Find the co-ordinates of the orthocentre of triangle ABC whose vertices are A(1,2), B(2,3) and C(4,3)

Slope of AB=3-22-1=1

Slope of BC=3-34-2=0

Slope of CA=3-24-1=13

Let AD, BE and CF be the altitudes in ABC


Slope of AD=-1Slope of BC=-10

Slope of BE=-1Slope of CA=-113=-3

Slope of CF=-1Slope of AB=-11=-1

We now have the vertices and slopes of AD, BE and CF. Let (x, y) be the orthocentre. So, by the equation y-y1=mx-x1, we have:-

AD: y-2=-10x-1   x=1    [the line is vertical]BE: y-3=-3x-2  y-3=-3x+6CF: y-3=-1x-4   y-3=-x+4

Solving these equations, we find x=1, y=6

Thus, the required orthocentre is (1, 6).

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