Calculate the distance from the point (3,5) to the line y = x + 4.

Question

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Answer 1

There are standard formulas for this type of problem.   However, it can also be solved by a combination of simple steps.

 

First,  the shortest distance from the point (3,5) to the line y = x +4 (line1) will be along a straight line perpendicular to line1.   Give  the perpendicular line the name line2.   Since the slope of line1 is 1, the slope of line2 will be -1. 

 

Second, since line2 must go through (3,5) and also have a slope of -1,  the point slope form can be used for line2:

  y - 5 = (-1) (x -3)  

So the equation of line2 is  y = -x +8.  

 

Third, the point of intersection of line1 and line2 can be found by solving the set of equations:

    y =  x +4

    y = -x + 8   

  The solution of this set of two equations is   x = 2,  y = 6  i.e. the point (2,6) .

 

Fourth, the distance formula can be used to find the distance between (3,5) and (2,6)  

  d = sqrt(  (3-2)² + (5-6)² )  = sqrt(2)    

  This is the desired distance.