10) A regular hexagon with side length $1 \mathrm{~m}$ is given, whose vertices, listed clockwise, are $A, B, C, D, E, F$. Let $X$ and $Y$ be the intersections of segment $A C$ with segments $B F$ and $B D$ respectively. Calculate the distance between $X$ and $Y$
(A) $\frac{1}{2} \mathrm{~m}$
(B) $\frac{\sqrt{3}}{3} \mathrm{~m}$
(C) $\frac{\sqrt{3}}{6} \mathrm{~m}$
(D) $\frac{2 \sqrt{3}}{3} \mathrm{~m}$
(E) $\frac{\sqrt{3}}{2} \mathrm{~m}$
10. The answer is $\mathbf{( B )}$.

Referring to the figure, consider the triangle $B X Y$. This is an equilateral triangle, and we can calculate its height $h$ as

$$
h=\frac{(B E-C D)}{2}=\frac{1}{2} \mathrm{~m}
$$

Thus, $X Y$ is the side of an equilateral triangle with height $1 / 2 \mathrm{~m}$. From this, we get $X Y=\frac{2}{\sqrt{3}} h=\frac{1}{\sqrt{3}} \mathrm{~m}=\frac{\sqrt{3}}{3} \mathrm{~m}$.

[Problem proposed by S. Mongodi]

![](https://cdn.mathpix.com/cropped/2024_04_17_ed8bf509241053ca04f5g-3.jpg?height=464&width=420&top_left_y=1183&top_left_x=1449)