Join extruded vertex with edge I need to join a vertex to edge, with snap to edge i can snap my vertex on the edge but i don't know how to join it.

Now my solution is:


snap vertex to edge
subdivide edge (this creates a new vertex on the edge)
merge the two vertices


But i think is too elaborate, so i look for a simpler solution.
Subdivide is a bad option in your situation because you get more verticals than you need.

The fastest way, in my opinion, is to manage this with a loop cut.


Loop Cut CTRL+R and slide to the position where you want to connect your vertex
Select your vertices and merge ALT+M them


 

You could the same with the knife tool (K). However you can't guarantee  for a good mesh flow if you do so. 
With Loop Cuts you are most of the time on the good side.