SOLUTION: Find the orthocenter of triangle KLM with vertices K(2,-2), L(4,6), M(8,-2)

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Question 823740: Find the orthocenter of triangle KLM with vertices K(2,-2), L(4,6), M(8,-2)
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!


The orthocenter is the point where the three (green)
altitudes intersect.  Altitudes are perpendicular to the
sides.  

We will find the equation of altitude AM and find
where it intersects the altitude LC

First we must find the slope of AM.  Since AM is
perpendicular to KL, we will find the slope of KL and
take its negative reciprocal:

We use the slope formula to find the slope of KL.

m = %28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29
where (x1,y1) = K(2,-2)
and where (x2,y2) = L(4,6)

m = %286-%28-2%29%29%2F%284-2%29%22%22=%22%22%286%2B2%29%2F2%22%22=%22%228%2F2 = 4.

Therefore the slope of MA is the negative reciprocal of
4, which is -1%2F4.

To find the equation of MA we use the point-slope formula:

y - y1 = m(x - x1)
where (x1,y1) = M(8,-2)

y - (-2) = -1%2F4(x - 8)

   y + 2 = -1%2F4x + 2

       y = -1%2F4x

The altitude LC is vertical (since KM is horizontal)
So the equation of LC is x = 4

So we substitute x = 4 in y = -1%2F4x and get

y = -1%2F4(4) = -1.

So the orthocenter O is (4,-1)

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Note:
We could have used the altitude KB instead of MA.

Edwin