Geometric interpretation of linear programs with both inequality and equality constraints

Question

I recently do a homework with linear programming, the standard form as:

$$\begin{array}{ll} \text{minimize} &\displaystyle \mathbf{c}^T\mathbf{x}+\mathbf{d}\\ \text{subject to} & \mathbf{A}\mathbf{x} \leq \mathbf{b}\\ & \mathbf{A_0}\mathbf{x} = \mathbf{b_0}\end{array}$$

Here we have the inequality and equality constraints. The homework says that the overall constraints is a polytope. Here are my questions:

1. The inequality $Ax \leq b$ is a polytope, which is clear for me. However, why this polytope intersects with $A_0x = b_0$ (this equality constraint seems to define a hyperplane) will become a hyperplane not a polytope. For example, consider intersection $ax \le b$ with $a_0x = b_0 $ is just a half line. Am I correct?

2. What is the geometric interpretation of a general $A_0x=b_0$?

Thanks a lot for the help.