A triangle has vertices #A(7 ,6 )#, #B(8 ,3 )# and #C(5 ,1 )#. What are the endpoints and length of the altitude going through vertex C?

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Answer 1 · 2018-06-01T08:41:18

One endpoint of the altitude is #C(5,1);# the other is the foot on AB, #(8.3,2.1),# giving a length #11/10 sqrt{10}.#

The direction vector #B-A=(1,-3)#; the line AB is

#(x,y)=A+t(B-A)=(7,6)+t(1,-3)#

The perpendicular direction is #(3,1).# The perpendicular through #C# is

#(x,y)=(5,1)+u(3,1)#

That meets AB when

#5 + 3u = 7 + t#

#1 + u = 6 - 3 t#

#9u=6 + 3t#

#10u=11#

#u = 11/10 #

Endpoint:

#(x,y)=(5,1)+11/10(3,1) = (8.3,2.1)#

Length:

# 11/10 \sqrt{3^2+1^2} = 11/10 sqrt{10}#