A triangle has corners A, B, and C located at #(4 ,3 )#, #(9 ,5 )#, and #(6 ,2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-07-07T06:36:06

The end-points are #=(156/29,103/29)# and the length is #=1.67#

The corners of the triangle are

#A=(4,3)#

#B=(9,5)#

#C=(6,2)#

The slope of the line #AB# is #m=(5-3)/(9-4)=2/5#

The equation of line #AB# is

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

#y-3=2/5x-8/5#

#y=2/5x+7/5#...........................#(1)#

#mm'=-1#

The slope of the line perpendicular to #AB# is #m'=-5/2#

The equation of the altitude through #C# is

#y-2=-5/2(x-6)#

#y-2=-5/2x+15#

#y=-5/2x+17#................................#(2)#

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

#-5/2x+17=2/5x+7/5#

#2/5x+5/2x=17-7/5#

#29/10x=78/5#

#x=78/5*10/29=156/29#

#y=2/5*156/29+7/5=515/145=103/29#

The end points of the altitude is #=(156/29,103/29)#

The length of the altitude is

#=sqrt((6-156/29)^2+(2-103/29)^2)#

#=sqrt((18/29)^2+(45/29)^2)#

#=sqrt(2349)/29#

#=1.67#