Question

A triangle has corners A, B, and C located at `(2 ,5 )`, `(7 ,4 )`, and `(2 ,3 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

The altitude basis coordinates over the side `AB` are `(2.38462, 4.92308)` and the altitude length is `1.96116`

Explanation:

Given `A, B, C` the altitude goes perpendicularly from `C` to the side `AB`. The altitude direction is obtained choosing a direction perpendicular to the segment `s=lambda A + (1-lambda)B` for `0 le lambda le 1`. The segment direction is given by the vector `v=B-A`.

A direction perpendicular to `v` is easily obtained, changing the `v` components and changing a sign for one of them.

Example:
The vector `v^T` perpendicular to `v = (v_x,v_y)` has components
`v^T = (v_y,-v_x)`
So the altitude straight is given by
`h = C + v^T mu`
The intersection point is obtained solving for `(lambda,mu)` the system
`lambda A + (1-lambda)B= C + v^T mu`
The altitude basis in side `AB` is `(2.38462, 4.92308)` and the altitude length is `1.96116`