A triangle has corners A, B, and C located at #(8 ,7 )#, #(4 ,5 )#, and #(6 , 3 )#, respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer
Ananda Dasgupta
Jul 29, 2016

#(6,3)# and #(24/5,27/5)#. Length = #6/5sqrt{5}#

Explanation:

One end point of the altitude must of course be #C=(6,3)#. The other point must lie on the line joining #(8,7)# and #(4,5)#. So, its coordinate must of the form

#(8alpha+4(1-alpha),7alpha+5(1-alpha) ) = (4+4alpha, 5+2alpha)#

Slope of this altitude must then be

#{5+2alpha-3}/{4+4alpha-6}={1+alpha}/{2alpha-1}#

Since the slope of #AB# is #{7-5}/{8-4}=1/2#, the slope of the altitude must be #-2#. Thus

#{1+alpha}/{2alpha-1}=-2#

or

#1+alpha=-4alpha+2#

and thus #alpha=1/5#

So, the coordinates of the second endpoint of the altitude is #(24/5,27/5)#

Thus the length of the altitude is

#sqrt{(6-24/5)^2+(3-27/5)^2}=1/5sqrt{6^2+12^2}=6/5sqrt{5}#

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