## 41. Knight's Armor

In the museum hall, there was a square sheet of iron (with a side of 3.414 m); such iron was once used to make knight's armor. The museum keeper decided that this exhibit would look more interesting (perhaps even reminiscent of a shield of an ancient knight) if the four corners of this sheet were tucked into four triangular niches in such a way that instead of a square, an octagon with equal sides would be visible. The master to whom this task was entrusted wondered what the dimensions of the triangular niches should be. Help him.
41. Each niche in the cross-section represents an isosceles right triangle with a leg length of $x$ (meters). The hypotenuse, therefore, is $\sqrt{2} x$. This hypotenuse is equal to one of the sides of the regular octagon.

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-176.jpg?height=748&width=734&top_left_y=734&top_left_x=661)

Thus, each side of the original square needs to be divided into three parts: two parts of length $x$, which are embedded in the niches, and one part of length $\sqrt{2} x$, corresponding to the side of the octagon.

Therefore, $(2+\sqrt{2}) x = 3.414$ m, from which, with a high degree of accuracy, $x \approx 1$ m. Thus, each triangular niche should have approximate dimensions of $1 \mathrm{~m}, 1$ m, and $1.414 \mathrm{~m}$.