Midpoint Calculator

Published on: September 22, 2024
Final Answer: Free • Full Steps: Plus

This Midpoint Calculator finds the point exactly halfway along the line joining two points on a coordinate plane, and shows the working line by line. The midpoint sits halfway in both directions, so each of its coordinates is the average of the matching coordinates of the two endpoints: add the pair, then halve the total. Following the same pattern every time is what makes the midpoint formula easy to check and repeat for any two points.

Line with midpoint labeled M

Enter the coordinates for \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\).

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the two pairs of coordinates into the formula.
  4. Add the two x-coordinates and the two y-coordinates.
  5. Halve each total to get the midpoint.

Formula:

\(M = \left(\tfrac{x_{1} + x_{2}}{2}, \tfrac{y_{1} + y_{2}}{2}\right)\)
See Example 1 Hide Example 1

Problem

\((x_{1}, y_{1}) = (2, 3),\; (x_{2}, y_{2}) = (10, 9)\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the two pairs of coordinates into the formula.
  4. Add the two x-coordinates and the two y-coordinates.
  5. Halve each total to get the midpoint.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the midpoint of the line joining \((2, 3)\) and \((10, 9)\).

\((x_{1}, y_{1}) = (2, 3),\; (x_{2}, y_{2}) = (10, 9)\)

Step 2

Step 2 - Write the formula.

In this problem: The midpoint sits halfway along the line, so each of its coordinates is the average of the matching coordinates of the two endpoints.

\(M = \left(\tfrac{x_{1} + x_{2}}{2}, \tfrac{y_{1} + y_{2}}{2}\right)\)

Step 3

Step 3 - Substitute the two pairs of coordinates into the formula.

In this problem: Replacing each coordinate in \(M = \left(\tfrac{x_{1} + x_{2}}{2}, \tfrac{y_{1} + y_{2}}{2}\right)\) gives \(M = \left(\tfrac{2 + 10}{2}, \tfrac{3 + 9}{2}\right)\).

\(M = \left(\tfrac{2 + 10}{2}, \tfrac{3 + 9}{2}\right)\)

Step 4

Step 4 - Add the two x-coordinates and the two y-coordinates.

In this problem: The top of each fraction is still written as an addition, so each one is worked out in turn.

\(M = \left(\tfrac{2 + 10}{2}, \tfrac{3 + 9}{2}\right)\)

Step 4a

Step 4a - Add the two x-coordinates and the two y-coordinates.

In this problem: Adding the two x-coordinates gives \(2 + 10 = 12\).

\(M = \left(\tfrac{12}{2}, \tfrac{3 + 9}{2}\right)\)

Step 4b

Step 4b - Add the two x-coordinates and the two y-coordinates.

In this problem: Adding the two y-coordinates gives \(3 + 9 = 12\).

\(M = \left(\tfrac{12}{2}, \tfrac{12}{2}\right)\)

Step 5

Step 5 - Halve each total to get the midpoint.

In this problem: Both totals are still written as halves, so each one is worked out in turn.

\(M = \left(\tfrac{12}{2}, \tfrac{12}{2}\right)\)

Step 5a

Step 5a - Halve each total to get the midpoint.

In this problem: Halving the x total gives \(12 \div 2 = 6\).

\(M = \left(6, \tfrac{12}{2}\right)\)

Step 5b

Step 5b - Halve each total to get the midpoint.

In this problem: Halving the y total gives \(12 \div 2 = 6\).

\(M = \left(6, 6\right)\)

Final Answer

\(M = \left(6, 6\right)\)
See Example 2 Hide Example 2

Problem

\((x_{1}, y_{1}) = (-5, 2),\; (x_{2}, y_{2}) = (7, -8)\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Substitute the two pairs of coordinates into the formula.
  4. Add the two x-coordinates and the two y-coordinates.
  5. Halve each total to get the midpoint.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the midpoint of the line joining \((-5, 2)\) and \((7, -8)\).

\((x_{1}, y_{1}) = (-5, 2),\; (x_{2}, y_{2}) = (7, -8)\)

Step 2

Step 2 - Write the formula.

In this problem: The midpoint sits halfway along the line, so each of its coordinates is the average of the matching coordinates of the two endpoints.

\(M = \left(\tfrac{x_{1} + x_{2}}{2}, \tfrac{y_{1} + y_{2}}{2}\right)\)

Step 3

Step 3 - Substitute the two pairs of coordinates into the formula.

In this problem: Replacing each coordinate in \(M = \left(\tfrac{x_{1} + x_{2}}{2}, \tfrac{y_{1} + y_{2}}{2}\right)\) gives \(M = \left(\tfrac{-5 + 7}{2}, \tfrac{2 + \left(-8\right)}{2}\right)\).

\(M = \left(\tfrac{-5 + 7}{2}, \tfrac{2 + \left(-8\right)}{2}\right)\)

Step 4

Step 4 - Add the two x-coordinates and the two y-coordinates.

In this problem: The top of each fraction is still written as an addition, so each one is worked out in turn.

\(M = \left(\tfrac{-5 + 7}{2}, \tfrac{2 + \left(-8\right)}{2}\right)\)

Step 4a

Step 4a - Add the two x-coordinates and the two y-coordinates.

In this problem: Adding the two x-coordinates gives \(-5 + 7 = 2\).

\(M = \left(\tfrac{2}{2}, \tfrac{2 + \left(-8\right)}{2}\right)\)

Step 4b

Step 4b - Add the two x-coordinates and the two y-coordinates.

In this problem: Adding the two y-coordinates gives \(2 + \left(-8\right) = -6\).

\(M = \left(\tfrac{2}{2}, \tfrac{-6}{2}\right)\)

Step 5

Step 5 - Halve each total to get the midpoint.

In this problem: Both totals are still written as halves, so each one is worked out in turn.

\(M = \left(\tfrac{2}{2}, \tfrac{-6}{2}\right)\)

Step 5a

Step 5a - Halve each total to get the midpoint.

In this problem: Halving the x total gives \(2 \div 2 = 1\).

\(M = \left(1, \tfrac{-6}{2}\right)\)

Step 5b

Step 5b - Halve each total to get the midpoint.

In this problem: Halving the y total gives \(-6 \div 2 = -3\).

\(M = \left(1, -3\right)\)

Final Answer

\(M = \left(1, -3\right)\)

Example Problems

Click one to load it into the box and solve it.