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 10:34, 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: