자다 = sleep
차다 = kick, attach, wear ...
But to be absolutely certain there is no confusion, you can use the full idiom 잠자다, to sleep.
차다 and 차 on the other hand are tricky words as they do carry various meanings. Best to consider the whole sentence than trying to dissect individual words.
In English, the "definite article" (the) is used to indicate a specific object [unlike the indefinite article (a or an) which means "one or any".].
When translated into Korean, the "definite/specific" function of "the" tends to be reflected in modifiers such as 이 (this); 그 (that there); 저 (that) .
Your sentence would look something like:
그 허리띠를 착용하는 것은 안 편해요 = Wearing the belt (there) is not comfortable.