# Distance formula

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

Proof:

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).

So

and

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

so

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)