Distance from a point to a line: Difference between revisions

Revision as of 08:53, 11 April 2024

Revision as of 22:27, 11 May 2024

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and finally obtain:<ref>{{harvnb|Ballantine|Jerbert|1952}}</ref>

:<math> |\overline{PR}| = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}.</math>

A variation of this proof is to place V at P and compute the area of the triangle ∆''UVT'' two ways to obtain that <math>D|\overline{TU}| = |\overline{VU}||\overline{VT}|</math>

where D is the altitude of ∆''UVT'' drawn to the hypotenuse of ∆''UVT'' from ''P''. The distance formula can then used to express <math>|\overline{TU}|</math>, <math>|\overline{VU}|</math>, and <math>|\overline{VT}|</math>in terms of the coordinates of P and the coefficients of the equation of the line to get the indicated formula.{{citation needed|date=April 2015}}

===A vector projection proof===