A TRACON Controller reported they thought an IFR aircraft was on a VFR flight plan and their trainee issued a vector below the Minimum Vectoring Altitude.

Narrative

I was training a trainee on the DR (Departure Radar) position. We took a handoff from ZZZ (Sector X) on a Cirrus. Neither my trainee nor I registered that the aircraft was on an IFR flight plan. We frequently take VFR aircraft from that sector in that general area. My trainee told me his plan; which was to vector the aircraft for a right base entry to RWY XX at ZZZ. That would not have been a normal entry for an IFR aircraft but neither one of us had figured out he was IFR. We vectored the aircraft south west of ZZZ at 090 and instructed them to enter a right base and to contact tower. Once the aircraft entered a 109 MVA the alarm went off. Since we had already switched the aircraft to tower I coordinated with the Local Controller to ensure the aircraft had the airport in sight and clear them for the visual approach to XX.I was sitting behind and to the left of my trainee and looking at the scope over his shoulder. I didn't notice the strip for that aircraft which would have identified him as IFR. This kind of confusion has happened here before and I usually put some kind of STARS (Standard Terminal Automation Replacement System) J ring on IFR aircraft who might be mistaken as VFR. In this case; I didn't notice it until it was too late. I advised my trainee that he should probably do the same in the future.