What is the orthocenter of a triangle with corners at #(7 ,3 )#, #(4 ,8 )#, and (6 ,8 )#?

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Answer 1 · 2017-06-19T12:52:56

The orthocenter is #=(7,42/5)#

Let the triangle #DeltaABC# be

#A=(7,3)#

#B=(4,8)#

#C=(6,8)#

The slope of the line #BC# is #=(8-8)/(6-4)=0/2=0#

The slope of the line perpendicular to #BC# is #=-1/0=-oo#

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

#x=7#...................#(1)#

The slope of the line #AB# is #=(8-3)/(4-7)=5/-2=-5/2#

The slope of the line perpendicular to #AB# is #=2/5#

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

#y-8=2/5(x-6)#

#y-8=2/5x-12/5#

#y-2/5x=28/5#...................#(2)#

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

#y-2/5*7=28/5#

#y-14/5=28/5#

#y=28/5-14/5=42/5#

The orthocenter of the triangle is #=(7,42/5)#