A triangle has corners A, B, and C located at #(5 ,3 )#, #(7 ,9 )#, and #(5 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-06-29T11:26:18

The end points of the altitude is #=(13/2,15/2)# and the length is #=1.58#

The corners of the triangle are

#A=(5,3)#

#B=(7,9)#

#C=(5,8)#

The slope of the line #AB# is #m=(9-3)/(7-5)=3#

The equation of line #AB# is

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

#y-3=3x-15#

#y-3x=-12#...........................#(1)#

#mm'=-1#

The slope of the line perpendicular to #AB# is #m'=-1/3#

The equation of the altitude through #C# is

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

#3y-24=-x+5#

#3y+x=5+24=29#................................#(2)#

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

#3*(3x-12)+x=29#

#9x-36+x=29#

#10x=29+36#

#x=65/10=13/2#

#y=(3*13/2-12)=15/2#

The end points of the altitude is #=(13/2,15/2)#

The length of the altitude is

#=sqrt((5-13/2)^2+(8-15/2)^2)#

#=sqrt((3/2)^2+(1/2)^2)#

#=sqrt(10)/2#

#=1.58#