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?
1 Answer
This turns out to be a right triangle, right angle B, so the altitude endpoints are C and B and its length BC
Explanation:
A
Let's do this one differently. From the Shoelace Theorem, the area of the triangle is
The base AB has length
So the altitude
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
F is a distance
where we need to choose the sign so we're going toward AB, decreasing
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.
Let's check