The Midpoint Formula -

Formula

M = \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Used to calculate:

The coordinates of the point exactly halfway between two points in a two-dimensional Cartesian coordinate plane.

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What Is the Midpoint Formula?

The midpoint formula calculates the coordinates of the point halfway between two endpoints of a line segment. It finds the arithmetic mean of the two x-coordinates and the arithmetic mean of the two y-coordinates.

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Formula and Variables

M = \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Symbol	Meaning	Unit
$M$	Midpoint between the two points	Coordinate units
$x_1$	x-coordinate of the first point	Depends on the coordinate system
$y_1$	y-coordinate of the first point	Depends on the coordinate system
$x_2$	x-coordinate of the second point	Depends on the coordinate system
$y_2$	y-coordinate of the second point	Depends on the coordinate system

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How to Use the Formula

Identify the coordinates of the two endpoints as ( $x_1$ , $y_1$ ) and ( $x_2$ , $y_2$ ). Add the two x-coordinates and divide by two to find the midpoint’s x-coordinate. Then add the two y-coordinates and divide by two to find the midpoint’s y-coordinate.

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When Is This Formula Used?

The midpoint formula is used to locate the point exactly halfway between two points in a two-dimensional Cartesian coordinate system.

Common uses:

– Finding the center of a line segment
– Solving coordinate geometry problems
– Dividing a line segment into two equal parts

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Important Notes

– The midpoint is equidistant from the two endpoints of the line segment.
– The formula assumes coordinates in a Cartesian coordinate system where corresponding coordinates can be averaged linearly.
– Corresponding coordinates should use compatible units and scales.
– A common point of confusion is averaging an x-coordinate with a y-coordinate. The x-coordinates must be averaged together, and the y-coordinates must be averaged together.

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Alternate Forms

Alternate form:

Separate coordinate forms:

x_M = \frac{x_1+x_2}{2}

y_M = \frac{y_1+y_2}{2}

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