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

1 Answer
Shwetank Mauria
Feb 23, 2016

Endpoints of altitude are #(8,7)# and #(4,3)# and length of altitude is #4sqrt2#.

Explanation:

As the triangle has corners A, B, and C located at #(5,2)#, #(2,5)#, and #(8,7)#, respectively and the altitude goes through C#(8,7)#, it is obvious it is perpendicular to line joining #(5,2)#, #(2,5)#.

The slope of line joining #(5,2)#, #(2,5)# is #(5-2)/(2-5)# or #-1# and hence slope of the required altitude is #1#.

Now using point-slope form of equation, equation of altitude from #(8.7)# is

#(y-7)=1*(x-8)# or #y-7=x-8# or #x-y=1#

Further, equation of line joining AB is given by #(y-2)/(x-5)=-1#

i.e. #y-2=-x+5# or #x+y=7#

As equation of line joining AB is #x+y=7# and equation of altitude is #x-y=1#, their solution gives us base point of altitude which can be easily got as #(4,3)# (by adding and subtracting two equations) and distance of #(4,3)# from #(8,7)# is given by #sqrt((7-3)^2+(8-4)^2# or #sqrt32# or #4sqrt2#.

Hence endpoints of altitude are #(8,7)# and #(4,3)# and length of altitude is #4sqrt2#.

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