Short Distance

Question

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

Answer 1

If we draw a line from the point (2, 3) to the line going through the other points, we would see that the line perpendicular will be closest to the line going through the other points.

 

First, find the slope between the two points the line goes through

 

slope = (-4 - 2) / (1 - (-3))

         = -6 / 4

         = - 3 / 2

 

Next, we find the point on the main line that makes a perpendicular line to the point (2, 3).  Before we do that, lets find the equation of the line going through the two points using point-slope form.

 

y = mx - mx₁ + y₁

y = -(3/2) - (-3/2)(1) - 4

y = -(3/2)x + (3/2) - 4

y = - (3/2)x - (5/2)

 

Next, we can find the other line using the negative reciprocal slope and the point (2, 3).

 

y = mx - mx₁ + y₁

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

y = (2/3)x - (4/3) + 3

y = (2/3)x + (5/3)

 

Now, we can find the point of the intersection between the two lines.  Call this point of intersection (x₂ , y₂).  Set up the equation to solve for x₂.

 

- (3/2)x - (5/2) = (2/3)x + (5/3)

 

Then find y₂ by plugging in the x value you found.

 

Finally, use the distance formula to find the distance between (2, 3) and (x₂ , y₂).

Answer 2

Use two point form to determine line through (-3,2) and (1, -4)

Take the negative reciprocal of the slope.

Use that slope and (2, 3) and point slope form for the equation of line perpendicular

Solve for common solution. 

Use distance formula from (2, 3) and the intersection.