A triangle has corners A, B, and C located at #(8 ,7 )#, #(4 ,5 )#, and #(6 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-06-21T06:50:48

The endpoint is #=(4.4,5.2)#
The length of the altitude is #=3.58#

The corners of the triangle are

#A=(8,7)#

#B=(4,5)#

#C=(6,2)#

The slope of #AB# is #=(7-5)/(8-4)=2/4=1/2#

The slope of the line through #C# and perpendicular to #BC# is #=-2#

The equation of the altitude is

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

#y-2=-2x+12#

#2x+y=14#....................#(1)#

The equation of the line #AB# is

#y-5=1/2(x-4)#

#2y-10=x-4#

#2y-x=6#......................#(2)#

Solving for #(x,y)# in equations #(1)# and #(2)#, we get the end point of the altitude

#2x+((x+6))/2=14#

#4x+6+x=28#

#5x=22#

#x=22/5=4.4#

#y=14-2*22/5==5.2#

The endpoint is #=(4.4,5.2)#

The length of the altitude is

#=sqrt((6-4.4)^2+(2-5.2)^2)#

#=sqrt(1.6^2+(-3.2)^2)#

#=sqrt(12.8)#

#=3.58#