Let H(Rn) denote the hyperspace of all non-empty compact subsets of Rn. The Hausdorff metric h provides a way to measure distances between two elements of H(Rn) and generates the complete metric space (H(Rn),h)). In this paper, we examine geometric properties of lines in H(Rn), as determined by the Hausdorff metric, and compare and contrast the properties of these lines with Euclidean lines in Rn. Several surprising properties of these objects will be highlighted.