2024 AMC 12A Problem 18

On top of a rectangular card with sides of length

1

and

2+\sqrt{3}

, an identical card is placed so that two of their diagonals line up, as shown (

\overline{AC}

, in this case). Continue the process, adding a third card to the second, and so on, lining up successive diagonals after rotating clockwise. In total, how many cards must be used until a vertex of a new card lands exactly on the vertex labeled

B

in the figure?

Two identical rectangular cards with successive diagonals aligned along AC

\text{(A)}\;6

\text{(B)}\;8

\text{(C)}\;10

\text{(D)}\;12

\text{(E)}\;\text{No new vertex will land on B.}