(으)려던 참이다 is used at the end of a sentence to show that you were about to do something:
너한테 전화를 하려던 참이었어요.
I was just about to call you.
지금 나가려던 참이었어요.
I was about to leave now.
-- 박물관에 같이 갈까요?
-- Shall we go to the museum together?
-- 안 그래도, 가려던 참이었어요. 같이 갈래요.
-- Actually, I was just about to go. Let's go together.
-- 지금 뭐 해요?
-- What are you doing now?
-- 얼굴을 세수하고 잠을 자려던 참이었어요.
-- I was about to wash my face and go to sleep.
-- 지금은 바빠요?
-- Are you busy right now?
-- 안 그래도, 밥을 먹으려던 참이었어요. 15분 후에 돌아오세요.
-- Actually, I was about to eat a meal. Come back in 15 minutes.
V + 려던 참에 (or 려는 참에) can be placed between clauses to show that you were about to do something when/so/but':
나가려던 참에 너는 사무실에 왔어요.
I was about to leave when you came to the office.
수영장에 가려던 참에 친구가 월요일에는 문을 닫는다고 말했어요.
I was about to go to the swimming pool when my friend told me it's closed on Mondays.
책을 사려던 참에 저는 돈이 없는 걸 알았어요.
I was about to buy the book when I realised I didn't have any money.
메시지를 촬영 하려던 참이었는데 카메라가 고장 난 것 같았어요.
I was about to record (film) a message but it seemed that my camera broke.
홍콩에 이사하려던 참에 먼저 소포를 많이 보내고 있어요.
I'm about to move to Hong Kong so I'm sending a lot of parcels beforehand.