8.2 Midpoint of a line

Analytical geometry; midpoint of a line

Sometimes, knowing the co-ordinates of the middle point or midpoint of a line is useful. For example, what is the midpoint of the line between point $P$ with co-ordinates $(2;1)$ and point $Q$ with co-ordinates $(-2;-2)$ .

The co-ordinates of the midpoint of any line between any two points $A$ and $B$ with co-ordinates $(x_1;y_1)$ and $(x_2;y_2)$ , is generally calculated as follows. Let the midpoint of $AB$ be at point $S$ with co-ordinates $(X;Y)$ . The aim is to calculate $X$ and $Y$ in terms of $(x_1;y_1)$ and $(x_2;y_2)$ .

Then the co-ordinates of the midpoint ( $S$ ) of the line between point $P$ with co-ordinates $(2;1)$ and point $Q$ with co-ordinates $(-2;-2)$ is:

It can be confirmed that the distance from each end point to the midpoint is equal. The co-ordinate of the midpoint $S$ is $(0;-0,5)$ .

and

It can be seen that $PS=QS$ as expected.

The following video provides a summary of the midpoint of a line.