35. Given a line $l$, two points $A$ and $B$ on the same side of it, and a segment $a$. Find a segment $X Y$ on the line $l$ of length $a$ such that the length of the broken line $A X Y B$ is the smallest (Fig. 42).
35. Since the length of the segment $X Y$ is equal to $a$, it is necessary that the sum $A X + B Y$ be minimal. Suppose that the segment $X Y$ is laid out. The sliding symmetry with the axis $l$ "of the parallel translation magnitude $a$ translates point $B$ to $B'$, and point $Y$ to $X$ (Fig. 144); therefore, $B Y = B' X$ and, consequently, $A X + B Y = A X + B' X$. Thus, it is required that the length of the broken line $A X B'$ be the smallest. From this, it follows that $X$ is the point of intersection of the line $l$ with the segment $A B'$.

![](https://cdn.mathpix.com/cropped/2024_05_21_42e14062cf36c74a4d8dg-176.jpg?height=383&width=332&top_left_y=760&top_left_x=104)

Fig. 144.

![](https://cdn.mathpix.com/cropped/2024_05_21_42e14062cf36c74a4d8dg-176.jpg?height=402&width=659&top_left_y=728&top_left_x=533)

Fig. 145.