"你吃了吗？" (Have you eaten?) is a traditional Chinese greeting, similar to "How are you?" - reflecting how central food is to Chinese culture. You can just reply "吃了" even if you haven't.
usage
吃 appears in many common collocations: 吃饭 (eat a meal), 吃早饭 (eat breakfast), 吃药 (take medicine), 吃亏 (suffer a loss). In chess, 吃 means to capture a piece.
Left mouth - the indexing radical. Eating is the prototypical mouth action, so 吃 sits at the heart of the mouth-verb family: 喝 (drink), 嚼 (chew), 咽 (swallow), 吐 (spit). Originally 吃 meant to stutter; the to-eat sense replaced an older 喫.
Right 乞 supplies the sound (qǐ → chī through a palatalization in this phonetic series). The beg meaning of 乞 also adds a faint semantic flavor: someone hungry asking for food, fitting the eating sense. Same phonetic family: 吃 itself, 屹 (towering), 迄 (until).
