Question 1110443: How can I find the orthocenter? I keep getting really big numbers like (30,15), but then the points would be far away from the triangle because the ordered pairs are (-6,3),(4,13) and (10,-5)
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website! I labeled the vertices A(-6,3), B(4,13), C(10,-5)
Side a connects B with C
Side b connects A with C
Side c connects A with B
Eqn for line coincident with side a: y=-3x+25 (1)
Eqn for line coincident with side b: y=(-1/2)x (2)
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Slope of perpendicular to (1): 1/3
y=(1/3)x+b (b here is y-intercept)
line passes through A(-6,3)
3 = (1/3)(-6) + b —> b = 5 —> y=(1/3)x + 5 (1')
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Slope of perpendicular to (2): 2
y=2x + b
line passes through B(4,13)
13=2(4)+b —> b = 5 —> y=2x+5 (2')
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Lines (1') and (2') meet at the orthocenter.
Where they meet, their x values AND y values must match:
2x+5 = (1/3)x+5
This only holds at x=0, and there y=5
[ Alternatively, observe that (1') and (2') have the same y-intercept, which occurs at x=0, y=b ]
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Ans:  is the orthocenter.
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