Midpoint of a Line Segment

What is the midpoint of a line segment?

A line segment is a straight line bounded by two endpoints, which we’ll call $ A $ and $ B $.

segment AB

The midpoint is the point located exactly in the middle of the segment, dividing it into two equal parts, or as we say in mathematical terms, into two “congruent” segments.

the midpoint of the segment

In other words, if the midpoint is $ M $, then $ AM = MB $.

This definition is more than just a geometric property: the midpoint represents a concept of balance and symmetry. It often appears in real-world applications, such as structural design or the equitable division of spaces and resources.

How do you find the midpoint?

The method to determine the midpoint of a segment varies slightly depending on the context:

1] On a number line

Suppose the points $ A $ and $ B $ have coordinates $ x_1 $ and $ x_2 $ on a number line. The midpoint is found by calculating the arithmetic mean of the coordinates:

$$ M = \frac{x_1 + x_2}{2} $$

For example, consider a segment $ AB $ on a line with endpoints $ A = 2 $ and $ B = 8 $.

example

We calculate the midpoint:

$$ M = \frac{2 + 8}{2} = 5 $$

The midpoint is at $ M = 5 $ on the line, exactly halfway along the segment.

the midpoint is at M=5

As you can see, point $ M $ is equidistant from the endpoints $ A $ and $ B $ of the segment.

It divides the segment $ AB $ into two smaller segments, $ AM $ and $ MB $, of equal length, or “congruent” segments.

$$ AM \cong MB $$

Two segments are considered congruent if they can be perfectly superimposed point by point, leaving no part uncovered.

2] In the Cartesian plane

If points $ A $ and $ B $ have coordinates $ (x_1, y_1) $ and $ (x_2, y_2) $, the midpoint is calculated by considering both the $ x $ and $ y $ coordinates:

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

For example, let’s assume two points with coordinates $ A(2, 3) $ and $ B(8, 7) $.

the midpoint of the segment

To find the midpoint $ M $, we calculate the average of the $ x $- and $ y $-coordinates of the points:

$$ M = \left( \frac{2+8}{2}, \frac{3+7}{2} \right) = (5, 5) $$

Thus, the midpoint $ M $ of the segment is located at the coordinates $ (5,5) $ in the plane.

the midpoint of the segment

Here too, the midpoint $ M = (5,5) $ lies exactly halfway along the segment $ AB $, equidistant from the endpoints $ A $ and $ B $.

The segments $ AM $ and $ MB $ have the same length and are congruent.

$$ AM \cong MB $$

Finding the midpoint of a segment reminds us that often, balance lies at the center.