Midpoint Calculator

The midpoint formula provides a simple way to find the exact center between two points in a coordinate plane. This concept is fundamental in geometry and has practical applications in many fields.

What is a midpoint

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.

The midpoint formula

If you have two points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint is calculated using:

For higher dimensions, the formula extends in the same pattern. For example, in 3D space with points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$, the midpoint is:

Example calculation

Let's find the midpoint between points $A(3, 5)$ and $B(7, 9)$:

1. Find the average of the x-coordinates: $\frac{3 + 7}{2} = \frac{10}{2} = 5$
2. Find the average of the y-coordinates: $\frac{5 + 9}{2} = \frac{14}{2} = 7$
3. The midpoint is $(5, 7)$

Properties of midpoints

• The midpoint is equidistant from both endpoints
• In a triangle, the line connecting a vertex to the midpoint of the opposite side is called a median
• The medians of a triangle intersect at the centroid, which divides each median in a 2:1 ratio

Applications

• Finding centers of objects
• Calculating average positions
• Constructing perpendicular bisectors
• Locating centers of mass in physics
• Creating proportional divisions in design

The midpoint concept can be extended to find points that divide a line segment in any ratio, using the concept of weighted averages.