Midpoint Calculator

Tutorial

This calculator solves two types of midpoint problems. The first type is to find the midpoint if two endpoints are given, while the second type is to find the endpoint if the midpoint and endpoint are given.

Midpoint formula

If A(x₁,y₂) and B(x₁,y₂) are two endpoints of a line segment, then the midpoint formula is:

M((x1+x2)/2 , (y1+y2)/2)

Midpoint problems

Example 1:

Find the midpoint coordinates of the line segment AB with endpoints A(1, 5) and B(7, 1).

In this example the constants x1, y1, x2, and y2 are x1 = 1, y1 = 5, x2 = 7 and y2 = 1.

M((x1+x2)/2, (y1+y2)/2)
M((1+7)/2, (5+1)/2)
M(8/2, 6/2)
M(4, 3)

Hence, the midpoint of the segment AB is M(4, 3).

The following video shows the complete solution.

Midpoint in 3D?

The midpoint formula in three dimensions is similar to the two-dimensional case.

M((x1+x2)/2 , (y1+y2)/2, (z1+z2)/2)

Example 3:

Given that the endpoints are A(-3, 4, 6) and B(2, 1, 12), find the midpoint coordinates.

Solution:

Since the points are in space, we need to find six constants: x1, y1, z1, x2, y2 and z2. x1 = -3, y1 = 4, z1 = 6, x2 = 2, y2 = 1, z2 = 12. Now we can apply midpoint formula.

M((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)
M((-3+2)/2, (4+1)/2, (6+12)/2)
M(1/2, 5/2, 9)

Hence, the midpoint is M(1/2, 5/2, 9).

How to find endpoint?

If M(x_m, y_m) is the midpoint and A(x_m, y_m) is one endpoint, then the formula for the second endpoint is

B(2x_m - x₁, 2x_m - y₁)

Example 3:

Find the endpoint of a line segment if the middle point is M(4,4) and the one endpoint is A(1,2).

B(2x_m-x₁, 2x_m-y₁)
B(2·4-1, 2·4-2)
B(8-1,8-2)
B(7,6)

Hence, the second endpoint is B(7, 6).