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Midpoint Calculator

Midpoint Calculator
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The midpoint of a line is a point that lies halfway between 2 points. The midpoint is the same distance from each endpoint.

How to Calculate the Midpoint

You can find the midpoint of a line segment given 2 endpoints, (x1, y1) and (x2, y2). Calculate the midpoint, (xM, yM) using the midpoint formula:

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

It's important to note that a midpoint is the middle point on a line segment. A true line in geometry is infinitely long in both directions. But a line segment has 2 endpoints so it is possible to calculate the midpoint. A ray has one endpoint and is infinitely long in the other direction.

Example: Find the Midpoint

Say you know two points on a line segment and their coordinates are (6, 3) and (12, 7). Find the midpoint using the midpoint formula.

\( (x_{M}, y_{M}) = \left(\dfrac {x_{1} + x_{2}} {2} , \dfrac {y_{1} + y_{2}} {2}\right) \)
  1. First, add the x coordinates and divide by 2. This gives you the x-coordinate of the midpoint, xM
  2. \( x_{M} = \dfrac {x_{1} + x_{2}} {2} \)
    \( x_{M} = \dfrac {6 + 12} {2} \)
    \( x_{M} = \dfrac {18} {2} \)
    \( x_{M} = {9} \)
  3. Second, add the y coordinates and divide by 2. This gives you the y-coordinate of the midpoint, yM
  4. \( y_{M} = \dfrac {y_{1} + y_{2}} {2} \)
    \( y_{M} = \dfrac {3 + 7} {2} \)
    \( y_{M} = \dfrac {10} {2} \)
    \( y_{M} = {5} \)
  5. Take each result to get the midpoint. In this example the midpoint is (9, 5).

How to Calculate Distance Between 2 Points

If you know the endpoints of a line segment you can use them to calculate the distance between the 2 points. Here you're actually finding the length of the line segment. Use the formula for distance between 2 points:

\( d = \sqrt {(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2} \)

The formula for distance between points is derived from the Pythagorean theorem, solving for the length of the hypotenuse. See our Pythagorean Theorem Calculator for a closer look.

Example: Find the Distance Between 2 Points

You know 2 points on a line segment and their coordinates are (13, 2) and (7, 10). Find the distance between the 2 points using the distance formula \( d = \sqrt {(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2} \)

  1. Insert your points (13, 2) and (7, 10) into the distance equation
  2. \( d = \sqrt {(7 - 13)^2 + (10 - 2)^2} \)
  3. Complete the subtraction first since they're in parentheses
  4. \( d = \sqrt {(-6)^2 + (8)^2} \)
  5. Find the square of each term
  6. \( d = \sqrt {36 + 64} \)
  7. Add the results
  8. \( d = \sqrt {100} \)
  9. Find the square root and you've found the distance between the 2 points
  10. \( d = 10 \)

Similar to this midpoint calculator is our Two Dimensional Distance Calculator. For distance between 2 points in 3 dimensions with (x, y, z) coordinates please see our 3 Dimension Distance Calculator.

How to Calculate Endpoint

If you know an endpoint and a midpoint on a line you can calculate the missing endpoint. Start with the midpoint formula from above and work out the coordinates of the unknown endpoint.

  1. First, take the midpoint formula:
  2. \( (x_{M}, y_{M}) = \left(\dfrac {x_{1} + x_{2}} {2} , \dfrac {y_{1} + y_{2}} {2}\right) \)
  3. And break it down so you have separate equations for the x and y coordinates of the midpoint
  4. \( x_{M} = \dfrac {x_{1} + x_{2}} {2} \)
    \( y_{M} = \dfrac {y_{1} + y_{2}} {2} \)
  5. Since you know the midpoint, insert its coordinates in place of xM and yM in each equation
  6. Insert the coordinates of your known endpoint into the values for x1 and y1
  7. Finally, solve each equation to find x2 and y2 which will be the coordinates of your missing endpoint

Example: Find the Endpoint

Using the steps above, let's find the endpoint of a line segment where we know one endpoint is (6, -4) and the midpoint is (1, 7). The endpoint is the (x1, y1) coordinate. The midpoint is the (xM, yM) coordinate.

  1. First, take the midpoint formula:
  2. \( (x_{M}, y_{M}) = \left(\dfrac {x_{1} + x_{2}} {2} , \dfrac {y_{1} + y_{2}} {2}\right) \)
  3. And break it down so you have separate equations
  4. \( x_{M} = \dfrac {x_{1} + x_{2}} {2} \)
    \( y_{M} = \dfrac {y_{1} + y_{2}} {2} \)
  5. Insert the coordinates of your midpoint (1, 7) in place of xM and yM in each equation
  6. \( 1 = \dfrac {x_{1} + x_{2}} {2} \)
    \( 7 = \dfrac {y_{1} + y_{2}} {2} \)
  7. Insert the coordinates of your known endpoint (6, -4) into the values for x1 and y1
  8. \( 1 = \dfrac {6 + x_{2}} {2} \)
    \( 7 = \dfrac {-4 + y_{2}} {2} \)
  9. Solve each equation to find x2 and y2. First multiply both sides of each equation by 2:
  10. \( 2 = {6 + x_{2}} \)
    \( 14 = {-4 + y_{2}} \)
  11. Add or subtract to isolate the variables on one side of each equation
  12. \( -4 = {x_{2}} \)
    \( 18 = {y_{2}} \)
  13. Your missing endpoint (x2, y2) is (-4, 18)
 

Cite this content, page or calculator as:

Furey, Edward "Midpoint Calculator"; CalculatorSoup, https://www.calculatorsoup.com - Online Calculators

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