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

StepIllustrationDescription
1 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Start with the points A and B.
2 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Draw the line segment AB with a line over the top..
3 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Draw a circle with center at A and radius AB.
4 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Draw a circle with center at B and radius AB
5 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Mark one intersection of the two circles as point C and the other intersection of the two circles as pointD.
6 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Draw the line segment CD with line over the top..
7 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Mark the point of intersection of line segment AB with a line over the top. and line segment CD with line over the top. as M. Point M is the midpoint.
Table 1: Constructing a midpoint.

Calculate a Midpoint in a 1-Dimensional Space

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
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 M=(A+B)/2. Click on the blue points in manipulative 1 and drag them to change the figure.

Calculate a Midpoint in a 2-Dimensional Space

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 2: Calculating a midpoint in 2-dimensional 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 2-dimensional Euclidean space such as a Cartesian coordinate system. The x-coordinate of the midpoint will be halfway between the x-coordinates of the two points, and the y-coordinate of the midpoint will be halfway between the y-coordinates of the two points. The formula for the midpoint of a line segment with end-points (x1, y1) and (x2, y2) is (((x2-x1)/2),((y2-y1)/2)).

Calculating a Midpoint in n-Dimensional Space

The algorithm for calculating an endpoint in 2-dimensional space can be generalized for n-Dimensional Space. Given two points A=(a_1,a_2,a_3,...,a_n) and B=(b_1,b_2,b_3,...,b_n) the midpoint is M=((a_1+b_1)/2,(a_2+b_2)/2,(a_3+b_3)/2,...,(a_n+b_n)/2).

Proof: If M is the midpoint of AB with a line over the top., then AM with a line over the top is congruent with MB with a line over the top.

This proof is a paragraph proof or informal proof.

The definition of the midpoint of a segment is that AM=MB. In other words, the lengths of the two segments are equal. By the definition of congruence, AM with a line over the top. is congruent with MB with a line over the top. if and only if AM with a line over the top. and MB with a line over the top. have the same measure. Since, by the definition of a midpoint, AM with a line over the top. and MB with a line over the top. have the same measure, AM with a line over the top is congruent with MB with a line over the top..

References

  1. midpoint. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 2011-01-08). 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. 2010-03-22. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/m/midpoint.html.

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Revision History


2010-03-22: Added information on constructing a midpoint, calculating a midpoint in 1 dimension, and calculating a midpoint in n dimensions. (McAdams, David.)
2009-12-18: Added "References" (McAdams, David.)
2008-10-07: Initial version (McAdams, David.)

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