Shortest distance between a point and a line

Here is Basic Geometry problem .Which will help you to understand ,finding a point of interaction and the Shortest distance between a point and a line .

We are using Distance Formula to solve this .




Find the Shortest Distance between to the given point and the Line
(2,3) , y=x-6

Answer:

The slope of Line y=x-6 is equal to 1
The slope of the perpendicular line is -1
The equation of the line joining the point (2,3) and the perpendicular line is given by

(y-y1)=m(x-x1)

Here m is the slope which is -1
Substitute x1 as 2 and y1 as 3 (given in question )

y-3 = -1(x-2)

y-3 = -x+2

x+y-5=0  -----1

Then solve the following system of equations to get the point of intersection

  y =x-6 we can write it as x-y =6
                   x+y=5 (1)
           (add) ________________

                   2x=11

                        11
                     X= --
                         2           

Substitute this in y = x-6

 

     11
y =  -----  -  6
      2


      - 1
y  =  -----
       2

So the point of intersection is { 11/2,-1/2}.
Now the shortest distance between This and (2,3) is given by using distance formula
d = √[(2-11/2)²+(3+1/2)²]
= 4.95
approximately 5.