Midpoint Calculator

Calculate the midpoint of two points with precision. Understand the midpoint of a line segment.

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 (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the midpoint is calculated using:

M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

For higher dimensions, the formula extends in the same pattern. For example, in 3D space with points (x1,y1,z1)(x_1, y_1, z_1) and (x2,y2,z2)(x_2, y_2, z_2), the midpoint is:

M=(x1+x22,y1+y22,z1+z22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)

Example calculation

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

  1. Find the average of the x-coordinates: 3+72=102=5\frac{3 + 7}{2} = \frac{10}{2} = 5
  2. Find the average of the y-coordinates: 5+92=142=7\frac{5 + 9}{2} = \frac{14}{2} = 7
  3. The midpoint is (5,7)(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.