At what point may you start a descent from your last assigned altitude on an ILS approach when being radar vectored?

Starting a descent from your last assigned altitude during an ILS (Instrument Landing System) approach, when being radar vectored, is permissible when you are established on a segment of a published route or an Instrument Approach Procedure (IAP). This is because being established means you are aligned with the prescribed path that ensures safe terrain clearance and properly sequenced descent.

When you reach this stage, you're in compliance with the established protocol for executing an approach, as this signifies that you are in the proper phase of flight where the aircraft can transition smoothly to the final approach segment without violating any safe altitude parameters. This approach is structured to prevent conflicts with terrain or obstacles and to ensure the aircraft is positioned correctly for the final descent.

When considering the other choices, reaching the last waypoint or being within 10 nautical miles of the airport does not necessarily mean you are safe to descend; these scenarios could vary significantly based on the specific approach and current air traffic conditions. Additionally, final approach instructions provide guidance on your descent but typically follow the condition of being established on the appropriate segment for safe operation. Therefore, the provision to start a descent hinges on being on the properly defined path as dictated by the IAP or published route, making the first choice the correct answer.