Lower bounds on non-central sections of isotropic convex bodies

For fixed $t_0\in [0,\sqrt{3}]$ we give asymptotically sharp lower bounds on the quantity $L_K{\text{vol}}_{d-1}(K\cap H)$, where $H$ is a hyperplane at distance $t_0{L}_K$ from the origin, $K$ is any symmetric isotropic convex body in ${\mathbb{R}}^d$, and $L_K$ stands for the isotropic constant of $K$.