SOLUTION: Analytic geometry Determine the shortest distance from the point H(5,2) to the line through points J(-6,4) and K(-2,-4).

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Analytic geometry Determine the shortest distance from the point H(5,2) to the line through points J(-6,4) and K(-2,-4).      Log On


   

Question 177791: Analytic geometry
Determine the shortest distance from the point H(5,2) to the line through points J(-6,4) and K(-2,-4).
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
To find the shortest distance from point H to line JK you need to find a line perpendicular to JK that passes through point H. Find the intersection point between the two lines and then calculate the distance.
First, find the slope of line JK.

Perpendicular lines have slopes that are negative reciprocals of each other.
m%5BJK%5Dm%5BP%5D=-1
-2m%5BP%5D=-1
m%5BP%5D=1%2F2
Use the point slope form of the line using this slope and point H.
y-y%5B0%5D=m%28x-x%5B0%5D%29
y-2=%281%2F2%29%28x-5%29
y-2=%281%2F2%29x-5%2F2
y=%281%2F2%29x-1%2F2

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Next step is to find the intersection point of the two lines.
The first line equation is,
y-4=-2%28x-%28-6%29%29
y-4=-2x-12
y=-2x-8
Set the two equations equal to each other and solve for x, then y.
%281%2F2%29x-1%2F2=-2x-8
x-1=-4x-16
5x=-15
x=-3
Then
y=-2x-8
y=-2%28-3%29-8
y=6-8
y=-2
The intersection between the two lines occurs at (-3,-2).

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Finally, use the distance formula to calculate the distance from H to the intersection point.
D%5E2=%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2
D%5E2=%28-3-5%29%5E2%2B%28-2-2%29%5E2
D%5E2=%28-8%29%5E2%2B%28-4%29%5E2
D%5E2=64%2B16
D%5E2=80
D=sqrt%2880%29