Midpoint of a Line Segment and the Midpoint Formula

Midpoint of Line Segment Formula top left.Midpoint of Line Segment Formula top right.
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The midpoint of a line segment is the point M on the segment such that:

Dist(P1, M) = Dist(P2, M)

“Dist” is an abbreviation for distance. P1 is a point on a line segment and P2 is a different point on the same line segment. M is a point ½ the distance between P1 and P2, and represents an exact same distance from P1 or P2.

Similar to the Distance Formula, to understand the Midpoint Formula we must consider a horizontal line segment, a vertical line segment and a line segment that has slope, is neither horizontal or vertical.

P1 = (x1, y1)

P2 = (x2, y2)

(Both P1 and P2 are Cartesian Coordinates)

Δx = Run = x2 − x1

Δy = Rise = y2 − y1

(Same definition as Linear Equation Slope where y = mx + b and the slope m = Rise / Run or m = Δy / Δx)

The Midpoint of a line segment is a middle between two different points on that line.

“To grasp the logic explaining the Midpoint Formula it is recommended to first understand the Distance Formula.”

Please review the Cartesian coordinate image (above right) that illustrates P1,and P2, on a sloped line segment.

Should y1 = y2, then y2 − y1 = 0, the line segment is horizontal with length x2 − x1:

M = ((x2 + x1) / 2, y1) where (x2 + x1) / 2 is the average of two x coordinate values.

Should x1 = x2, then x2 − x1 = 0, the line segment is vertical with length y2 − y1:

M = (x1, (y2 + y1) / 2) where (y2 + y1) / 2 is an average of the two y coordinate values.

For a sloped line the midpoint is ½ the distance between point P1 and point P2, and is given as the point defined by coordinates:

M = ((x2 + x1) / 2, (y2 + y1) / 2)

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