A (0,0), B(8,10), C(5,-10) First, find the slope of AB and AC, slope of AB is 5/4, slope of AC is -2 next, find the negative reciprocals of the slopes: -4/5 and 1/2 respectively so the altitude of AB is a line with a slope of -4/5 passing through (5,-10) the altitude of AC is a line with a slope of 1/2 passing through (8,10) find these two lines: Plug (5,-10) in y=(-4/5)x+b to find b, b=-6, so y=(-4/5)x -6 plug (8,10) in y=(1/2)x +b to find b, b=6, so y=(1/2) x +6
the point where these two lines meet is the orthocenter. y=(-4/5)x -6 y=(1/2) x +6 (-4/5)x -6=(1/2) x +6 x=-120/13 y=(1/2)*(-120/13)+6=18/13 so the orthocenter is (-120/13, 18/13) these numbers look odd, but I've run the vertex through an orthodox calculator, and got the same result.
Given that the first number shows up as four. The second number should be six so that the sum would be ten. Therefore, its probability should be 1/6.
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