"Tengo un vaso en la mano."
Translation:I have a glass in my hand.
Yep! Whenever you know the person doing an action and to whom (themselves is assumed if not otherwise stated) the body part is never possessive. Examples:
my leg hurts = me duele la pierna
I wash my face = me lavo la cara
I comb my hair = me peino el pelo (or just "me peino")
She washes my hair = Ella me lava el pelo
Jenny shaves her legs = Jenny se afeita las piernas
They scratch their feet = Se rascan los pies.
****You DO use possessives when it is unclear who's body parts are who's especially when two different subject's body parts are involved in the same sentence.
I love when her lips touch my chest. = Me encanta cuando sus labios tocan mi pecho.
Googling phrases for result number comparisons and trying various combos in google translator has convinced me that duolingo is right here. Not sure if there is a rule that explains why. It is not a stretch for me to see how "I have a glass in the hand" could mean "I have a glass in my hand" I did read this discussion thread on it where some people discus it, http://www.spanishdict.com/answers/202651/mi-mano-la-mano-los-partes-del-cuerpo-y-adjectivos-posesivos