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Coordinate geometry

Midpoint formula

The point exactly halfway between two points: average the x-coordinates and average the y-coordinates.

M=(x1+x22,y1+y22)
Missing endpointx2=2Mx−x1, y2=2My−y1

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

−12−10−8−6−4−224681012−6−4−2246Δx = 10Δy = −6ABM
  • Distance AB = √(100 + 36) ≈ 11.662
  • Midpoint M = (3, 2)
  • Slope = Δy/Δx = −0.6

What the symbols mean

M
the midpoint of segment AB
Mx
the x-coordinate of the midpoint
My
the y-coordinate of the midpoint
x1
the x-coordinate of endpoint A
y1
the y-coordinate of endpoint A
x2
the x-coordinate of endpoint B
y2
the y-coordinate of endpoint B
B
in the calculator's second mode, the unknown endpoint found from A and the midpoint

When to use it

Use it to find the centre of a circle from the ends of a diameter, to bisect a segment, to find where the diagonals of a parallelogram cross, or to build the perpendicular bisector of a segment (the midpoint plus the perpendicular slope). Run backwards, it finds a missing endpoint when you know one end and the middle.

Worked example

Find the midpoint of A(−2, 5) and B(8, −1).

  1. Start from the midpoint formula: average the x-coordinates and average the y-coordinates.

    M=(x1+x22,y1+y22)

  2. x-coordinate of the midpoint:

    Mx=x1+x22
    =−2+82
    =62
    =3

  3. y-coordinate of the midpoint:

    My=y1+y22
    =5+(−1)2
    =42
    =2

Answer M = (3, 2)

Why it works

Halfway along the segment you have gone half of the horizontal change and half of the vertical change: x1+12(x2−x1)=x1+x22, and the same for y.

Common mistakes

  • Subtracting instead of adding: the midpoint averages the coordinates, it does not use their difference.
  • Halving only one coordinate. Both the x and y sums are divided by 2.
  • For a missing endpoint, using (x1+Mx)/2. That gives the point a quarter of the way along; the endpoint is 2Mx−x1.