idiomatic sense of 면 트다
Came across this one though failed to find it under Naver. Knowing 트다 is the basis where 터지다 comes from i can roughly guess its meaning, though i prefer having a confirmation straight from the horse's mouth.

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Answer

면(을) 트다 is not commonly used nowadays. it means "to make acquaintance with someone"
It is an alternate form is 안면을 트다

면(面),안면(顔面) : 얼굴, 낯"face"

트다 means "to open passage physically or figuratively"
1. 막혀 있던 것을 치우고 통하게 하다.
e.g.
길을 트다	
벽을 트다

2. 서로 스스럼없이 사귀는 관계가 되다.
e.g.
친구와 마음을 트고 지내다

터지다 is passive verb of 트다