Distance from a point to a straight line

Let us define an arbitrary point P(x₀; y₀) and a straight line MN: Ax + By + C = 0. We obtain the formula for determining the distance d from the point P to the line MN. Let's write the equation of a straight line as an equation with an angular coefficient:

y = - A ∕ B ⋅ x - C ∕ B.

We find the coordinates of the points of intersection of this straight line with the coordinate axes:

M: x=0  =>   y = - C ∕ B  =>  M(0; - C ∕ B).

N: y=0  =>   x = - C ∕ A  =>  N(- C ∕ A; 0).

Let's draw a straight line EF || MN through the point P. Using the equation of a straight line passing through a given point, and the condition of parallelism of straight lines, we obtain the equation of a straight line EF in the form:

y = - A ∕ B ⋅ x + y₀ + A ∕ B ⋅ x₀.

The distance between the parallel lines MN and EF is equal to the distance from the point P to the line MN, that is PQ=MK=d. Let's ∠NOH=α, then ∠NOH=∠MEK=α, as angles with respectively perpendicular sides. Since ΔEKM - is rectangular, by definition the sine of the angle is

sinα = MK / ME   ⇔   MK = ME⋅sinα.

Coordinates of the point M(0;- C ∕ B), and of the point E(0; y₀ + A ∕ B ⋅ x₀), therefore, the length of the segment ME is:

To find sinα, consider the right triangle ΔMON. According to the Pythagoras' theorem: The rectangular ΔMON is similar to the rectangular ΔHON, therefore,

∠OMN=∠HON=α;    sinα=ON/MN.

Substituting the values of the lengths of the segments ON and MN, we obtain


Thus, for the distance MK we have:


Finally, the formula for determining the distance from the point P(x₀; y₀) to the straight line Ax + By + C = 0 is: