9) In the Wild West, two straight rivers flow parallel to each other at a distance of $10 \mathrm{~km}$. The medicine man of an Indian village, who is in his hut, must collect a vial of water from each of the two rivers and bring the vials to the village for the prayer ceremony to Manitou. What is the minimum distance of the path he must travel, knowing that the hut and the village are equidistant from the two rivers and are $48 \mathrm{~km}$ apart?
(A) 48
(B) 52
(C) 53
(D) 58
(E) $\sqrt{48^{2}+10^{2}}$
9) The answer is (B). Let $C A B V$ be the broken line representing the wizard's path. By performing a reflection with respect to the straight bank passing through $A$, the broken line $A B V$ is transformed into $A B^{\prime} V^{\prime}$; by performing a second reflection with respect to the line passing through $B^{\prime}$ parallel to the banks, the segment $B^{\prime} V^{\prime}$ is transformed into $B^{\prime} V^{\prime \prime}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_cc98646a8b0a08f1c276g-2.jpg?height=382&width=829&top_left_y=1750&top_left_x=1106)
We observe that the length of the broken line $C A^{\prime} B^{\prime} V^{\prime \prime}$ is equal to that of the broken line $C A B V$ and that the position of $V^{\prime \prime}$ does not depend on the path chosen by the wizard. Indeed, $V V^{\prime \prime} \perp C V$ and $V V^{\prime \prime}=20 \mathrm{~km}$. The length of the path will therefore be minimal when the points $C, A^{\prime}, B^{\prime}$, and $V^{\prime \prime}$ are aligned, i.e., all belong to the segment $C V^{\prime \prime}$. This segment has a length of $\sqrt{20^{2}+48^{2}} \mathrm{~km}=52 \mathrm{~km}$, which is the minimum path required.