topology

You and your beloved have a pizza and a cheesecake for dinner. You love him or her but things must be fair! Prove that there are at least one ways to cut the pizza and the cheesecake in halves of equal areas by one straight cut. (By pizzas and cheesecakes, we mean compact connected subsets of R^2)

let me give you some idea: the area is continuous change with respectively to x. the pizza is bounded by rectangle. move the line to another position called x bar. the area between two position change little bit. prove that the given area is less than epsilon. if M*delta x < epsilon. please help