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Determine an endpoint given the midpoint and another endpoint

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: x2=2xM−x1x_2=2x_M-x_1 and y2=2yM−y1y_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.

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:

xM=x1+x22,yM=y1+y22x_M=\frac{x_1+x_2}{2}, \qquad y_M=\frac{y_1+y_2}{2}

To find a missing endpoint, reverse the averaging:

x2=2xM−x1,y2=2yM−y1x_2=2x_M-x_1, \qquad y_2=2y_M-y_1

Example: One endpoint is A(−2,5)A(-2,5), and the midpoint is M(3,1)M(3,1). Find the other endpoint B(x2,y2)B(x_2,y_2).

First find the xx-coordinate:

x2=2(3)−(−2)=6+2=8x_2=2(3)-(-2)=6+2=8

Then find the yy-coordinate:

y2=2(1)−5=2−5=−3y_2=2(1)-5=2-5=-3

Therefore, the other endpoint is

B(8,−3)\boxed{B(8,-3)}

Check the midpoint:

(−2+82,5+(−3)2)=(62,22)=(3,1)\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.

Learn by doing: Determine an endpoint given the midpoint and another endpoint

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

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