midpointm

The midpoint formula is a fundamental concept in coordinate geometry used to find the coordinates of the midpoint of a line segment. A midpoint is the point that divides a line segment into two equal parts.

To find the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) in a two-dimensional Cartesian coordinate system, the midpoint (xm, ym) is calculated by averaging the x-coordinates and averaging the y-coordinates of the endpoints. The formula is: xm = (x1 + x2) / 2 and ym = (y1 + y2) / 2. This means the x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints, and the y-coordinate of the midpoint is the average of the y-coordinates of the two endpoints.

This concept can be extended to higher dimensions. For a line segment in three-dimensional space with endpoints (x1, y1, z1) and (x2, y2, z2), the midpoint (xm, ym, zm) is found using the formula: xm = (x1 + x2) / 2, ym = (y1 + y2) / 2, and zm = (z1 + z2) / 2. The principle remains the same: average the corresponding coordinates of the endpoints.

The midpoint formula has numerous applications in geometry, such as finding the center of a circle given two points on its diameter, or in various geometric proofs and constructions. It is a straightforward but essential tool for analyzing and manipulating geometric figures within a coordinate system.