Classical computational geometry algorithms handle geometric constructs whose shapes and locations are exact. However, many real-world applications require modeling and computing with geometric uncertainties, which are often coupled and mutually dependent. In this paper we address distance problems and orthogonal range queries in the plane, subject to geometric uncertainty. Point coordinates and range ncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), a general and computationally efficient worst-case, first-order linear approximation of geometric uncertainty that supports dependence among uncertainties. We present algorithms for closest pair, diameter and bounding box problems, and efficient algorithms for uncertain range queries: uncertain range/nominal points, nominal range/uncertain points, uncertain range/uncertain points, with independent/dependent uncertainties.