In this paper we investigate a family of matroids introduced by Ardila and Billey to study one-dimensional intersections of complete flag arrangements of $\mathbb{C}^n$. The set of lattice points $P_n$ inside the equilateral triangle $S_n$ obtained by intersecting the nonnegative cone of $\mathbb{R}^3$ with the affine hyperplane $x_1 + x_2 + x_3 = n-1$ is the ground set of a matroid $\mathcal{T}_n$ whose independent sets are the subsets $S$ of $P_n$ satisfying that $|S \cap P| \le k$ for each translation $P$ of the set $P_k$. Here we study the structure of the matroids $\mathcal{T}_n$ in connection with tilings of $S_n$ into unit triangles, rhombi, and trapezoids. First, we characterize the independent sets of $\mathcal{T}_n$, extending a characterization of the bases of $\mathcal{T}_n$ already given by Ardila and Billey. Then we explore the connection between the rank function of $\mathcal{T}_n$ and the tilings of $S_n$ into unit triangles and rhombi. Then we provide a tiling characterization of the circuits of $\mathcal{T}_n$. We conclude with a geometric characterization of the flats of $\mathcal{T}_n$.