10) A spider wants to inspect the external surface of a square-based pyramid, whose lateral faces are equilateral triangles. Starting from the center of one lateral face, it wants to touch the centers of all the other lateral faces, following the shortest possible path. Knowing that an edge of the pyramid measures 2, find the total length of the path.
(A) 4
(B) $2 \sqrt{3}$
(C) 3
(D) $\frac{3}{2} \sqrt{2}$
(E) none of the above.
10) The answer is $(\mathbf{B})$. The shortest path that connects the centers of two adjacent lateral faces of the pyramid can be found by considering on the plane two equilateral triangles sharing a side and connecting the two respective centroids with a segment.

This segment lies on the heights of the two triangles relative to the common side. Since the centroid is $\frac{1}{3}$ of the height from the base, the segment will be $\frac{2}{3}$ of the height $h$ of an equilateral triangle with side length 2. Knowing that $h=2 \cdot \frac{\sqrt{3}}{2}$, the segment is $\frac{2}{3} \sqrt{3}$ long. Finally, it is observed that the spider's path is as long as 3 of these segments (sufficient to connect the four centers). The spider therefore travels a distance of $2 \sqrt{3}$.

Note. The shortest path among those that pass from one center to an adjacent one has been sought. It is easy to observe that any other order of vertices to be traversed results in longer paths than the one found.

![](https://cdn.mathpix.com/cropped/2024_04_17_d4fc67ca786af90b2fcdg-3.jpg?height=814&width=505&top_left_y=153&top_left_x=1432)