Show that if M is the midpoint of the line segment with endpoints P(x1, y1) and Q(x2, y2), then

step 1 Okay, so we are discussing the midpoint formula and we have that. If m is the midpoint of the li

Show that if M is the midpoint of the line segment with...

Show that if $M$ is the midpoint of the line segment with endpoints $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right),$ then
$$
d(P, M)+d(M, Q)=d(P, Q) \quad \text { and } \quad d(P, M)=d(M, Q)
$$

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Key Concepts

Euclidean Distance Formula

This formula provides the distance between two points in the plane by calculating the square root of the sum of the squares of the differences in their corresponding coordinates. It forms the basis for measuring lengths in coordinate geometry and is essential in proving relationships like the ones given in the problem.

Midpoint Formula

The midpoint formula calculates the coordinates of the point that divides a segment into two equal parts by averaging the corresponding coordinates of the endpoints. This concept is crucial here as it guarantees that the distances from the midpoint to each endpoint are equal.

Segment Partitioning and Linearity

The property that a segment can be split into parts that sum up to the whole is fundamental in geometry. When a segment is divided by its midpoint, it confirms that the sum of the two equal half-lengths equals the entire segment, which is the underlying idea in showing that the combined distances from the endpoints to the midpoint equals the distance between the endpoints.

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