Measurements of the midrapidity transverse-energy distribution, dE(T)/d eta, are presented for p + p, d+Au, and Au+Au collisions atv root s(NN) = 200 GeV and additionally for Au+Au collisions atv root s(NN) = 62.4 and 130 GeV. The dE(T)/d eta distributions are first compared with the number of nucleon participants N-part, number of binary collisions N-coll, and number of constituent-quark participants N-qp calculated from a Glauber model based on the nuclear geometry. For Au+Au, < dE(T)/d eta >/N-part increases with N-part, while < dE(T)/d eta >/N-qp is approximately constant for all three energies. This indicates that the two-component ansatz, dE(T)/d eta alpha (1 - x)N-part/2 + xN(coll), which was used to represent E-T distributions, is simply a proxy for N-qp, and that the N-coll term does not represent a hard-scattering component in E-T distributions. The dE(T)/d eta distributions of Au+Au and d+Au are then calculated from the measured p + p E-T distribution using two models that both reproduce the Au+Au data. However, while the number-of-constituent-quark-participant model agrees well with the d+Au data, the additive-quark model does not.