# 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

$$(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}).$$ |

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
$$, and their opposite numbers, if $$, in any case we can write

$${x}_{0}-{x}_{1}={x}_{2}-{x}_{0}.$$ |

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

Title | coordinates of midpoint |
---|---|

Canonical name | CoordinatesOfMidpoint |

Date of creation | 2013-03-22 17:31:07 |

Last modified on | 2013-03-22 17:31:07 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Result |

Classification | msc 51N20 |

Classification | msc 51M15 |

Classification | msc 51-00 |

Related topic | ConjugateDiametersOfEllipse |

Related topic | CentreOfMassOfPolygon |

Related topic | Midpoint4 |