Pertinently tells you that something is said or done in a way that directly relates to what is being considered. If someone asks pertinently, the question is not a side track; it goes to the matter at hand. If they remark pertinently, they have put their finger on the useful point. It is the adverb of pertinent, and it has the same formal, careful feel. You meet it in writing and in polite discussion more than in the corridor. In ordinary speech people say relevantly, or they just say she asked a good question. Though actually, pertinently is a little sharper than “in a related way”: the related way can still wander; the pertinent way stays on the thing you are trying to settle.