There will be 3 parts in this study. In Part 1, healthy adult participants will be randomized to receive blinded MK-1092 subcutaneously (SC) or glargine SC, as a single dose under the euglycemic clamp. Once a safe and tolerated dose that achieves GIRmax is identified in Part 1, Part 2 will start. In Part 2, 4 different healthy adult participants will be enrolled in a single panel and receive open-label MK-1092 SC as a single dose under the euglycemic clamp and also receive an intravenous infusion of Humalog®. In Part 3, adult participants with T1DM will be randomized to receive blinded MK-1092 SC or glargine SC, as a single dose under the euglycemic clamp.