Determine the midpoint of a line segment

The midpoint of a line segment in the Cartesian plane is the point halfway between its endpoints, with equal distance to each endpoint; its coordinates are found by averaging the corresponding $x$ -coordinates and the corresponding $y$ -coordinates, $M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ , including results that may be fractions or decimals. This understanding supports coordinate proofs, distance relationships, and geometric transformations; the scope is limited to two-dimensional Cartesian coordinates, not three-dimensional or more abstract vector generalizations.

Detailed Explanation: Determine the midpoint of a line segment

The midpoint is the point exactly halfway between the two endpoints of a line segment. To find it, average the two $x$ -coordinates and average the two $y$ -coordinates:

M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Example

Find the midpoint of the segment with endpoints $A(-3,5)$ and $B(7,-1)$ .

Step 1: Identify the coordinates.

x_1=-3,\quad y_1=5,\quad x_2=7,\quad y_2=-1

Step 2: Average the $x$ -coordinates.

\frac{x_1+x_2}{2} =\frac{-3+7}{2} =\frac{4}{2} =2

Step 3: Average the $y$ -coordinates.

\frac{y_1+y_2}{2} =\frac{5+(-1)}{2} =\frac{4}{2} =2

Step 4: Write the midpoint.

\boxed{M(2,2)}

So, the midpoint of the segment from $(-3,5)$ to $(7,-1)$ is $(2,2)$ . Always average corresponding coordinates: the $x$ -coordinates together and the $y$ -coordinates together.

Learn by doing: Determine the midpoint of a line segment

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Line Segment (Points) - Find Midpoint (Coordinate)

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