Problem 101

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Problem 101: (2016 AMC 10b, 23)

In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$? $\textbf{(A)}\ \frac{1}{3}\qquad$ $\textbf{(B)}\ \frac{10}{27}\qquad$ $\textbf{(C)}\ \frac{11}{27}\qquad$ $\textbf{(D)}\ \frac{4}{9}\qquad$ $\textbf{(E)}\ \frac{13}{27}$

Solution

Exercise 101: Solution

In regular hexagon $ABCDEF$, points $W,X,Y,Z$ lie on sides $\overline{BC},\overline{CD},\overline{EF},\overline{FA}$, respectively. Lines $AB$, $ZW$, $YX$, and $ED$ are parallel, with the distance from $AB$ to $ZW$ equal to the distance from $YX$ to $ED$, and each of these distances twice the distance from $ZW$ to $YX$. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$?

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