Also known as the Midpoint Theorem:
The coordinates
of the
midpoint
of a
line segment
are the average of the coordinates of its endpoints.

Try this
Adjust the line segment below by dragging an orange dot at point A or B. The point C is the midpoint.
You can also drag the origin point at (0,0).

A line segment on the
coordinate plane
is defined by two endpoints whose
coordinates
are known.
The
midpoint
of this line is exactly halfway between these endpoints and its location can be found using the Midpoint Theorem,
which states:

The x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints.

Likewise, the y-coordinate is the average of the y-coordinates of the endpoints.

To see this graphically, in the figure above turn off the grid and turn on the pointers.
As you drag either A or B, you will see that on each axis, the black pointers from the midpoint C
are always exactly halfway between the orange pointers from the endpoints A and B.

Worked example

In this example we find the coordinates of the midpoint C of the line segment AB.

In the figure above, press 'reset'.

First find the x-coordinate of C.
This is the average of the x-coordinates of A and B.
The coordinates of A are (10,20) so the x-coordinate is 10, the first number of the pair. Similarly, the x-coordinate of B is 50.
To find the average of these, add them together and divide the result by two:

Next, find the y-coordinate of C.
This is the average of the y-coordinates of A and B.
The coordinates of A are (10,20) so the y-coordinate is 20, the second number of the pair. Similarly, the y-coordinate of B is 10.
To find the average of these, add them together and divide the result by two:

So now we know that the midpoint C has the coordinates (30,15). Verify this in the figure above.

Things to try

In the above diagram, Drag the points A and B around and notice how the midpoint is always in the center of the line,
and its coordinates are continuously calculated as you drag them.

Click "hide details". Drag A and B to some new locations and calculate the midpoint yourself.
Then click "show details" and see how close you got.
(Note: the coordinates in the diagram above are rounded off to whole numbers for clarity).

Limitations

In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place.
This can cause calculatioons to be slightly off.