0580

Coordinate Geometry

Coordinate Geometry and Graphs

Averaging the two ends

  • The midpoint is the point exactly half way along, so it is the same distance from each end
  • Finding it means averaging the two ends one direction at a time: average the x values, then average the y values
  • For ends (x₁, y₁) and (x₂, y₂) the midpoint is ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
  • This formula is not printed anywhere in the paper, so it has to be known
  • Each coordinate is worked out independently, and each one is worth a mark, so a slip in one does not cost the other
  • A half in the answer is perfectly normal and should be left as 0.5 or ½ rather than rounded away
Exam tip

The midpoint is the average of the two x values and the average of the two y values, so it lies between the ends: if your answer is outside the segment you have subtracted where you should have added. The midpoint formula is not in the List of formulas, so it has to be known.

Worked example

Midpoint from two given points

A is the point (−7, 2) and B is the point (3, 9). Find the coordinates of the midpoint of AB.

Solution:

  • Average the x values: (−7 + 3) ÷ 2 = −4 ÷ 2 = −2
  • Average the y values: (2 + 9) ÷ 2 = 11 ÷ 2 = 5.5
  • The midpoint of AB is (−2, 5.5)

Working backwards from the midpoint

  • A question may give one end and the midpoint and ask for the other end
  • Whatever the step is from the given end to the midpoint, the same step continues to the far end
  • In numbers, double the midpoint value and subtract the known end, in each direction separately
  • Checking is quick: average your answer with the given end and confirm the midpoint comes back
Worked example

Finding the far end

M(7, 4) is the midpoint of AB, and A is the point (3, −1). Find the coordinates of B.

Solution:

  • In the x direction: 2 × 7 − 3 = 14 − 3 = 11
  • In the y direction: 2 × 4 − (−1) = 8 + 1 = 9
  • So B is (11, 9)
  • Check: the midpoint of (3, −1) and (11, 9) is ((3 + 11) ÷ 2, (−1 + 9) ÷ 2) = (7, 4)