2001 AMC 10 Problems/Problem 20

Problem

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?

$\textbf{(A)}\ \frac{1}{3}(2000) \qquad \textbf{(B)}\ 2000(\sqrt2-1) \qquad \textbf{(C)}\ 2000(2-\sqrt2) \textbf{(D)}\ 1000 \qquad \textbf{(E)}\ 1000\sqrt2$

Solution

$[asy] draw((0,0)--(0,10)--(10,10)--(10,0)--cycle); draw((0,7)--(3,10)); draw((7,10)--(10,7)); draw((10,3)--(7,0)); draw((3,0)--(0,3)); label("$x$",(0,1),W); label("$x\sqrt{2}$",(1.5,1.5),NE); label("$2000-2x$",(5,0),S);[/asy]$

$2000 - 2x = x\sqrt2$

$2000 = x(2 + \sqrt2)$

$x = \frac {2000}{2 + \sqrt2} = \frac {2000(2 - \sqrt2)}{2} = 1000(2 - \sqrt2)$

$x\sqrt2 = 1000(2\sqrt {2} - 2) = \boxed{\textbf{(B)}\ 2000(\sqrt2-1)}$ .

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2001 AMC 10 Problems/Problem 20

Problem

Solution