Minimum area of tarp to cover a carpet
To solve this, consider what happens when exactly $k$ corners of $C$ lie inside $T_1$, for $k\in \{0,1,2,3,4\}$, where $C\setminus T_1$ denotes the area of $C$ not covered by $T_1$:


If $0$ corners of $C$ lie in $T_1$, then $T_1$ doesn't cover any corner of $C$. So any rectangle covering $C\setminus T_1$ must cover all corners of $C$, so the smallest of these rectangles is $C$ itself.
If exactly $1$ corner of $C$ lies in $T_1$, the $C\setminus T_1$ contains $3$ corners of $C$ and the smallest rectangle covering this area is again $C$ itself.
If exactly $2$ corners of $C$ lie in $T_1$, then $C\setminus T_1$ is a rectangle, so the smallest rectangle covering it is itself. More precisely, this is the rectangle with as corners the two intersection points of the edges of $C$ and $T_1$ and the two corners of $C$ not covered by $T_1$.
If $3$ corners of $C$ lie in $T_1$, then $4$ corners of $C$ lie in $T_1$, so $k$ cannot be $3$.
If $4$ corners of $C$ lie inside $T_1$, then $C$ is fully covered by $T_1$, so the smallest rectangle covering the remainder has area $0$.


So, to determine the size of the smallest rectangle covering $C\setminus T_1$, we simply count the number of corners of $C$ inside $T_1$, apply the proper case and we're done.