Midpoint
Pronunciation: /ˈmɪdˌpɔɪnt/ ?
A midpoint is a point
equidistant
between two points. The midpoint is on the line segment joining the two points
and divides the line segment exactly in half. The exact mathematical definition is:
A midpoint M between points A and B
is a point such that AM = MB.
Constructing a Midpoint
Step  Illustration  Description 
1 
 Start with the points A and B. 
2 
 Draw the line segment . 
3 
 Draw a circle with center at A and radius AB. 
4 
 Draw a circle with center at B and radius AB 
5 
 Mark one intersection of the two circles as point C and the other intersection of the two circles as pointD. 
6 
 Draw the line segment . 
7 
 Mark the point of intersection of line segment and line segment as M. Point M is the midpoint. 
Table 1: Constructing a midpoint. 
Calculate a Midpoint in a 1Dimensional Space
 Manipulative 1: Midpoint on a number line. Click on the blue points and drag them to change the figure. Double click on the manipulative to view it full screen. 

The formula for a midpoint in a one dimensional space between A and B is
. Click on the
blue points in manipulative 1 and drag them to change the figure.

Calculate a Midpoint in a 2Dimensional Space
 Manipulative 2: Calculating a midpoint in 2dimensional space.


A midpoint divides the line segment exactly in half. This fact can be used
to figure out the formula for a midpoint of a line segment in a 2dimensional
Euclidean space such as a Cartesian coordinate system. The xcoordinate of
the midpoint will be halfway between the xcoordinates of the two points, and
the ycoordinate of the midpoint will be halfway between the ycoordinates of
the two points. The formula for the midpoint of a line segment with endpoints
and
is
.

Calculating a Midpoint in nDimensional Space
The algorithm for calculating an endpoint in 2dimensional
space can be generalized for nDimensional Space. Given two points
and
the midpoint is
.
Proof: If M is the midpoint of
, then
This proof is a paragraph proof or informal proof.
The definition of the midpoint of a segment is that .
In other words, the lengths of the two segments are equal. By the definition of
congruence,
is congruent with
if and only if
and
have the same measure. Since, by the definition of a midpoint,
and
have the same measure,
.
References
 midpoint. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 20110108). http://wordnetweb.princeton.edu/perl/webwn?s=midpoint&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.
Cite this article as:
Midpoint. 20100322. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/m/midpoint.html.
Translations
Image Credits
Revision History
20100322: Added information on constructing a midpoint, calculating a midpoint in 1 dimension, and calculating a midpoint in n dimensions. (
McAdams, David.)
20091218: Added "References" (
McAdams, David.)
20081007: Initial version (
McAdams, David.)