# Difference between revisions of "Distance formula"

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− | + | ==Shortest distance from a point to a line== | |

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the distance between the line <math>ax+by+c = 0</math> and point <math>(x_1,y_1)</math> is | the distance between the line <math>ax+by+c = 0</math> and point <math>(x_1,y_1)</math> is | ||

<cmath>\dfrac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}</cmath> | <cmath>\dfrac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}</cmath> | ||

− | + | ===Proof=== | |

The equation <math>ax + by + c = 0</math> can be written as <math>y = -\dfrac{a}{b}x - \dfrac{c}{a}</math> | The equation <math>ax + by + c = 0</math> can be written as <math>y = -\dfrac{a}{b}x - \dfrac{c}{a}</math> | ||

Thus, the perpendicular line through <math>(x_1,y_1)</math> is: | Thus, the perpendicular line through <math>(x_1,y_1)</math> is: |

## Revision as of 11:34, 22 October 2015

The **distance formula** is a direct application of the Pythagorean Theorem in the setting of a Cartesian coordinate system. In the two-dimensional case, it says that the distance between two points and is given by . In the -dimensional case, the distance between and is

## Shortest distance from a point to a line

the distance between the line and point is

### Proof

The equation can be written as Thus, the perpendicular line through is: where is the parameter.

will be the distance from the point along the perpendicular line to . So and

This meets the given line , where:

, so:

Therefore the perpendicular distance from to the line is: