转机 zhuǎnjī (to take) a turn for the better

How to remember 转机: 转 to turn + 机 machine = (to take) a turn for the better.

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Two distinct meanings that context separates cleanly: 'a turn for the better' (病情出现转机 - the illness took a turn for the better) versus 'to change planes' (在香港转机 - connect through Hong Kong).
Learners often only learn the flight-connection sense and miss the more literary 'turning point' sense.
As 'turn for the better,' it's fairly formal/written, used about situations, illness, or negotiations improving.
As 'connecting flight,' it's everyday travel vocabulary.