How to Find the Midpoint of a Line Segment

Midpoint Between Two Points Video Lesson:

What is the Midpoint?

The midpoint of a line segment is the point that divides the line segment in two. The midpoint is located equally far from each end point. It can be calculated by finding the average of the coordinates of the end points. Add the ๐‘ฅ and y coordinates of the two end points and divide them by two.

A line segment is a portion of a line that has two end points. Unlike a line, a line segment has a fixed length and does not continue on for ever.

The midpoint is found in the middle of a line segment.

If a line segment has two end points at (1, 1) and (9, 7), the midpoint of the line segment is found at (5, 4).

definition of the midpoint of a line segment

The midpoint is always equally far from each end point.

The midpoint (5, 4) is 4 right and 3 up from the end point of (1, 1). It it is 4 left and 3 down from the other end point of (9, 7).

The midpoint is the average of the two end point coordinates.

the midpoint is equally far from either endpoint

The midpoint divides the line segment in the ratio 1:1. This means that the length of the line segment is the same on each side of the midpoint.

How to Find the Midpoint Between 2 Points

To find the midpoint between 2 points, add the x-coordinates of the points and the y-coordinates of the points separately and then divide the result by 2. For example, the midpoint of the points (2, 4) and (6, 8) is found by adding the coordinates to get (8, 12) and then dividing by 2 to get (4, 6).

  1. Add the coordinates

The ๐‘ฅ-coordinates are the first numbers written in each pair of coordinates.

In (2, 4) the ๐‘ฅ-coordinate is 2 and in (6, 8) the ๐‘ฅ-coordinate is 6.

Adding these ๐‘ฅ-coordinates, 2 + 6 = 8

The y-coordinates are the second numbers written in each pair of coordinates.

In (2, 4) the y-coordinate is 4 and in (6, 8) the y-coordinate is 8.

Adding these y-coordinates, 4 + 8 = 12

Adding the coordinates of (2, 4) and (6, 8), we obtain (8, 12).

2. Divide by 2

Dividing both numbers in (8, 12) by 2, we obtain (4, 6).

The midpoint of (2, 4) and (6, 8) is therefore (4, 6).

how to find the midpoint of 2 points

Examples of Finding the Midpoint of 2 Points

Point 1Point 2Sum of CoordinatesMidpoint
(3, 1)(9, 7)(12, 8)(6, 4)
(10, 3)(4, 7)(14, 10)(7, 5)
(4, 4)(0, 0)(4, 4)(2, 2)
(5, 1)(2, 3)(7, 4)(3.5, 2)
(-6, -2)(8, 10)(2, 8)(1, 4)
(-2, 3)(-12, -2)(-14, 1)(-7, 0.5)

Midpoint of a Line Segment Formula

The formula to calculate the midpoint of a line segment is (๐‘ฅm, ym) = ( (๐‘ฅ1+๐‘ฅ2)/2, (y1+y2)/2 ), where (๐‘ฅ1, y1) and (๐‘ฅ2, y2) are the coordinates of each end point. The formula for the midpoint is used to calculate an average of the end points.

open paren x sub m comma y sub m close paren equals open paren the fraction with numerator x sub 1 plus x sub 2 and denominator 2 comma the fraction with numerator y sub 1 plus y sub 2 and denominator 2 close paren

The formula for the midpoint of a line segment

open paren x sub m comma y sub m close paren are the coordinates of the midpoint
open paren x sub 1 comma y sub 1 close paren are the coordinates of the first end point
open paren x sub 2 comma y sub 2 close paren are the coordinates of the second end point

midpoint of a line segment formula

Example

Use the midpoint formula to find the midpoint of (-1, 4) and (7, 2).

  • x sub 1 equals negative 1
  • x sub 2 equals 7
  • y sub 1 equals 4
  • y sub 2 equals 2

The midpoint formula is open paren x sub m comma y sub m close paren equals open paren the fraction with numerator x sub 1 plus x sub 2 and denominator 2 comma the fraction with numerator y sub 1 plus y sub 2 and denominator 2 close paren.

Substituting in the above values, open paren x sub m comma y sub m close paren equals open paren the fraction with numerator negative 1 plus 7 and denominator 2 comma the fraction with numerator 4 plus 2 and denominator 2 close paren.

Evaluating each calculation, open paren x sub m comma y sub m close paren equals open paren six halves comma six halves close paren and so, the midpoint is found at open paren x sub m comma y sub m close paren equals open paren 3 comma 3 close paren.

