Skill: Determine an endpoint given the midpoint and another endpoint

Explanation and Free Practice Resources

The midpoint of a segment is the coordinate-wise average of its endpoints, so when one endpoint and the midpoint are known, the other endpoint is found by reversing that averaging: $x_2=2x_M-x_1$ and $y_2=2y_M-y_1$ . This reasoning applies to positive, negative, integer, and simple fractional coordinates on the Cartesian plane, emphasizing equal horizontal and vertical displacements; three-dimensional, vector-based, and more abstract generalizations are outside this scope.

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Detailed Explanation: Determine an endpoint given the midpoint and another endpoint

The midpoint is halfway between the two endpoints, so its coordinates are the averages of the endpoint coordinates:

x_M=\frac{x_1+x_2}{2}, \qquad y_M=\frac{y_1+y_2}{2}

To find a missing endpoint, reverse the averaging:

x_2=2x_M-x_1, \qquad y_2=2y_M-y_1

Example: One endpoint is $A(-2,5)$ , and the midpoint is $M(3,1)$ . Find the other endpoint $B(x_2,y_2)$ .

First find the $x$ -coordinate:

x_2=2(3)-(-2)=6+2=8

Then find the $y$ -coordinate:

y_2=2(1)-5=2-5=-3

Therefore, the other endpoint is

\boxed{B(8,-3)}

Check the midpoint:

\left(\frac{-2+8}{2},\frac{5+(-3)}{2}\right) = \left(\frac{6}{2},\frac{2}{2}\right) = (3,1)

The midpoint is correct because it is equally far from both endpoints horizontally and vertically.

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Line Segment (Points) - Find Endpoint from Midpoint (Coordinate)

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