A triangle has corners A, B, and C located at #(2 ,3 )#, #(5 ,8 )#, and #(4 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-07-16T15:02:20

The end points are #=(71/34,107/34)# and the length of the altitude is #=2.23#

The corners of the triangle are

#A=(2,3)#

#B=(5,8)#

#C=(4,2)#

The slope of the line #AB# is #m=(8-3)/(5-2)=5/3#

The equation of line #AB# is

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

#y-3=5/3x-10/3#

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

#mm'=-1#

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

The equation of the altitude through #C# is

#y-2=-3/5(x-4)#

#y-2=-3/5x+12/5#

#y=-3/5x+22/5#................................#(2)#

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

#5/3x-1/3=-3/5x+22/5#

#3/5x+5/3x=1/3+22/5#

#34/15x=71/15#

#x=71/34#

#y=-3/5*71/34+22/5=535/170=107/34#

The end points of the altitude is #=(71/34,107/34)#

The length of the altitude is

#=sqrt((4-71/34)^2+(2-107/34)^2)#

#=sqrt((65/34)^2+(-39/34)^2)#

#=sqrt(5746)/34#

#=2.23#