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.
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. |
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 . Click on the blue points in manipulative 1 and drag them to change the figure.
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 and is .
The algorithm for calculating an endpoint in 2-dimensional space can be generalized for n-Dimensional Space. Given two points and the midpoint is .
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, .
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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