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 $P({x}_{1},{y}_{1})$ and a line $Ax+By+C=0$, the length of the perpendicular from the point to the line is given by the formula:
$$\text{LengthofPerpendicular}=\frac{\mathrm{\mid}A{x}_{1}+B{y}_{1}+C\mathrm{\mid}}{\sqrt{{A}^{2}+{B}^{2}}}$$
Steps to Find the Length of Perpendicular:
 Calculate Numerator:
 Substitute the coordinates of the point into the equation of the line to find $A{x}_{1}+B{y}_{1}+C$.
 Calculate Denominator:
 Use the coefficients of the line equation ($A$and $B$) to calculate $\sqrt{{A}^{2}+{B}^{2}}$
 .
 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 $3x+4y5=0$ and a point $P(2,1)$.

Calculate Numerator:
 Substitute point coordinates into the line equation:
$$3(2)+4(1)5=6+45=5$$
 Substitute point coordinates into the line equation:

Calculate Denominator:
 Use the coefficients of the line equation: $\sqrt{{3}^{2}+{4}^{2}}=\sqrt{9+16}=\sqrt{25}=5$$ =5$

Find Length of Perpendicular:
 $\text{LengthofPerpendicular}=\frac{\mathrm{\mid}5\mathrm{\mid}}{5}=1$
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.