Shortest Distance from a Point to a Line

Question

Determine the Shortest distance from the point E  (-3,-2) to the line through points F (0,12) and G (6,0).

Answer 1

Shortest distance from the point E  (-3,-2) to the line

through points F (0,12) and G (6,0)

 

First find the equation of the line through (0,12), (6,0)

slope:  m = (12-0)/(0-6) = 12/(-6) = -2

 

You have a slope of m=-2 and two points..

You can use the point slope formula

y-y₁ = m(x-x₁) with either of the points.

I will use point G (6,0)

 

y-0 = -2(x-6)

y = -2x + 12

 

The shortest distance will be a perpendicular line

from point (-3,-2) through this line.

Since perpendicular lines have slopes that are

negative reciprocals, the slope of our new line

will be -1/m using m=-2

Our perpendicular line has slope m = -1/-2 = 1/2

 

The equation can be found the same way using

m=1/2 and point (-3, -2)

 

y-(-2) = (1/2)(x-(-3))

y+2 = (x+3)/2

2y + 4 = x+3

2y = x - 1

 

At this point we have the equation of two perpendicular

lines.  The original line through points F and G

and the perpendicular line through point E.

We need to know the point where the lines intersect.

 

1)  y = -2x + 12

2) 2y = x - 1

 

Let's put them in standard format ax + by = c

 

1) 2x + y = 12

2) x - 2y = 1

 

multiply equation 1 by 2 and add to eliminate y

 

1)  4x + 2y = 24

2)    x - 2y = 1

   -----------------

       5x      = 25

        x = 5

 

Now that we have the x value plug into either

equation to find the y value.

 x-2y = 1

5-2y = 1

-2y = -4

y = 2

 

The intersecting point is (5,2)

 

The final part of the problem is to find the distance

from point E (-3,-2) to the intersecting point (5,2)

 

Use the distance formula

d = √[(x₂-x₁)²+(y₂-y₁)²]  using (-3,-2) and (5,2)

 

d = √[(5-(-3))² + (2-(-2))²]

 

d = √(8² + 4²) = √(64+16) = √80 = √(16(5)) = 4√5