Perpendicular Distance from a Point to a Line

What is Perpendicular Distance?

The perpendicular distance from a point $P(x_1, y_1)$ to a line $ax + by + c = 0$ is the shortest distance between the point and the line. This distance is always measured along the line perpendicular to the given line.

Formula for Perpendicular Distance

The perpendicular distance dd from a point $P(x_1, y_1)$ to the line $ax + by + c = 0$ is:

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

Components of the Formula

$a, b, c$ : Coefficients from the equation of the line.
$x_1, y_1$ : Coordinates of the given point.
$| \cdot |$ : Absolute value ensures the distance is non-negative.
$\sqrt{a^2 + b^2}$ : Normalizes the direction of the line.

Derivation of the Formula

Start with the equation of the line: $ax + by + c = 0$
Identify the perpendicular line passing through $P(x_1, y_1)$ , which has slope $-\frac{a}{b}$ (negative reciprocal of $\frac{b}{a}$ ).
Find the intersection of the two lines.
Use the distance formula to compute the shortest distance between $P(x_1, y_1)$ and the line.

Worked Examples

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Example 1: Find the Perpendicular Distance

Problem: Find the perpendicular distance from $P(3, -2)$ to the line $4x - 3y + 5 = 0$

Solution:

Step 1: Using the formula:

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

Step 2: Substitute $a = 4, b = -3, c = 5, x_1 = 3, y_1 = -2$

d = \frac{|4(3) + (-3)(-2) + 5|}{\sqrt{4^2 + (-3)^2}} = \frac{|12 + 6 + 5|}{\sqrt{16 + 9}}

= \frac{|23|}{\sqrt{25}} = \frac{23}{5}

Answer: The perpendicular distance is $\frac{23}{5}$ or $4.6$ units.

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Example 2: Check Perpendicularity

Problem: Determine if the point $P(1, 1)$ lies on the line $2x - y - 1 = 0$

Solution:

The perpendicular distance will be zero if the point lies on the line.

Use the formula:

d = \frac{|2(1) - 1(1) - 1|}{\sqrt{2^2 + (-1)^2}} = \frac{|2 - 1 - 1|}{\sqrt{4 + 1}} = \frac{|0|}{\sqrt{5}} = 0

Answer: The point $P(1, 1)$ lies on the line.

Summary

Perpendicular distance formula:

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

The distance is the shortest path between a point and a line.
If $d = 0$ , the point lies on the line.
Practice applying the formula to compute distances efficiently.

Perpendicular Distance from a Point to a Line Simplified Revision Notes for Leaving Cert Mathematics