A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length $2000$. What is the length of each side of the octagon? $\textbf{(A) } \frac{1}{3}(2000) \qquad$ $ \textbf{(B) } {2000(\sqrt{2}-1)} \qquad$ $ \textbf{(C) } {2000(2-\sqrt{2})} \qquad$ $ \textbf{(D) } {1000} \qquad$ $ \textbf{(E) } {1000\sqrt{2}}$
Solution
Let $x$ be the length of each leg of the four congruent isosceles right triangles cut from the square's corners. Each diagonal cut is a side of the octagon, so it has length $x\sqrt2$.
Each original square side loses length $x$ at both ends. The remaining segment is therefore $2000-2x$, and this is also a side of the regular octagon.
Equating the two side lengths gives $x\sqrt2=2000-2x$, so $x=\dfrac{2000}{2+\sqrt2}$.
The octagon's side length is $x\sqrt2=\dfrac{2000\sqrt2}{2+\sqrt2}=2000(\sqrt2-1)$, answer (B).
A regular hexagon is formed by cutting a congruent equilateral triangle from each of the three corners of an equilateral triangle with side length $12$. What is the length of each side of the hexagon?
