AMC 10 2014a Problem 23

A rectangular piece of paper whose length is $\sqrt3$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $\frac{B}{A}$?

$\textbf{(A)}\ \frac{1}{2} \textbf{(B)}\ \frac{3}{5} \textbf{(C)}\ \frac{2}{3} \textbf{(D)}\ \frac{3}{4} \textbf{(E)}\ \frac{4}{5}$