# Midpoint. Coordinates of midpoint

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Definition.

**Midpoint**is a point that is located on the segment and is equidistant from the segment endpoints.

In geometric problems can often be faced with the need to find the midpoint of segment that defined by coordinates of its endpoints, for example in search problems median, midline, etc.

Each coordinate of the midpoint is equal to half the sum of the corresponding coordinates of the endpoints.

## Midpoint formulas:

*The formula for calculating the coordinates of midpoint*of a line segment with endpoints A(x_{a}, y_{a}) and B(x_{b}, y_{b}) on plane:x _{c}=x _{a}+ x_{b}y _{c}=y _{a}+ y_{b}2 2 *The formula for calculating the coordinates of midpoint*of a line segment with endpoints A(x_{a}, y_{a}, z_{a}) and B(x_{b}, y_{b}, z_{b}) in space:x _{c}=x _{a}+ x_{b}y _{c}=y _{a}+ y_{b}z _{c}=z _{a}+ z_{b}2 2 2

## Examples of tasks with midpoint

### Examples of tasks with coordinates of midpoint of line segment on the plane

Example 1.

Find the midpoint, point C, which is between the two endpoints A(-1, 3) and B(6, 5).
**Solution.**

x = _{c} |
x + _{a}x _{b} |
= | -1 + 6 | = | 5 | = 2.5 |

2 | 2 | 2 |

y = _{c} |
y + _{a}y _{b} |
= | 3 + 5 | = | 8 | = 4 |

2 | 2 | 2 |

**Answer:** С(2.5, 4).

Example 2.

Find the Coordinates of endpoint, point B, when midpoint is C(1; 5) and other endpoint A(-1, 3).
**Solution.**

x = _{c} |
x + _{a}x _{b} |
=> x = 2_{b}x - _{c}x = 2·1 - (-1) = 2 + 1 = 3_{a} |

2 |

y = _{c} |
y + _{a}y _{b} |
=> y = 2_{b}y - _{c}y = 2·5 - 3 = 10 - 3 = 7_{a} |

2 |

**Answer:** B(3, 7).

### Examples of tasks with coordinates of midpoint of line segment on the space

Example 3.

Find the midpoint, point C, which is between the two endpoints A(-1, 3, 1) and B(6, 5, -3).
**Solution.**

x = _{c} |
x + _{a}x _{b} |
= | -1 + 6 | = | 5 | = 2.5 |

2 | 2 | 2 |

y = _{c} |
y + _{a}y _{b} |
= | 3 + 5 | = | 8 | = 4 |

2 | 2 | 2 |

z = _{c} |
z + _{a}z _{b} |
= | 1 + (-3) | = | -2 | = -1 |

2 | 2 | 2 |

**Answer:** С(2.5, 4, -1).

Example 4.

Find the Coordinates of endpoint, point B, when midpoint is C(1, 5, 2) and other endpoint A(-1, 3, 10).
**Solution.**

x = _{c} |
x + _{a}x _{b} |
=> x = 2_{b}x - _{c}x = 2·1 - (-1) = 2 + 1 = 3_{a} |

2 |

y = _{c} |
y + _{a}y _{b} |
=> y = 2_{b}y - _{c}y = 2·5 - 3 = 10 - 3 = 7_{a} |

2 |

z = _{c} |
z + _{a}z _{b} |
=> z = 2_{b}z - _{c}z = 2·2 - 10 = 4 - 10 = -6_{a} |

2 |

**Answer:** B(3, 7, -6).

Analytic geometry: Introduction and contentsDistance between two pointsMidpoint. Coordinates of midpointEquation of a lineEquation of a planeDistance from point to planeDistance between two planesDistance from a point to a line - 2-DimensionalDistance from a point to a line - 3-DimensionalAngle between two planesAngle between line and plane

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