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(x1,y1) and a line Ax+By+C=0, the length of the perpendicular from the point to the line is given by the formula:
    Length of Perpendicular=Ax1+By1+CA2+B2

Steps to Find the Length of Perpendicular:

  1. Calculate Numerator:
    • Substitute the coordinates of the point into the equation of the line to find Ax1+By1+C.
  2. Calculate Denominator:
    • Use the coefficients of the line equation (Aand B) to calculate A2+B2
    • .
  3. Divide Numerator by Denominator:
    • Divide the absolute value of Ax1+By1+C by A2+B2
    • to find the length of the perpendicular.

Example:

Given a line 3x+4y5=0 and a point P(2,1).

  1. Calculate Numerator:

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

    • Use the coefficients of the line equation: 32+42=9+16=25=5

 

  1.  
  2. Find Length of Perpendicular:

    • Length of Perpendicular=55=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.