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?

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

Solution 1 (video solution)

https://youtu.be/B1OXVB5GDjk

Solution 2

Solution 3 (Longer solution-credit: Ileytyn)

First, realize that each triangle is congruent, a right triangle and that the two legs are equal. Also, each side of the octagon is equal, because of the definition of regular shapes. Let $s$ be the length of a leg of the isosceles right triangle. In terms of $s$ , the hypotenuse of the isosceles right triangle, which is also the length of a side of the regular octagon, is $s \sqrt{2}$ . Since the length of each side of the square is 2000, the length of each side of the regular octagon is equal to the length of a side of the square ( $2000$ ) subtracted by $2$ times the length of a leg of the isosceles right triangle ( the total length of the side is $2s+ o$ , $o$ being the length of a side of the regular octagon), which is the same as $2s$ . As an expression, this is $2000-2s$ , which we can equate to $s \sqrt{2}$ , ( since the octagon is regular, meaning all of the side's lengths are congruent) giving us the following equation: $2000-2s = s \sqrt{2}$ . By isolating the variable and simplifying the right side, we get the following: $2000 = s(2 + \sqrt{2})$ . Dividing both sides by $(2 + \sqrt{2})$ , we arrive with $\frac{2000}{2 + \sqrt{2}} = s$ , now, to find the length of the side of the octagon, we can plug in $s$ and use the equation $2000-2s = o$ , $o$ being the length of a side of the octagon, to derive the value of a side of the octagon. After plugging in the values, we derive $2000-2(\frac{2000}{2 + \sqrt{2}})$ , which is the same as $2000-(\frac{4000}{2 + \sqrt{2}})$ , factoring out a $2000$ , we derive the following: $2000(1-(\frac{2}{2 + \sqrt{2}}))$ , by rationalizing the denominator of $\frac{2}{2 + \sqrt{2}}$ , we get $2000(1-(2 - \sqrt{2}))$ , after expanding, finally, we get $\boxed{\textbf{(B) }2000(\sqrt{2} -1)}$ !(not a factorial symbol, just an exclamation point)

2001 AMC 10 (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2001 AMC 10 Problems/Problem 20

Contents

Problem

Solution 1 (video solution)

Solution 2

Solution 3 (Longer solution-credit: Ileytyn)

See Also