A triangle has corners A, B, and C located at #(2 ,7 )#, #(7 ,4 )#, and #(1 , 8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-07-03T05:49:18

The enpoints are #=(14/17,131/17)# and the length is #=0.34#

The corners of the triangle are

#A=(2,7)#

#B=(7,4)#

#C=(1,8)#

The slope of the line #AB# is #m=(4-7)/(7-2)=-3/5#

The equation of line #AB# is

#y-7=-3/5(x-2)#

#y-7=-3/5x+6/5#

#y+3/5x=6/5+7=41/5#...........................#(1)#

#mm'=-1#

The slope of the line perpendicular to #AB# is #m'=5/3#

The equation of the altitude through #C# is

#y-8=5/3(x-1)#

#y-8=5/3x-5/3#

#y-5/3x=8-5/3=19/3#................................#(2)#

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

#-3/5x+41/5=5/3x+19/3#

#5/3x+3/5x=41/5-19/3#

#34/15x=28/15#

#x=14/17#

#y=5/3*14/17+19/3=393/51=131/17#

The end points of the altitude is #=(14/17,131/17)#

The length of the altitude is

#=sqrt((1-14/17)^2+(8-131/17)^2)#

#=sqrt((3/17)^2+(5/17)^2)#

#=sqrt(34)/17#

#=0.34#