Plural of flatfoot; feet with fallen arches that touch the ground completely along the sole, or informal slang for police officers.
Compound of 'flat' (Old English 'flet') and 'feet' (Old English 'fēt'), used medically since the 17th century. The slang meaning for police emerged in early 20th-century American English, possibly from the notion of walking a beat.
Police were called 'flatfoots' because they literally walked the streets constantly as part of their patrol duties—after hours of walking a beat, officers' feet probably did get pretty flat!
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