To continue the move towards miniaturization in technology, developing new methods for fabricating micro- and nanoscale objects has become increasingly important. One potential method, called capillary origami, consists of placing a small drop of liquid on a thin, inextensible sheet. In this article, we model the static configurations of this system in an idealized two-dimensional setting and describe how the plate will fold due to capillary forces. We do this by minimizing the total energy of the system, which consists of bending and interfacial components whose relative importance can be measured by a dimensionless parameter . The deflection of the plate is characterized in terms of bifurcation diagrams, where the bifurcation parameter is the drop's size. This allows us to consider the quasi-static evolution of the system in the presence of evaporation. Variations in this bifurcation diagram for various are then studied, leading us to organize the physical description of the system's behavior into three different regimes. The present approach provides a general framework for the study of capillary origami that can be extended to three-dimensional settings.