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Determine an endpoint from a midpoint and another endpoint

The midpoint of a segment is the point whose coordinates are the averages of the corresponding endpoint coordinates. Given a midpoint M(xm,ym)M(x_m,y_m) and one endpoint A(x1,y1)A(x_1,y_1), the other endpoint is B(2xm−x1,  2ym−y1)B(2x_m-x_1,\;2y_m-y_1), reflecting equal horizontal and vertical displacements on the coordinate plane; coordinates may include integers, decimals, or simple fractions, without requiring vector or higher-dimensional generalizations.

Detailed Explanation: Determine an endpoint from a midpoint and another endpoint

The midpoint of a segment is found by averaging the corresponding coordinates of its endpoints.

If the midpoint is (M(xm,ym))(M(x_m,y_m)) and one endpoint is (A(x1,y1))(A(x_1,y_1)), then the other endpoint BB is

B(2xm−x1,  2ym−y1).B(2x_m-x_1,\;2y_m-y_1).

This works because the midpoint must be halfway between the two endpoints in both the horizontal and vertical directions.

Example: Find endpoint BB if the midpoint is (M(4,3))(M(4,3)) and one endpoint is (A(1,7))(A(1,7)).

  1. Use the formula:

B(2xm−x1,  2ym−y1) B(2x_m-x_1,\;2y_m-y_1)

  1. Substitute the coordinates:

B(2(4)−1,  2(3)−7) B(2(4)-1,\;2(3)-7)

  1. Simplify each coordinate:

B(8−1,  6−7) B(8-1,\;6-7) B(7,−1) B(7,-1)

Therefore, the missing endpoint is

B(7,−1).\boxed{B(7,-1)}.

You can check the answer by averaging the endpoints:

(1+72,7+(−1)2)=(82,62)=(4,3),\left(\frac{1+7}{2},\frac{7+(-1)}{2}\right) = \left(\frac{8}{2},\frac{6}{2}\right) = (4,3),

which is the given midpoint.

Learn by doing: Determine an endpoint from a midpoint and another endpoint

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

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