What is the orthocenter of a triangle with corners at #(9 ,7 )#, #(4 ,1 )#, and (8 ,2 )#?

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Answer 1 · 2017-06-22T12:14:38

The orthocenter of the triangle is #=(206/19,-7/19)#

Let the triangle #DeltaABC# be

#A=(9,7)#

#B=(4,1)#

#C=(8,2)#

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

The slope of the line perpendicular to #BC# is #=-4#

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

#y-7=-4(x-9)#...................#(1)#

#y=-4x+36+7=-4x+43#

The slope of the line #AB# is #=(1-7)/(4-9)=-6/-5=6/5#

The slope of the line perpendicular to #AB# is #=-5/6#

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

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

#y-2=-5/6x+20/3#

#y+5/6x=20/3+2=26/3#...................#(2)#

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

#-4x+43=26/3-5/6x#

#4x-5/6x=43-26/3#

#19/6x=103/3#

#x=206/19#

#y=26/3-5/6x=26/3-5/6*206/19=26/3-1030/114=-42/114=-7/19#

The orthocenter of the triangle is #=(206/19,-7/19)#