SOLUTION: find the distance from (6,-4) to the line defined by y=2x-6

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Question 749634: find the distance from (6,-4) to the line defined by y=2x-6
Answer by tommyt3rd(5050) About Me  (Show Source):
You can put this solution on YOUR website!
Typically we ask for the shortest distance because the distance from the line varies. So here we go...

There is a line perpendicular to our line that passes through (6,-4). Since it is perpendicular it has the slope -1/2

%0D%0Agraph%28300%2C300%2C-10%2C10%2C-10%2C10%2C2%2Ax-6%2C2%2Ax-16%2C-.5%2Ax-1%29%0D%0A


%0D%0Ay-%28-4%29=%28-1%2F2%29%28x-6%29%0D%0A

After some simplifying we can get

%0D%0Ay=%28-1%2F2%29x-1%0D%0A



Notice that this new line must intersect our original at a point exactly parallel to (6,-4) (why?)

Therefore if we find the point of intersection we can use that point to find our distance


y=2x-6=%28-1%2F2%29x-1
and this leads us to the x-coordinate which is x=2 and so y=2(2)-6=-2

Next we find the distance between (6,-4) and (2,-2)


d=sqrt%28%286-2%29%5E2%2B%28-4-%28-2%29%29%5E2%29
and this gives us

d=2sqrt%285%29%0D%0A



Actually in precalculus there is a formula:




:)