John M. answered • 01/01/17
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Analytical assistance -- Writing, Math, and more
Juliette,
The orthocenter of a triangle is the point at which the three altitudes of a triangle intersect. This involves creating 3 lines, one from each vertex (K, L, and M) perpendicular to the line segment opposite it, so K⊥LM, L⊥KM, and M⊥KL.
Because these three lines intersect at one point, you only need to do this procedure for two of the three altitudes. Because K and M are on the same Y coordinate (-4), lets do this for LM and KL
First, we need to find the slope of the perpendicular to each line segment, so we need the slope of LM and KL.
The slope is the change in Y over the change in X
Slope LM = 6 -(-4) / 2-7 = 10/-5 = -2
Slope KL = -4-(6) / 4-2 = -10/2 = -5
The slope of a perpendicular line is the negative reciprocal of the slope of the given line, so for
the slope of K⊥LM has to be 1/2 and
the slope of M⊥KL has to be 1/5
Second, the vertex has to be on a line with that slope. So using the point-slope formula
(y-y1) = m(x-x1) , where x1,y1 is a point on the line for each line gives us
K⊥LM: y-(-4) = 1/2(x-4)
M⊥KL: y-(-4) = 1/5(x-7)
Third, find the point on both lines, are where the two equations are equal
So, 1/2(x)-2 = 1/5(x) - 7/5;
5x - 20 = 2x - 14
3x = 6 --> x=2
y+4=-1 --> y=-5
(2,-5)
