  We consider a random graph G(n,p) whose vertex set V has been randomly
embedded in the unit square and whose edges are given weight equal to the
geometric distance between their end vertices. Then each pair {u,v} of vertices
have a distance in the weighted graph, and a Euclidean distance. The stretch
factor of the embedded graph is defined as the maximum ratio of these two
distances, over all u,v in V. We give upper and lower bounds on the stretch
factor (holding asymptotically almost surely), and show that for p not too
close to 0 or 1, these bounds are best possible in a certain sense. Our results
imply that the stretch factor is bounded with probability tending to 1 if and
only if n(1-p) tends to 0, answering a question of O'Rourke.
