Triangle ABC has the vertices (1,2), (3,5), and (6,3) Find the coordinates of the orthocenter.
+128475 We only need the equations of two lines to do this
Find the slope between AB = (1,2) and (3,5) = (5-2) /( 3 -1) = 3/2
The equation of the line perpendicular to segment AB through ( 6,3) is
y = (-2/3)(x - 6) + 3
y = (-2/3)x + 4 + 3
y = (-2/3)x + 7 (1)
Next
Find the slope between AC = ( 1,2) and (6,3) = (3 -2) /( 6 - 1) = 1/5
The equation of the line perpendicular to segment AC through B is
y = -5 ( x -3)+ 5
y = -5x + 20 (2)
Find the x coordinate of the orthocenter by setting (1) = (2)
(-2/3)x + 7 = - 5x + 20 multiply through by 3
-2x + 21 = -15x + 60
13x = 39
x = 3
And the y coordinate is -5(3) + 20 = 5
So....the orthocenter is actually B = (3,5)
