What is the orthocenter of a triangle with corners at #(4 ,5 )#, #(8 ,3 )#, and #(5 ,9 )#?

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Answer 1 · 2017-06-27T07:04:41

The orthocenter is #=(8/3,13/3)#

Let the triangle #DeltaABC# be

#A=(4,5)#

#B=(8,3)#

#C=(5,9)#

The slope of the line #BC# is #=(9-3)/(5-8)=-6/3=-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-5=1/2(x-4)#...................#(1)#

#2y=x-4+10=x+6#

The slope of the line #AB# is #=(3-5)/(8-4)=-2/4=-1/2#

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

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

#y-9=2(x-5)#

#y-9=2x-10#

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

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

#4x-2=x+6#

#4x-x=6+2#

#3x=8#

#x=8/3#

#y=2x-1=2*8/3-1=13/3#

The orthocenter of the triangle is #=(8/3,13/3)#