Distance of a Point From a Line

Summary

This math recipe will help you find the perpendicular distance of point from a line in the XY plane.

Skill Level

Easy

Time

Approx. 2 min

Ingredients

Coordinates of the point: \(A(x_1, y_1)\)

Equation of the Line: ax + by + c = 0

Method

To find the distance d of the point from the line, we’ll use the following formula:

\(d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2+b^2}}\)

Examples

Example 1 Find the distance of the point P(2, 3) from the line 3x – 4y – 9 = 0.

Solution The figure shows the point P and the line.

Distance of a Point from a Line

To find the distance d, we’ll use the above formula directly.

\(d = \frac{|3(2) – 4(3) – 9|}{\sqrt{3^2+(-4)^2}}\)

\(= \frac{|6 – 12 – 9|}{\sqrt{9+16}}\)

\(= \frac{|-15|}{\sqrt{25}}\)

\(= \frac{15}{5}\)

= 3 units

Example 2 Find the distance of the point \(A(3, –3)\) from the line \(3x + y – 6 = 0\).

Solution To find the distance \(d\), we’ll use the above formula directly.

\(d = \frac{|3(3) + (–3) – 6|}{\sqrt{3^2+1^1}}\)

\(= \frac{|9 – 3 – 6|}{\sqrt{9+1}}\)

\(= \frac{|0|}{\sqrt{10}}\)

= 0 units

This means that the point A lies on the given line. Here’s a figure showing the point and the line.

Distance of a Point from a Line

That’s it for this recipe. Hope you found it helpful.

For more recipes, please visit www.doubleroot.in/recipes.

You can follow me on Instagram, Twitter, or Facebook to get all updates.

Scroll to Top