Midpoint of a Line

Definition of Midpoint

The midpoint of a line segment is the point exactly halfway between the two endpoints. It divides the line into two equal parts, so the distances from the midpoint to each endpoint are the same.

If you have two points with coordinates $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint lies directly between them on the coordinate plane.

Midpoint Formula

To find the midpoint, you calculate the average of the x-coordinates and the average of the y-coordinates of the two endpoints. This gives the coordinates of the midpoint.

The formula is:

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

This means you add the x-values of the two points, divide by 2, and do the same for the y-values.

For instance, if you want to find the midpoint between $A(2, 5)$ and $B(6, 9)$, you calculate:

$$x_M = \frac{2 + 6}{2} = \frac{8}{2} = 4, \quad y_M = \frac{5 + 9}{2} = \frac{14}{2} = 7$$

So, the midpoint is $M(4, 7)$.

For example, the midpoint between points $(1, 3)$ and $(5, 7)$ is $\left( \frac{1+5}{2}, \frac{3+7}{2} \right) = (3, 5)$.

Applications of Midpoint

The midpoint is useful in many coordinate geometry problems, such as:

• Finding the centre of a line segment
• Dividing a line segment into two equal parts
• Working out coordinates for bisecting lines

It is often a key step in problems involving shapes, symmetry, and coordinate proofs.