them. The run is the difference in x and the rise is the difference in y.
  • Gradient m = (y₂ − y₁) ÷ (x₂ − x₁).
  • Distance d = √((x₂ − x₁)² + (y₂ − y₁)²).
  • Midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).
Worked example

From A(1, 2) to B(7, 10)

  • Find the run and rise first: both are used twice.
  1. run = 7 − 1 = 6, rise = 10 − 2 = 8
  2. m = 8 ÷ 6 = 4/3rise over run
  3. d = √(36 + 64) = √100 = 10Pythagoras
  4. midpoint = (4, 6)average each coordinate

Common mistakes

  • Subtracting in a different order on the top and the bottom.
    Use the same point first in both, or the sign of the gradient flips.
  • Subtracting the coordinates to find a midpoint.
    A midpoint adds and halves, because it is an average.

Your turn

  • 1. Find the gradient from P(−2, 3) to Q(4, −5).
  • 2. Find the length of PQ.
  • 3. Find the midpoint of PQ.
Show answer

1. −8 ÷ 6 = −4/3 2. √(36 + 64) = 10 3. (1, −1)

Summary

What to take away

  • Draw the triangle: run and rise give you everything.
  • Gradient divides, distance uses Pythagoras, midpoint averages.

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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