  Let $\mathcal{O}_c$ be the category of finite-length central-charge-$c$
modules for the Virasoro Lie algebra whose composition factors are irreducible
quotients of reducible Verma modules. Recently, it has been shown that
$\mathcal{O}_c$ admits vertex algebraic tensor category structure for any
$c\in\mathbb{C}$. Here, we determine the structure of this tensor category when
$c=13-6p-6p^{-1}$ for an integer $p>1$. For such $c$, we prove that
$\mathcal{O}_{c}$ is rigid, and we construct projective covers of irreducible
modules in a natural tensor subcategory $\mathcal{O}_{c}^0$. We then compute
all tensor products involving irreducible modules and their projective covers.
Using these tensor product formulas, we show that $\mathcal{O}_c$ has a
semisimplification which, as an abelian category, is the Deligne product of two
tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig
categories for affine $\mathfrak{sl}_2$ at levels $-2+p^{\pm 1}$. Next, as a
straightforward consequence of the braided tensor category structure on
$\mathcal{O}_c$ together with the theory of vertex operator algebra extensions,
we rederive known results for triplet vertex operator algebras
$\mathcal{W}(p)$, including rigidity, fusion rules, and construction of
projective covers. Finally, we prove a recent conjecture of Negron that
$\mathcal{O}_c^0$ is braided tensor equivalent to the
$PSL(2,\mathbb{C})$-equivariantization of the category of
$\mathcal{W}(p)$-modules.
