Motivated by the algebraic geometry and combinatorics of Cayley-Menger varieties, we seek to understand the relationship among the algebraic matroids of a variety and its secant varieties. The coordinates of the Cayley-Menger variety CM(n,d) represent pairwise distances among n points in R^d and its defining ideal consists of the polynomial relations these distances must satisfy. Its algebraic matroid captures the data of which of these distances are independent and which must satisfy a polynomial relation. From the point of view of classical algebraic geometry, CM(n,1) is the quadratic Veronese P^{n-2}, and CM(n,d) is its d-secant variety. We seek to understand the combinatorics of algebraic matroids of a variety and its secant varieties more broadly, and will discuss several classes of examples. This is joint work with Fatemah Mohammadi and Louis Theran.