A triangle has corners A, B, and C located at #(5 ,6 )#, #(3 ,9 )#, and #(1 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-07-16T14:40:46

The end points are #=(73/13,198/39)# and the length of the altitude is #=5.55#

The corners of the triangle are

#A=(5,6)#

#B=(3,9)#

#C=(1,2)#

The slope of the line #AB# is #m=(9-6)/(3-5)=-3/2#

The equation of line #AB# is

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

#y-9=-3/2x+9/2#

#y+3/2x=27/2#...........................#(1)#

#mm'=-1#

The slope of the line perpendicular to #AB# is #m'=2/3#

The equation of the altitude through #C# is

#y-2=2/3(x-1)#

#y-2=2/3x-2/3#

#y=2/3x+4/3#................................#(2)#

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

#-3/2x+27/2=2/3x+4/3#

#2/3x+3/2x=27/2-4/3#

#13/6x=73/6#

#x=73/13#

#y=2/3*73/13+4/3=198/39#

The end points of the altitude is #=(73/13,198/39)#

The length of the altitude is

#=sqrt((1-73/13)^2+(2-198/39)^2)#

#=sqrt((-60/13)^2+(-120/39)^2)#

#=sqrt(46800)/39#

#=5.55#