Midpoint Calculator 2D: Find the Center of a Line Segment

What is the Midpoint of a Line Segment?

The midpoint of a line segment is the point that divides the segment into two equal parts. It's located exactly halfway between the two endpoints of the line segment.

How to Calculate the Midpoint

To find the midpoint of a line segment, we use the coordinates of its endpoints. The process involves finding the average of the x-coordinates and the average of the y-coordinates.

Formula

The formula for finding the midpoint (x, y) of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is:

\[ x = \frac{x_1 + x_2}{2}, \quad y = \frac{y_1 + y_2}{2} \]

Where:

• (x, y) are the coordinates of the midpoint
• (x₁, y₁) are the coordinates of the first endpoint
• (x₂, y₂) are the coordinates of the second endpoint

Calculation Steps

1. Identify the coordinates of the two endpoints
2. Add the x-coordinates and divide by 2
3. Add the y-coordinates and divide by 2
4. The resulting (x, y) is the midpoint

Example and Visual Representation

Let's find the midpoint of a line segment with endpoints A(1, 2) and B(5, 8):

Step 1: We have (x₁, y₁) = (1, 2) and (x₂, y₂) = (5, 8)

Step 2: x = (x₁ + x₂) / 2 = (1 + 5) / 2 = 6 / 2 = 3

Step 3: y = (y₁ + y₂) / 2 = (2 + 8) / 2 = 10 / 2 = 5

Step 4: The midpoint M is (3, 5)

In this diagram, you can see the line segment AB in blue, with endpoints A and B in red. The green point M represents the midpoint we calculated. Notice how M is exactly halfway between A and B, both horizontally and vertically.

Understanding the midpoint formula allows us to easily find the center of any line segment, which is crucial in many geometric calculations and real-world applications, from construction to computer graphics.