What is the orthocenter of a triangle with corners at #(2 ,3 )#, #(5 ,1 )#, and (9 ,6 )#?

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Answer 1 · 2016-10-17T17:04:11

The Orthocenter is #(121/23, 9/23)#

Find the equation of the line that goes through the point #(2,3)# and is perpendicular to the line through the other two points:

#y - 3 = (9 - 5)/(1 -6)(x - 2)#

#y - 3 = (4)/(-5)(x - 2)#

#y - 3 = -4/5x + 8/5#

#y = -4/5x + 23/5#

Find the equation of the line that goes through the point #(9,6)# and is perpendicular to the line through the other two points:

#y - 6 = (5 - 2)/(3 - 1)(x - 9)#

#y - 6 = (3)/(2)(x - 9)#

#y - 6 = 3/2x - 27/2#

#y = 3/2x - 15/2#

The orthocenter is at the intersection of these two lines:

#y = -4/5x + 23/5#
#y = 3/2x - 15/2#

Because y = y, we set the right sides equal and solve for the x coordinate:

#3/2x - 15/2 = -4/5x + 23/5#

Multiply by 2:

#3x - 15 = -8/5x + 46/5#

Multiply by 5

#15x - 75 = -8x + 46#

#23x = + 121#

#x = 121/23

#y = 3/2(121/23) - 15/2#

#y = 363/46 - 345/46#

#y = 9/23#

The Orthocenter is #(121/23, 9/23)#