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All courses Math II · G-GPE.6 44 of 73
Find the point that partitions a directed segment between two points in a given ratio.

Find the midpoint of a segment

Problem
Find the midpoint of the segment with endpoints (0,0) and (6,8).
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Hint

Average 0 and 6 for the x-coordinate, and average 0 and 8 for the y-coordinate.

The midpoint formula averages matching coordinates: ((x1+x2)/2, (y1+y2)/2).

Solution walkthrough

01

Use the coordinate midpoint theorem

\[M=((x_1+x_2)/2,(y_1+y_2)/2)\]

The midpoint is halfway in both coordinate directions, so each coordinate is the arithmetic mean of the matching endpoint coordinates.

02

Average the x-coordinates

\[x_M=(0+6)/2=6/2=3\]

The endpoint x-values are 0 and 6; their halfway value is 3.

03

Average the y-coordinates

\[y_M=(0+8)/2=8/2=4\]

The endpoint y-values are 0 and 8; their halfway value is 4.

04

Combine and check

\[M=(3,4);~M-(0,0)=<3,4\ge~(6,8)-M\]

The displacement from the first endpoint to M equals the displacement from M to the second endpoint, confirming that M bisects the segment.

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Another way

  1. Take half of the displacement <6,8>, obtaining <3,4>, and add it to the starting endpoint.

!

Common mistake

Add and then divide each coordinate sum by 2. The coordinate total (6,8) is the full displacement, not the midpoint.