What is the orthocenter of a triangle with corners at #(5 ,9 )#, #(4 ,3 )#, and (1 ,2 )#?

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Answer 1 · 2017-07-24T13:13:27

The orthocenter of the triangle is #=(131/17,15/17)#

Let the triangle #DeltaABC# be

#A=(5,9)#

#B=(4,3)#

#C=(1,2)#

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

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

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

#y-9=-3(x-5)#...................#(1)#

#y=-3x+15+9=-3x+24#

The slope of the line #AB# is #=(3-9)/(4-5)=-6/-1=6#

The slope of the line perpendicular to #AB# is #=-1/6#

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

#y-2=-1/6(x-1)#

#y=-1/6x+1/6+2#

#y=-1/6x+13/6#...................#(2)#

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

#-3x+24=-1/6x+13/6#

#3x-1/6x=24-13/6#

#17/6x=131/6#, #=>#, #x=131/17#

#y=-3*131/17+24=15/17#

The orthocenter of the triangle is #=(131/17,15/17)#