A triangle has corners A, B, and C located at #(4 ,3 )#, #(7 ,4 )#, and #(2 ,5 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-07-16T18:07:39

The endpoints are #(2,5)# and #(14/5,13/5)#. The length is #=2.53#

The corners of the triangle are

#A=(4,3)#

#B=(7,4)#

#C=(2,5)#

The slope of the line #AB# is #m=(4-3)/(7-4)=1/3#

The equation of line #AB# is

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

#y-3=1/3x-4/3#

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

#mm'=-1#

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

The equation of the altitude through #C# is

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

#y-5=-3x+6#

#y=-3x+11#................................#(2)#

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

#1/3x+5/3=-3x+11#

#1/3x+3x=11-5/3#

#10/3x=28/3#

#x=28/10=14/5#

#y=1/3*14/5+5/3=39/15=13/5#

The end points of the altitude is #=(14/5,13/5)#

The length of the altitude is

#=sqrt((2-14/5)^2+(5-13/5)^2)#

#=sqrt((-4/5)^2+(12/5)^2)#

#=sqrt(160)/5#

#=2.53#