Coordinates, midpoints and shapes on a grid

Coordinates turn geometry into arithmetic. Midpoints, lengths and missing corners can all be worked out from the numbers.

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Midpoints

The midpoint of (x1, y1) and (x2, y2) is ((x1 + x2)2, (y1 + y2)2): the mean of the x's and the mean of the y's.
Midpoint of (2, 5) and (8, −3) = (5, 1).
Working backwards: if M(4, 7) is the midpoint of A(1, 3) and B, then B = (7, 11): A to M is +3, +4, so M to B is +3, +4 again.

Lengths

Make a right-angled triangle and use Pythagoras.
(−2, 1) to (4, 9): across 6, up 8, so the length is √(62 + 82) = 10.

Missing corners

In a parallelogram ABCD, the step from B to A is the same as the step from C to D.
A(1, 1), B(5, 2), C(7, 6): B to A is −4, −1, so D = (7 − 4, 6 − 1) = (3, 5).
Sketch the points first; it shows you which way to go.
Worked example

Find the midpoint of (2, 5) and (8, −3).

  1. x: (2 + 8) ÷ 2 = 5
  2. y: (5 + (−3)) ÷ 2 = 1

Answer: (5, 1)

Key idea

Midpoint: average the x's and average the y's. Length: Pythagoras on the across and up distances. For a missing corner, repeat the step along the parallel side.

Common mistakes
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Extra practice

2 more questions

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  1. Find the x-coordinate of the midpoint of (−1, 4) and (9, −6).

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    Answer: 4

    (−1 + 9) ÷ 2 = 4.

  2. Find the y-coordinate of the midpoint of (−6, 1) and (8, 3).

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    Answer: 2

    (1 + 3) ÷ 2 = 2.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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