Understanding the concept of midpoint coordinates is fundamental in the study of coordinate geometry. It involves finding a point that divides a line segment into two equal parts. This point, known as the midpoint, is crucial in various applications such as geometry, design, and even navigation. The midpoint formula provides a straightforward method for calculating the coordinates of this point using the coordinates of the endpoints of the line segment.

The Midpoint Formula

The midpoint (M) of a line segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by the formula:

This formula calculates the average of the x-coordinates and the y-coordinates of the endpoints to find the coordinates of the midpoint.

Image courtesy of Chilimath

Key Concepts

• Midpoint: The point that divides a line segment into two equal parts.
• Endpoint: The points at either end of a line segment.
• Coordinate: The numerical values that determine the position of a point in a plane, usually defined by x (horizontal) and y (vertical) axes.

Worked Examples

Let's go through a few examples to understand how to apply the midpoint formula in practical scenarios.

Example 1: Basic Calculation

Given a line segment with endpoints $A(2, 3)$ and $B(4, 7)$, find the midpoint.

Solution:

Applying the midpoint formula:

Hence, the midpoint $M$ of the line segment $AB$ is at coordinates $(3, 5)$.

Example 2: Real-world Application

Suppose you want to find the central point between two cities on a map with coordinates: City $A(-8, 4)$ and City $B(12, -6)$.

Solution:

By applying the midpoint formula:

Therefore, the midpoint $M$, or the central point between the two cities, is at $(2, -1)$.

Example 3: Further Practice

Find the midpoint of a line segment with endpoints $C(-5, -1)$ and $D(3, 9)$.

Solution:

Using the formula:

The midpoint $M$ is located at $(-1, 4)$.