Skyhawk 172/Cutlass 172
Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Commercial
Flight Crew Last 90 Days 105; Flight Crew Total 1105; Flight Crew Type 900
Communication Breakdown; Training / Qualification
Conflict NMAC
Flight Crew Took Evasive Action; Flight Crew Executed Go Around / Missed Approach
Procedure; Airport; Human Factors
My student and I were on an IFR flight plan. ATC had cleared us on the VOR-B Approach. As we descended to 2;500 feet; we cancelled our flight plan and switched to the CTAF frequency. My student made his first call stating he was approximately 8 miles to a fix on the VOR-B. Our TCAS lit up and made us aware that someone was approaching from the west and they were 100 feet beneath us. We could not see them and waited probably 30 seconds to see if we could find them or they would turn. We were not able to see them so we began to climb. As we were climbing; we caught sight of them passing directly beneath us. If we had not climbed; there would have been a distinct possibility we would have had a direct mid-air collision; so aligned the flight paths of our planes were. As they passed beneath us and appeared off our right side we began to descend back down to our approach altitude of 2500 feet only to see them then turn northeast bound to merge on the VOR-B next to us. We were then at the exact same altitude; wingtip to wingtip about 300 feet apart. I could see into their cockpit. We made another call on the CTAF but got no response. After flying like this for 60 seconds and hoping they would catch sight of us and break off of the approach; we decided they were unaware and it was no longer safe to be where we were. We executed a left circling 360 degree turn and re-joined the approach course a mile behind them. It was then that they made their first call; announcing themselves and that they were at the final approach fix. We announced ourselves a mile behind them.This plane - Aircraft Y - was not making calls on the CTAF. They were not aware of their surroundings or maintaining visual separation. Most importantly; federal regulations state that when two aircraft approach each other; the aircraft on the right has the right of way and the aircraft on the left is supposed to turn right to avoid them.
C172 Instructor Pilot reported a NMAC with another aircraft which was not on CTAF frequency at a non-towered airport.