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
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Shortest distance from a point to a line: the distance between the line and point is
The equation can be written:
So the perpendicular line through (x1,y1) is:
---- = ---- = where t is a parameter. a b
t will be the distance from the point along the perpendicular line to (x,y).
This meets the given line ax+by+c = 0 where:
a(x1+a.t/sqrt(a^2+b^2)) + b(y1+b.t/sqrt(a^2+b^2)) + c = 0
ax1 + by1 + c + t(a^2+b^2)/sqrt(a^2+b^2) + c = 0
ax1 + by1 + c + t.sqrt(a^2+b^2) = 0
t.sqrt(a^2+b^2) = -(ax1+by1+c)
t = -(ax1+by1+c)/sqrt(a^2+b^2)
Therefore the perpendicular distance from (x1,y1) to the line ax+by+c = 0 is:
ax1 + by1 + c |t| = ------------- sqrt(a^2+b^2)