Question

What is the formula to find the distance between a point and a line? (in the coordinate plane)
And how do you prove that formula?

Answer 1

The distance from a point to a line is the shortest distance from a point to a line.
For a line in the plane given by the equation ax + by + c = 0, the distance from the line to a point (x0,y0) is\[\frac{ \left| a x_{0}+b y_{0}+c \right| }{ \sqrt{a^{2}+b^{2}}}\]

Answer 2

let the equation of the line be AX +BY + c = 0 and the point be (x0,y0), the the distance be \[d = \frac{ \left| Ax _{o}+By _{0}+c \right| }{ \sqrt{A ^{2}+B ^{2}} }\]
This can be proven using distance formula for two points where the tow points are : one that is given and the other is founded to be on the line that pass through the given point and perpendicular to the given line.