F.3 coordinate

Since x-axis is one of the three perpendicular bisector of triangle OAB,
the orthocenter is on x-axis.
Let the coordinates of the orthocenter be C(x, 0)

Let the mid-pt of OA be M,
thus the coordinates of M is (9, -12)

Since OA is perpendicular to CM,
(the slope of CM) x (the slope of OA) = -1
[(0 + 12) / (x - 9)] x [(0 + 24) / (0 - 18)] = -1
[12 / (x - 9)] x (-4/3) = -1
16 = (x - 9)
x = 25

therefore. the x-coordinate of the orthocenter is 25

The orthocenter must be on the x - axis because the x - axis is the perpendicular to AB and pass through the origin, the vertex of the triangle.
Let the orthocenter be C(x , 0).
By definition of orthocenter, BC is perpendicular to OA
slope of BC = (24 - 0)/(18 - x) = 24/(18 - x)
slope of OA = [0 - (-24)]/(0 - 18) = 24/(-18)
so [24/(18 - x)][24/(-18)] = - 1
(24)(24) = 18(18 - x)
576 = 324 - 18x
x = - 252/18 = - 14
so orthocenter is (-14,0).
Note : Answer provided by 50418129 is the circum- center, not orthocenter.


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