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

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Answer 1 · 2017-07-21T12:48:23

The end points are #=(164/61,654/183)# and the length is #=2.05u#

The corners of the triangle are

#A=(2,3)#

#B=(8,8)#

#C=(4,2)#

The slope of the line #AB# is #m=(8-3)/(8-2)=5/6#

The equation of line #AB# is

#y-3=5/6(x-2)#

#y-3=5/6x-5/3#

#y=5/6x+4/3#...........................#(1)#

#mm'=-1#

The slope of the line perpendicular to #AB# is #m'=-6/5#

The equation of the altitude through #C# is

#y-2=-6/5(x-4)#

#y-2=-6/5x+24/5#

#y=-6/5x+34/5#................................#(2)#

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

#5/6x+4/3=-6/5x+34/5#

#5/6x+6/5x=34/5-4/3#

#61/30x=82/15#

#x=164/61#

#y=5/6*164/61+4/3=410/183+4/3=654/183#

The end points of the altitude is #=(164/61,654/183)#

The length of the altitude is

#=sqrt((4-164/61)^2+(2-654/183)^2)#

#=sqrt((1.311)^2+(-1.574)^2)#

#=sqrt(4.197)#

#=2.05#