Solve each problem. Prove that the midpoint M of the line...

Solve each problem. Prove that the midpoint $M$ of the line segment joining endpoints $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right) has coordinates $$ \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) $$ by showing that the distance between P and M is equal to the distance between M and Q and that the sum of these distances is equal to the distance between P and Q.

Solve each problem.
Prove that the midpoint $M$ of the line segment joining endpoints $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right) has coordinates
$$
\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)
$$
by showing that the distance between P and M is equal to the distance between M and Q and that the sum of these distances is equal to the distance between P and Q.

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Key Concepts

Midpoint Formula

The midpoint formula is a fundamental concept in coordinate geometry that provides the coordinates of the point exactly in the middle of a segment connecting two points. By averaging the corresponding coordinates of the endpoints, the formula ((x1 + x2) / 2, (y1 + y2) / 2) ensures that the resulting point is equidistant from both endpoints.

Distance Formula

The distance formula is used to calculate the Euclidean distance between two points in the Cartesian plane. Derived from the Pythagorean theorem, it computes the distance as the square root of the sum of the squared differences of corresponding coordinates, which is essential for verifying that segments are of equal length.

Verification of Equal Distances

Verifying equal distances involves demonstrating that the distance from the midpoint to one endpoint is the same as the distance from the midpoint to the other endpoint. This concept relies on applying the distance formula to both halves of a segment and ensuring that their equality, along with their sum equaling the total distance, confirms the accuracy of the midpoint formula.

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