2022 AMC 12B Problem 10

Regular hexagon ABCDEF has side length 2. Let G be the midpoint of \overline{AB}, and let H be the midpoint of \overline{DE}. What is the perimeter of GCHF?

\textbf{(A) }\ 4\sqrt3 \qquad \textbf{(B) }\ 8 \qquad \textbf{(C) }\ 4\sqrt5 \qquad \textbf{(D) }\ 4\sqrt7 \qquad \textbf{(E) }\ 12

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.