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Question 1091708: Find orthocentre of triangle with vertices (-2,-1),(6,-1),(2,5)
Found 2 solutions by htmentor, ikleyn:
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! By inspection, we see that the base of the triangle formed by the vertices (-2,-1) and (6,-1) is symmetric about the line x=2,
so this altitude goes through the point (2,-1), and we know the x coordinate of the orthocentre is 2.
To find the y coordinate, we use the fact that an altitude will be perpendicular
to the line formed by any two vertices, and will pass through the 3rd vertex.
Using (6,-1) and (2,5), the line is
y + 1 = ((5+1)/(2-6)(x - 6) -> y = -3/2x + 8
And, since perpendicular lines have negative reciprocal slope, the line for the altitude is
y + 1 = 2/3(x + 2) -> y = 2/3x + 1/3
The intersection point of x = 2 and y = 2/3x + 1/3 gives the orthocentre
y = 2/3*2 + 1/3 = 5/3
Ans: (2,5/3)
Answer by ikleyn(52794)  (Show Source):
You can put this solution on YOUR website! .
Find orthocentre of triangle with vertices (-2,-1),(6,-1),(2,5)
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The orthocenter is the point where all three altitudes of the triangle intersect.
Notice that the side of the triangle, connecting the vertices (-2,-1) and (6,-1), is horizontal line parallel to x-axis.
Therefore, the altitude drawn to this side is vertical line x = const, and since it passes through the point (2,5), this constant is equal to 2,
and the equation of this altitude is x = 2.
We will find the orthocenter as the intersection point of this altitude with the other altitude drawn from the vertex (6,-1).
This altitude is perpendicular to the side of the triangle connecting two other points, (-2,-1) and (2,5).
The slope of this side/segment is m =  =  =  =  .
Since the altitude from the point (6,-1) is perpendicular to this side, it has the slope  .
And since it passes through the point (6,-1), its equation is
y - (-1) =  , or
y + 1 =  , or y =  .
Its intersection with the line x = 2 has y-coordinate y =  =   .
Thus the orthocenter is the point (x,y) = ( ,).
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