
Midpoint of a Line Segment(Coordinate Geometry)
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 xcoordinate of the midpoint is the average of the xcoordinates of the two endpoints.
 Likewise, the ycoordinate is the average of the ycoordinates 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 xcoordinate of C.
This is the average of the xcoordinates of A and B.
The coordinates of A are (10,20) so the xcoordinate is 10, the first number of the pair. Similarly, the xcoordinate of B is 50.
To find the average of these, add them together and divide the result by two:
 Next, find the ycoordinate of C.
This is the average of the ycoordinates of A and B.
The coordinates of A are (10,20) so the ycoordinate is 20, the second number of the pair. Similarly, the ycoordinate 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.
For more see
Teaching Notes
Other Coordinate Geometry entries
(C) 2009 Copyright Math Open Reference. All rights reserved

