Determine the midpoint of a line segment

The midpoint of a line segment is the unique point halfway between its endpoints, so it is equidistant from both and divides the segment into two congruent parts. In the Cartesian plane, its coordinates are found by averaging the corresponding coordinates: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ , including negative, fractional, horizontal, vertical, and diagonal cases. This supports coordinate proofs and geometric reasoning; higher-dimensional or generalized section formulas are not included.

Detailed Explanation: Determine the midpoint of a line segment

The midpoint of a line segment is the point exactly halfway between its two endpoints. To find it, average the $x$ -coordinates and average the $y$ -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 $A(-4,5)$ and $B(6,-1)$ .

Step 1: Average the $x$ -coordinates.

\frac{-4+6}{2}=\frac{2}{2}=1

Step 2: Average the $y$ -coordinates.

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

Step 3: Write the midpoint.

\boxed{M(1,2)}

So, the midpoint of $\overline{AB}$ is $(1,2)$ . It is halfway between the endpoints, meaning the distance from $A$ to $M$ equals the distance from $M$ to $B$ .

Learn by doing: Determine the midpoint of a line segment

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Line Segment (Points) - Find Midpoint (Coordinate)

Earned ?