How to find the distance between a point and a line?

The shortest distance between a point and a line is known as the perpendicular distance between them.

Answer: The distance of a point from the line is given by the formula: (a\(x_1\) + b\(y_1\) + c) / √(a² + b²).

Let's look into the solution.

Explanation:

The distance between a point P and a line AB is the shortest distance between P and AB i.e. it is the minimum length required to move from point P to a point on AB.

To find the distance between a point and the line, we need to first find the shortest distance between the point and the line. Such a line can be found by drawing a perpendicular from the point on the line.

Let's look at the image given below to understand the plotting of this perpendicular line.

A line L perpendicular to point P.

Let's understand why is the perpendicular distance known as the shortest distance.

We know that the longest side in a right triangle is the hypotenuse. If we draw the foot of the perpendicular from the point to the line and draw any other segment joining the point to the line, this segment will always be the longest side of the right-angled triangle, formed, which is nothing but the hypotenuse.

Let's look into the formula for the distance between the point and a line.

Let a point be P with the coordinates (\(x_1\),\(y_1\)), and we need to know its distance from the line represented by ax + by + c = 0

The distance of a point from the line is given by the formula: (a\(x_1\)+ b\(y_1\) + c) / √(a² + b²).

Thus, the distance of a point from the line is given by the formula: (a\(x_1\)+ b\(y_1\)+ c) / √(a² + b²).