Midpoint Formula

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In these lessons, 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.

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

For example:

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

   

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.

How to use the formula for finding the midpoint of two points?
Example:
Find the midpoint of the two points (5, 8) and (-5, -6).

• Show Step-by-step Solutions

How to use the midpoint formula given coordinates in fractions?
Example:
Determine the midpoint of the two points (2/3, 1/4) and (11/6, 7/9).

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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)

How to find a missing endpoint when given the midpoint and another endpoint?
Example:
In the standard xy-coordinate plane, point M with coordinates (5,4) is the midpoint of AB, and B has coordinates (7, 3). What are the coordinates of A?

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How to solve problems using the Midpoint Formula?
Example:
For a line segment DE, one endpoint is D(6, 5) and the midpoint M(4, 2). Find the coordinates of the other endpoint, E.

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Proof of the Midpoint Formula

How to derive the midpoint formula by finding the midpoint of a line segment?

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How to use the Pythagorean theorem to prove the midpoint formula?
The following video gives a proof of the midpoint formula using the Pythagorean Theorem.
Step 1: Use the distance formula to show the midpoint creates two congruent segments. Step 2: Use the slope formula to show that the coordinate of the midpoint is located on the line segment.

• Show Step-by-step Solutions

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.

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

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