Hi welcome, Your questions and examples are correct. In writing, sometimes instead of expressing two or more thoughts in a full sentence, we often reduce them by using adverbial phrase which may appear in various forms. Preposition and present particple phrases are among the advberial category.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.
gretelliubroken cup
It s the cup is/was broken ?
Or the cup have/had been broken? If you broke a cup, that cup is broken (by you - is omitted)
In this case, "broken" is a past participle adjective. Other similar verbla adjectival phrases " Boiled egg", "frozen waffle", and "burned toast", All have passive connotation.