1957 AHSME Problems/Problem 31

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Problem

A regular octagon is to be formed by cutting equal isosceles right triangles from the corners of a square. If the square has sides of one unit, the leg of each of the triangles has length:

$\textbf{(A)}\ \frac{2 + \sqrt{2}}{3} \qquad \textbf{(B)}\ \frac{2 - \sqrt{2}}{2}\qquad \textbf{(C)}\ \frac{1+\sqrt{2}}{2}\qquad\textbf{(D)}\ \frac{1+\sqrt{2}}{3}\qquad\textbf{(E)}\ \frac{2-\sqrt{2}}{3}$


Solution

[asy]  real x = 8*(2-sqrt(2))/2;  // Square draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0));  // Corners draw((0,x)--(x,0)); draw((8-x,0)--(8,x)); draw((8,8-x)--(8-x,8)); draw((x,8)--(0,8-x));  [/asy]

$\boxed{\textbf{(B) }\frac{2-\sqrt{2}}2}$.

See Also

1957 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 30
Followed by
Problem 32
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