Midpoint

Pronunciation: /ˈmɪdˌpɔɪnt/ Explain

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 of a midpoint is:

A midpoint M between points A and B is a point on the line AB such that AM = MB.

Constructing a Midpoint

StepIllustrationDescription
1
Start with the points A and B.
2
Draw the line segment AB with a line over the top..
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 CD with line over the top..
7
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

Click on the blue points and drag them to change the figure.

What happens is B is to the left of A?
Manipulative 8 - Calculating Midpoint in One Dimension Created with GeoGebra.

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

Click on the blue points and drag them to change the figure.

Manipulative 9 - Calculating Midpoint in Two Dimensions Created with GeoGebra.

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..

Cite this article as:

McAdams, David E. Midpoint. 9/3/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/m/midpoint.html.

Image Credits

Revision History

9/3/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
3/22/2010: Added information on constructing a midpoint, calculating a midpoint in 1 dimension, and calculating a midpoint in n dimensions. (McAdams, David E.)
12/18/2009: Added "References". (McAdams, David E.)
10/7/2008: Initial version. (McAdams, David E.)

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