Coordinates
Mid-Points of Line Segments

The coordinates of the mid-point between two other points may be found by drawing or by calculation.

Consider the line segment that joins the point A which has coordinates (2, 2) and the point B (6, 8). The mid-point of the line segment AB is shown in the diagram below.

The value of the x-coordinate of the mid-point of the line segment AB is the mean value of the two x-coordinates of the end points A and B.

Similarly for the y-coordinate of the mid-point, it is the mean of the y-coordinates of the end points A and B.

The coordinates of the mid-point could have been calculated directly as shown below.

⎛
⎝, ⎞
⎠		 = ⎛
⎝, ⎞
⎠
= (4, 5)

Generally, for any two points, the coordinates of the mid-point of the line segment joining the points (a, b) and (c, d) is given by ⎛
⎝, ⎞
⎠.

Worked Examples

1

The diagram shows the points A, B, C and D.

Find the coordinates of the mid-point of the line segment:

(a)

AB

Show me

The coordinates of A are (2, 7).

The coordinates of B are (5, 3).

The coordinates of the mid-point of AB = ⎛
⎝, ⎞
⎠
= ⎛
⎝, ⎞
⎠
= (3.5, 5)
(b)

AC

Show me

The coordinates of C are (–5, 3).

The coordinates of the mid-point of AC = ⎛
⎝, ⎞
⎠
= ⎛
⎝, ⎞
⎠
= (–1.5, 5)
(c)

BD

Show me

The coordinates of D are (−4, −5).

The coordinates of the mid-point of BD = ⎛
⎝, ⎞
⎠
= ⎛
⎝, ⎞
⎠
= (0.5, –1)

Exercises

The diagram shows the points A, B, C, D and E.

Find the coordinates of the mid-point of the line segment:

(a)
AB (, )
(b)
AC (, )
(c)
AD (, )
(d)
AE (, )
(e)
BE (, )
(f)
CD (, )
(g)
DE (, )
(h)
CE (, )

Determine the coordinates of the mid-point of the line segment joining the two points given in each case.

(a)
(4, 7) (8, 11) (, )
(b)
(6, 2) (18, 8) (, )
(c)
(3, 2) (9, 4) (, )
(d)
(6, 3) (10, 11) (, )
(e)
(4, 1) (3, 4) (, )
(f)
(6, 6) (1, 7) (, )
(g)
(2, 15) (13, 2) (, )
(h)
(24, 2) (13, 3) (, )

The diagram shows the points A, B, C, D and E.

Determine the coordinates of the mid-point of the line segments below.

(a)
AB (, )
(b)
AE (, )
(c)
BD (, )
(d)
BC (, )
(e)
AC (, )
(f)
DC (, )
(g)
DE (, )
(h)
CE (, )

Determine the coordinates of the mid-point of the line segment joining each pair of points listed below.

(a)
(2, 3) (4, –6) (, )
(b)
(–2, 1) (3, –6) (, )
(c)
(–2, –3) (–8, –10) (, )
(d)
(–2, 4) (5, –6) (, )
(e)
(–3, –2) (4, 7) (, )
(f)
(6, –2) (8, –10) (, )
(g)
(2, –6) (7, –5) (, )
(h)
(8, –3) (–10, –11) (, )

The diagram shows a cuboid.

The coordinates of the vertices are listed below.

A (0, 0, 0) B (7, 0, 0)
C (7, 6, 0) D (0, 6, 0)
E (0, 0, 5) F (7, 0, 5)
G (7, 6, 5) H (0, 6, 5)

Determine the coordinates of the mid-point of each of the following line segments.

(a)
AB (, , )
(b)
CD (, , )
(c)
DH (, , )
(d)
FG (, , )

The points A, B and C have coordinates (4, 2, 1), (6, 10, 7) and (4, 8, 11).

(a)

Determine the coordinates of the mid-points of AB and AC.

AB: (, , )

AC: (, , )

(b)

The mid-point of AB is joined to the mid-point of AC by a line segment. Determine the coordinates of the mid-point of this line segment.

(, , )