A triangle has corners A, B, and C located at #(3 ,5 )#, #(6 ,9 )#, and #(9 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-06-25T06:30:24

The end points of the altitude is #=(33/5,49/5)# and the length is #=3#

The corners of the triangle are

#A=(3,5)#

#B=(6,9)#

#C=(9,8)#

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

The equation of line #AB# is

#y-5=4/3(x-3)#

#3y-15=4x-12#

#3y-4x=3#...........................#(1)#

#mm'=-1#

The slope of the line perpendicular to #AB# is #m'=-3/4#

The equation of the altitude through #C# is

#y-8=-3/4(x-9)#

#4y-32=-3x+27#

#4y+3x=32+27=59#................................#(2)#

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

#4*(3+4x)/3+3x=59#

#12+16x+9x=177#

#25x=177-12=165#

#x=165/25=33/5#

#y=1/3*(3+4*33/5)=147/15=49/5#

The end points of the altitude is #=(33/5,49/5)#

The length of the altitude is

#=sqrt((9-33/5)^2+(8-49/5)^2)#

#=sqrt((12/5)^2+(-9/5)^2)#

#=sqrt(225)/5#

#=15/5=3#