Midpoint Formula



In this lesson, we will learn

  • the midpoint formula.
  • how to find the midpoint given two endpoints.
  • how to find one endpoint given the midpoint and another endpoint.
  • how to proof the midpoint formula.

We have included a midpoint calculator at the end of this lesson.

Related Topics:
More Algebra Lessons

More Geometry Lessons

The Midpoint Formula

Some coordinate geometry questions may require you to find the midpoint of line segments in the coordinate plane. To find a point that is halfway between two given points, get the average of the x-values and the average of the y-values.

The midpoint between the two points (x1,y1) and (x2,y2) is given by the formula

midpoint

For example:

The midpoint of the points A(1,4) and B(5,6) is


    midpoint

Find the midpoint given two endpoints

We can use the midpoint formula to find the midpoint when given two endpoints.

Example:
Find the midpoint of the two points A(1, -3) and B(4, 5).

Solution:
Midpoint = = (2.5, 1)

 

Worksheet 1, Worksheet 2 to calculate the midpoint.



This video gives the formula for finding the midpoint of two points and does one example to find the midpoint.



This video gives another example of the midpoint formula given coordinates in fractions.



Find an endpoint when given a midpoint and another endpoint

We can use the midpoint formula to find an endpoint when given a midpoint and another endpoint.

Example:
M(3, 8) is the midpoint of the line AB. A has the coordinates (-2, 3), Find the coordinates of B.

Solution:
Let the coordinates of B be (x, y)

Coordinates of B = (8, 13)

The following video gives an example of finding a missing endpoint if given the midpoint and another endpoint.



The following video shows how to find the other endpoint using the Midpoint Formula.



Proof of the Midpoint Formula

Midpoint of a line segment (derivation)
This video shows how to derive the midpoint formula.



We can use the Pythagorean theorem to prove the midpoint formula.
The following video gives a proof of the midpoint formula using the Pythagorean Theorem.
In the following video we prove the Midpoint Formula by using the distance formula to show the midpoint creates two congruent segments & the slope formula to show that the coodinate of the midpoint is located on the line segment.





Midpoint Calculator

Enter the coordinates of two points and the midpoint calculator will give the midpoint of the two points. Use this to check your answers.





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