Sometimes, knowing the coordinates of the middle point or midpoint of a line is useful. For example, what is the midpoint of the line between point PP with coordinates (2;1)(2;1) and point QQ with coordinates (2;2)(2;2).
The coordinates of the midpoint of any line between any two points AA and BB with coordinates (x1;y1)(x1;y1) and (x2;y2)(x2;y2), is generally calculated as follows. Let the midpoint of ABAB be at point SS with coordinates (X;Y)(X;Y). The aim is to calculate XX and YY in terms of (x1;y1)(x1;y1) and (x2;y2)(x2;y2).
X
=
x
1
+
x
2
2
Y
=
y
1
+
y
2
2
âˆ´
S
x
1
+
x
2
2
;
y
1
+
y
2
2
X
=
x
1
+
x
2
2
Y
=
y
1
+
y
2
2
âˆ´
S
x
1
+
x
2
2
;
y
1
+
y
2
2
(1)Then the coordinates of the midpoint (SS) of the line between point PP with coordinates (2;1)(2;1) and point QQ with coordinates (2;2)(2;2) is:
X
=
x
1
+
x
2
2
=

2
+
2
2
=
0
Y
=
y
1
+
y
2
2
=

2
+
1
2
=

1
2
âˆ´
S
is
at
(
0
;

1
2
)
X
=
x
1
+
x
2
2
=

2
+
2
2
=
0
Y
=
y
1
+
y
2
2
=

2
+
1
2
=

1
2
âˆ´
S
is
at
(
0
;

1
2
)
(2)It can be confirmed that the distance from each end point to the midpoint is equal. The coordinate of the midpoint SS is (0;0,5)(0;0,5).
P
S
=
(
x
1

x
2
)
2
+
(
y
1

y
2
)
2
=
(
0

2
)
2
+
(

0
.
5

1
)
2
=
(

2
)
2
+
(

1
.
5
)
2
=
4
+
2
.
25
=
6
.
25
P
S
=
(
x
1

x
2
)
2
+
(
y
1

y
2
)
2
=
(
0

2
)
2
+
(

0
.
5

1
)
2
=
(

2
)
2
+
(

1
.
5
)
2
=
4
+
2
.
25
=
6
.
25
(3)and
Q
S
=
(
x
1

x
2
)
2
+
(
y
1

y
2
)
2
=
(
0

(

2
)
)
2
+
(

0
.
5

(

2
)
)
2
=
(
0
+
2
)
)
2
+
(

0
.
5
+
2
)
)
2
=
(
2
)
)
2
+
(

1
.
5
)
)
2
=
4
+
2
.
25
=
6
.
25
Q
S
=
(
x
1

x
2
)
2
+
(
y
1

y
2
)
2
=
(
0

(

2
)
)
2
+
(

0
.
5

(

2
)
)
2
=
(
0
+
2
)
)
2
+
(

0
.
5
+
2
)
)
2
=
(
2
)
)
2
+
(

1
.
5
)
)
2
=
4
+
2
.
25
=
6
.
25
(4)It can be seen that PS=QSPS=QS as expected.
The following video provides a summary of the midpoint of a line.