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

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Answer 1 · 2017-07-02T09:10:32

The orthocenter of the triangle is #=(9,27/2)#

Let the triangle #DeltaABC# be

#A=(2,3)#

#B=(9,1)#

#C=(6,3)#

The slope of the line #BC# is #=(3-1)/(6-9)=2/-3=-2/3#

The slope of the line perpendicular to #BC# is #=3/2#

The equation of the line through #A# and perpendicular to #BC# is

#y-3=3/2(x-2)#...................#(1)#

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

The slope of the line #AB# is #=(1-3)/(9-2)=-2/7#

The slope of the line perpendicular to #AB# is #=7/2#

The equation of the line through #C# and perpendicular to #AB# is

#y-3=7/2(x-6)#

#y=7/2x-21+3#

#y=7/2x-18#...................#(2)#

Solving for #x# and #y# in equations #(1)# and #(2)#

#3/2x=7/2x-18#

#7/2x-3/2x=18#

#2x=18#, #=>#, #x=9#

#y=(3*9)/2=27/2#

The orthocenter of the triangle is #=(9,27/2)#