A triangle has corners A, B, and C located at #(5 ,5 )#, #(7 ,9 )#, and #(9 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

A#(5,5)#, B#(7,9)#, C#(9,8)#. We call them vertices.

Let's do this one differently. From the Shoelace Theorem, the area of the triangle is

#mathcal{A} = 1/2 |5(9) - 5(7) + 7(8)-9(9) +9(5)-5(8)| = 5#

The base AB has length # c=\sqrt{(7-5)^2+(9-5)^2}=2sqrt{5}#

So the altitude #h# from AB to C satisfies

#mathcal{A} = 1/ 2 c h #

#h = {2 mathcal{A}}/c= {2 (5)}/{2 \sqrt{5}} = sqrt{5}#

We got the length of the altitude without ever calculating the other endpoint.

We need the endpoints of the altitude from vertex C. The vertex itself, C#(9,8)#, is obviously one of them. Call the other end F, the foot.

F is a distance #h# from C along the perpendicular to AB. #B-A=(2,4)# so the perpendicular has direction vector #P(4,-2)# and we have

#F = C pm h P/|P| #

where we need to choose the sign so we're going toward AB, decreasing #x#.

# F = (9,8) - \sqrt{5} cdot {(4,-2)}/sqrt{4^2 + 2^2} = (7,9)=B #

That's a surprise. The foot of the altitude is another vertex. That means we have a right triangle, where B is the right angle.

Check.

Let's check B is a right angle through the zero dot product, our surprising conclusion.

#(B-A)cdot(B-C) = (2,4) cdot(-2,1)=2(-2) + 4(1) = 0 quad sqrt #

Let's check #h=BC#, also our surprising conclusion.

# BC=\sqrt{(9-7)^2+(8-9)^2}=sqrt{5} = h quad sqrt #