What is the orthocenter of a triangle with corners at #(7 ,8 )#, #(3 ,2 )#, and (5 ,6 )#?

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Answer 1 · 2017-06-27T07:50:52

The orthocenter is #=(-13,18)#

Let the triangle #DeltaABC# be

#A=(7,8)#

#B=(3,2)#

#C=(5,6)#

The slope of the line #BC# is #=(6-2)/(5-3)=4/2=2#

The slope of the line perpendicular to #BC# is #=-1/2#

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

#y-8=-1/2(x-7)#...................#(1)#

#2y=-x+7+16=-x+23#

The slope of the line #AB# is #=(2-8)/(3-7)=-6/-4=3/2#

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

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

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

#3y-18=-2x+10#

#3y=-2x+28#...................#(2)#

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

#-3x+69=-4x+56#

#4x-3x=56-69#

#x=-13#

#y=(13+23)/2=18#

The orthocenter of the triangle is #=(-13,18)#