(3, 3) is the midpoint of (-1, 4) and (7, 2).

example of finding the midpoint between 2 points

Midpoint of a Line Segment Calculator

How to Find the Midpoint of a Line Segment in Steps

The steps to find the midpoint of a line segment are:
  1. Add the x-coordinates of the two end points.
  2. Divide this by 2 to find the x-coordinate of the midpoint.
  3. Add the y-coordinates of the two end points.
  4. Divide this by 2 to find the y-coordinate of the midpoint.

For example, find the midpoint of the line segment between (3, 10) and (1, 12).

1. Add the x-coordinates of the two end points

The x-coordinates are 3 and 1.

3 + 1 = 4

2. Divide this by 2 to find the x-coordinate of the midpoint

4 รท 2 = 2 and so, the x-coordinate of the midpoint is 2.

3. Add the y-coordinates of the two end points

The y-coordinates are 10 and 12.

10 + 12 = 22

4. Divide this by 2 to find the y-coordinate of the midpoint

22 รท 2 = 11 and so, the y-coordinate of the midpoint is 11.

Therefore the midpoint of (3, 10) and (1, 12) is (2, 11).

Midpoint of a Horizontal Line Segment

The midpoint of a horizontal line segment will have the same y-coordinate as any other point on the line segment. The x-coordinate will be half way in between the x-coordinates at each end of the line segment. Simply divide the length of the line segment by 2 and count this distance from either of the end points.

For example, the line segment from (2, 6) to (10, 6) is 8 long.

Therefore the midpoint is found 4 places in from either end point.

The midpoint is at (6, 6). This has the same y-coordinate as each end point.

the midpoint of a horizontal line segment

Midpoint of a Vertical Line Segment

The midpoint of a vertical line segment will have the same x-coordinate as any other point on the line segment. The y-coordinate will be half way in between the y-coordinates at each end of the line segment. Simply divide the length of the line segment by 2 and count this distance from either of the end points.

For example, the line segment from (8, 1) to (8, 7) is 6 long.

Therefore the midpoint is found 3 places in from either end point.

The midpoint is at (8, 4). This has the same y-coordinate as each end point.

midpoint of a vertical line segment

How to Find the End point When Given the Midpoint

Find the difference between the start point of the line segment and the midpoint for both the x and y coordinates. Add this difference on to the midpoint coordinates to find the coordinates of the end point of the line segment.

Example

Find the end point of a line segment if the start point is (1, 1) and the midpoint is (5, 4).

The difference from (1, 1) to (5, 4) is (4, 3).

4 has been added to to the x-coordinate and 3 has been added to the y-coordinate.

To find the end point, add the distance from the start point to the midpoint on to the midpoint.

(4, 3) is added to (5, 4) to obtain (9, 7).

The end point is located at (9, 7).

how to calculate the end point when given the mid point
how to find the endpoint from the midpoint

How to Find the Midpoint with 3D Coordinates

In 3D, the midpoint between the two coordinates (x1, y1, z1) and (x2, y2, z2) is located at (x1+x2)/2, (y1+y2)/2, (z1+z2)/2.

open paren x sub m comma y sub m comma z sub m close paren equals open paren the fraction with numerator x sub 1 plus x sub 2 and denominator 2 comma the fraction with numerator y sub 1 plus y sub 2 and denominator 2 comma the fraction with numerator z sub 1 plus z sub 2 and denominator 2 close paren

Formula for the midpoint of a line segment in three dimensions

Example

Find the midpoint between the two 3D coordinates (3, 0, 6) and (5, 4, 10).

  • x sub 1 equals 3
  • x sub 2 equals 5
  • y sub 1 equals 0
  • y sub 2 equals 4
  • z sub 1 equals 6
  • z sub 2 equals 10

Therefore open paren x sub m comma y sub m comma z sub m close paren equals open paren the fraction with numerator x sub 1 plus x sub 2 and denominator 2 comma the fraction with numerator y sub 1 plus y sub 2 and denominator 2 comma the fraction with numerator z sub 1 plus z sub 2 and denominator 2 close paren becomes open paren the fraction with numerator 3 plus 5 and denominator 2 comma the fraction with numerator 0 plus 4 and denominator 2 comma the fraction with numerator 6 plus 10 and denominator 2 close paren.

Therefore the midpoint is found at open paren 4 comma 2 comma 8 close paren.

Alternatively, simply add the coordinates of (3, 0, 6) and (5, 4, 10) to obtain (8, 4, 16). Then halve this to obtain (4, 2, 8).

how to find the midpoint of a line segment in three dimensions