How do I split an cylinder in half, and mirror it?

I am fairly new to blender so I apologize if may seem like a basic question, but I have been having difficulties in trying to evenly split a cylinder down its middle.

By middle I am referring to the “face” or circles at its end. Essentially I am trying to create a new vertex in the center of the face, and delete everything on one side of that midpoint

Use triangle fan instead of the Ngon.
If you are fine with creating new cylinder then Add Cylinder operator has Triangle fan (or None) as Cap Fill Type.
If you have cylinder already then either inset its top and / or bottom face and merge vertices in the middle, or remove the Ngon completely and select edges on the sides forming central edge loop. Hide them, select linked geometry on one of the sides, remove it, unhide hidden loop and add Mirror modifier.

Thank you very much