coordinates of midpoint

The coordinates of the midpoint of a line segment are the arithmetic means of the coordinates of the endpoints of the segment. Thus, if the endpoints are  $(x_{1},\,y_{1})$  and  $(x_{2},\,y_{2})$,  then the midpoint is

 $\left(\frac{x_{1}\!+\!x_{2}}{2},\,\frac{y_{1}\!+\!y_{2}}{2}\right)\!.$

For justifying the above coordinates of the midpoint, we know that its abscissa $x_{0}$ halves on the $x$-axis the line segment between $x_{1}$ and $x_{2}$. Since the lengths of the half-segments are $x_{0}\!-\!x_{1}$ and $x_{2}\!-\!x_{0}$, if  $x_{1},  and their opposite numbers, if  $x_{2},  in any case we can write

 $x_{0}-x_{1}=x_{2}-x_{0}.$

Solving this equation for $x_{0}$ yields:  $\displaystyle x_{0}=\frac{x_{1}\!+\!x_{2}}{2}$.  result is gotten for the ordinate of the midpoint.

Title coordinates of midpoint CoordinatesOfMidpoint 2013-03-22 17:31:07 2013-03-22 17:31:07 pahio (2872) pahio (2872) 5 pahio (2872) Result msc 51N20 msc 51M15 msc 51-00 ConjugateDiametersOfEllipse CentreOfMassOfPolygon Midpoint4