LINE THRU DISTANCE

Question

Find the shortest distance from point (2, 3) to the line through (-3, 2) and (1, -4).

Answer 1

Find the shortest distance from point (2, 3) to the line through (-3, 2) and (1, -4).

Two points determine a line. This problem mentions “the line through” points (-3,2) and (1,-4). First, determine the equation for that line.

The “shortest distance” from a point to that line creates another line that is perpendicular to the original line. Now, remember that perpendicular lines have slopes that are the negative reciprocal of each other [that is, m and (-1/m)].

With the slope of the perpendicular line and a point on that line, you may determine the equation of the perpendicular line.

 

The intersection point MUST satisfy the equations for both lines. Therefore, you must solve the system of equations to find the intersection point.

Finally, the distance formula (using the original point and the intersection point) calculates the distance from the point to the original line.

 

Now, PLZ allow me to quote Mark M. from yesterday (and PLZ follow along, considering the above information):
   “Using the two point form determine the equation of the line through (-3, 2) and (1, -4)
    Note the slope and take the negative reciprocal.
    Use the negative reciprocal for the slope of the line passing through (2, 3) - use point-slope form.
    Solve the system of the two equations to determine the point of intersection.
    Determine the distance from (2, 3) to the point of intersection.”

O.K., here are some results:

 

Using the two point form determine the equation of the line through (-3, 2) and (1, -4)
                    y = (-3/2)x – 11/2

Note the slope and take the negative reciprocal.
                    Slope=(-3/2), negative reciprocal=2/3

Use the negative reciprocal for the slope of the line passing through (2, 3) - use point-slope form.

                Final equation of perpendicular line:  y = (2/3)x   5/3

Solve the system of the two equations to determine the point of intersection.

               Point of intersection: (-25/13,5/13)

Determine the distance from (2, 3) to the point of intersection.
                d = 17*√(13) / 13

= = = = = =

Now, according to Wikipedia (Distance from a Point to a Line):

 

If a line passes through two points P₁=(x₁,y₁) and P₂=(x₂,y₂) then the distance from (x₀,y₀) from the line is:
        d(P₁,P₂,(x₀,y₀)) = | (y₂ – y₁)x₀ – (x₂ – x₁)y₀ + x₂y₁ – y₂x₁ | / √( (y₂-y₁)² + (x₂-x₁)² )

[full disclosure: When I put values into a spreadsheet, I couldn’t get them to match. So, there’s work for you to do.]

Answer 2

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