terminer may not be followed by a prepositional phrase (i.e. it is never a transitive verb) finir can be used for anything that terminer can be used for, plus a lot more e.g. as a reprimand "are you finished?" "will you stop it?" e.g. to die "to finish at the hospital" e.g. to arrive at an end point, e.g. "the road ends at the bridge" Source: Larousse En-Fr Dictionary. Hope that helps.
In the meantime (that is at some point in the past 3 years), we decided that in such sentences, the owner should be the same person as the verb's subject.
If it were "his" letter, chances that a French person would say: "elle finit sa lettre à lui" so that there is no ambiguity as to whose letter it is.
Ah, okay. That makes sense, I won't report such again. (I've been interpreting sa as his and son as hers to reinforce in my own head that "his/hers" in French takes the gender of the object/noun being "possessed," not the person doing the "possessing." Duo doesn't always approve, but I've finished the tree, and I'm doing it for my own reference.)
"son" means his or her or its, in front of a masculine singular noun or a feminine singular noun starting with a vowel sound.
"sa" means his or her or its, in front of a feminine noun starting with a consonant sound.
- his/her letter = sa lettre (lettre is feminine)
- his/her pen = son stylo (stylo is masculine)
- his/her water = son eau (eau is feminine)
By no means can "sa" (his/her/its) mean "you" (tu/vous).
In French, the possessive adjectives agree with the possession.
- his/her/its father (possessive) = son père (masculine noun)
- his/her/its letter (possessive) = sa lettre (feminine noun)
- his/her/its parents (possessive) = ses parents (plural noun)
So, "la femme termine sa lettre" is "the woman finishes her letter".