Midpoint of a Line Segment(Coordinate Geometry)

A line segment on the coordinate plane is defined by two endpoints whose coordinates are known. The midpoint of this line is exactly halfway between these endpoints and its location can be found using the Midpoint Theorem, which states:

• The x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints.
• Likewise, the y-coordinate is the average of the y-coordinates of the endpoints.

To see this graphically, in the figure above turn off the grid and turn on the pointers. As you drag either A or B, you will see that on each axis, the black pointers from the midpoint C are always exactly halfway between the orange pointers from the endpoints A and B.

Worked example

1. In the figure above, press 'reset'.
2. First find the x-coordinate of C.
   This is the average of the x-coordinates of A and B. The coordinates of A are (10,20) so the x-coordinate is 10, the first number of the pair. Similarly, the x-coordinate of B is 50. To find the average of these, add them together and divide the result by two:
3. Next, find the y-coordinate of C.
   This is the average of the y-coordinates of A and B. The coordinates of A are (10,20) so the y-coordinate is 20, the second number of the pair. Similarly, the y-coordinate of B is 10. To find the average of these, add them together and divide the result by two:
   
4. So now we know that the midpoint C has the coordinates (30,15). Verify this in the figure above.

Things to try

1. In the above diagram, Drag the points A and B around and notice how the midpoint is always in the center of the line, and its coordinates are continuously calculated as you drag them.
2. Click "hide details". Drag A and B to some new locations and calculate the midpoint yourself. Then click "show details" and see how close you got. (Note: the coordinates in the diagram above are rounded off to whole numbers for clarity).

Limitations

In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place. This can cause calculatioons to be slightly off.

For more see Teaching Notes

Other Coordinate Geometry topics

• Introduction to coordinate geometry
• The coordinate plane
• The origin of the plane
• Axis definition
• Coordinates of a point
• Distance between two points

• Introduction to Lines
  in Coordinate Geometry
• Line (Coordinate Geometry)
• Ray (Coordinate Geometry)
• Segment (Coordinate Geometry)
• Midpoint Theorem

• Distance from a point to a line
• - When line is horizontal or vertical
• - Using two line equations
• - Using trigonometry
• - Using a formula

• Intersecting lines

• Cirumscribed rectangle (bounding box)
• Area of a triangle (formula method)
• Area of a triangle (box method)
• Centroid of a triangle
• Incenter of a triangle
• Area of a polygon
• Algorithm to find the area of a polygon
• Area of a polygon (calculator)
• Rectangle
• Definition and properties, diagonals
• Area and perimeter
• Square
• Definition and properties, diagonals
• Area and perimeter
• Trapezoid
• Definition and properties, altitude, median
• Area and perimeter
• Parallelogram
• Definition and properties, altitude, diagonals

• Print blank graph paper