Find the orthocenter for the triangle described by each set of vertices. k (4, −4), l (2, 6), m (7, −4)

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 XSlope LM = 6 -(-4) / 2-7 = 10/-5 = -2Slope 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 forthe slope of K⊥LM has to be 1/2 andthe 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 usK⊥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 equalSo, 1/2(x)-2 = 1/5(x) - 7/5; 5x - 20 = 2x - 143x = 6 --> x=2y+4=-1 --> y=-5(2,-5)

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