What is the orthocenter of a triangle with corners at #(4 ,9 )#, #(3 ,4 )#, and (5 ,1 )#?

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Answer 1 · 2017-06-13T10:33:48

The orthocenter of the triangle is #=(-5,3)#

Let the triangle #DeltaABC# be

#A=(4,9)#

#B=(3,4)#

#C=(5,1)#

The slope of the line #BC# is #=(1-4)/(5-3)=-3/2#

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

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

#y-9=2/3(x-4)#

#3y-27=2x-8#

#3y-2x=19#...................#(1)#

The slope of the line #AB# is #=(4-9)/(3-4)=-5/-1=5#

The slope of the line perpendicular to #AB# is #=-1/5#

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

#y-1=-1/5(x-5)#

#5y-5=-x+5#

#5y+x=10#...................#(2)#

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

#3y-2(10-5y)=19#

#3y-20+10y=19#

#13y=20+19=39#

#y=39/13=3#

#x=10-5y=10-15=-5#

The orthocenter of the triangle is #=(-5,3)#