# Length of the Perpendicular from a Point to a Line

#### Introduction:

• The length of the perpendicular from a point to a line measures the shortest distance between the point and the line in a coordinate plane.

#### Formula for the Length of Perpendicular:

• Given a point $P\left({x}_{1},{y}_{1}\right)$ and a line $Ax+By+C=0$, the length of the perpendicular from the point to the line is given by the formula:

#### Steps to Find the Length of Perpendicular:

1. Calculate Numerator:
• Substitute the coordinates of the point into the equation of the line to find $A{x}_{1}+B{y}_{1}+C$.
2. Calculate Denominator:
• Use the coefficients of the line equation ($A$and $B$) to calculate $\sqrt{{A}^{2}+{B}^{2}}$
• .
3. Divide Numerator by Denominator:
• Divide the absolute value of $A{x}_{1}+B{y}_{1}+C$ by $\sqrt{{A}^{2}+{B}^{2}}$
• to find the length of the perpendicular.

#### Example:

Given a line $3x+4y-5=0$ and a point $P\left(2,1\right)$.

1. Calculate Numerator:

• Substitute point coordinates into the line equation:
$3\left(2\right)+4\left(1\right)-5=6+4-5=5$
2. Calculate Denominator:

• Use the coefficients of the line equation: $\sqrt{{3}^{2}+{4}^{2}}=\sqrt{9+16}=\sqrt{25}=5$
1.
2. Find Length of Perpendicular:

#### Importance:

• Determines the shortest distance between a point and a line.
• Crucial in optimization problems, such as finding the minimum distance between objects or locations.

#### Summary:

• The length of the perpendicular from a point to a line is given by a formula involving the coefficients of the line equation and the coordinates of the point.
• This measurement provides essential spatial information and aids in solving various geometric problems involving distances in a coordinate plane.