Difference between revisions of "2001 AMC 10 Problems/Problem 20"
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<!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length <math> 2000 </math>. What is the length of each side of the octagon?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length <math> 2000 </math>. What is the length of each side of the octagon?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> | ||
− | <math> \textbf{(A)} \frac{1}{3}(2000) \qquad \textbf{(B)} {2000(\sqrt{2}-1)} \qquad \textbf{(C)} {2000(2-\sqrt{2})} | + | <math> \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}} </math> | + | \qquad \textbf{(D) } {1000} \qquad \textbf{(E) } {1000\sqrt{2}} </math> |
== Solution 1 (video solution) == | == Solution 1 (video solution) == | ||
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https://youtu.be/B1OXVB5GDjk | https://youtu.be/B1OXVB5GDjk | ||
− | == Solution 2== | + | == Solution 2 (Longer solution)== |
− | Let <math> | + | 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 <math>s</math> be the length of a leg of the isosceles right triangle. In terms of <math>s</math>, the hypotenuse of the isosceles right triangle, which is also the length of a side of the regular octagon, is <math>s \sqrt{2}</math>. 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 (<math>2000</math>) subtracted by <math>2</math> times the length of a leg of the isosceles right triangle ( the total length of the side is <math>2s+ o</math>, <math>o</math> being the length of a side of the regular octagon), which is the same as <math> 2s </math>. As an expression, this is <math>2000-2s</math>, which we can equate to <math>s \sqrt{2}</math>, ( since the octagon is regular, meaning all of the side's lengths are congruent) giving us the following equation:<math>2000-2s = s \sqrt{2}</math>. By isolating the variable and simplifying the right side, we get the following: <math>2000 = s(2 + \sqrt{2})</math>. Dividing both sides by <math>(2 + \sqrt{2})</math>, we arrive with <math>\frac{2000}{2 + \sqrt{2}} = s</math>, now, to find the length of the side of the octagon, we can plug in <math>s</math> and use the equation <math>2000-2s = o </math>, <math>o</math> 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 <math>2000-2(\frac{2000}{2 + \sqrt{2}})</math>, which is the same as <math>2000-(\frac{4000}{2 + \sqrt{2}})</math>, factoring out a <math> 2000 </math>, we derive the following: <math> 2000(1-(\frac{2}{2 + \sqrt{2}}))</math>, by rationalizing the denominator of <math> \frac{2}{2 + \sqrt{2}} </math>, we get <math> 2000(1-(2 - \sqrt{2})) </math>, after expanding, finally, we get <math>\boxed{\textbf{(B) }2000(\sqrt{2} -1)}</math> !(not a factorial symbol, just an exclamation point) |
− | + | ~Ileytyn | |
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== See Also == | == See Also == |
Revision as of 09:27, 25 June 2024
Problem
A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length . What is the length of each side of the octagon?
Solution 1 (video solution)
Solution 2 (Longer solution)
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 be the length of a leg of the isosceles right triangle. In terms of
, the hypotenuse of the isosceles right triangle, which is also the length of a side of the regular octagon, is
. 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 (
) subtracted by
times the length of a leg of the isosceles right triangle ( the total length of the side is
,
being the length of a side of the regular octagon), which is the same as
. As an expression, this is
, which we can equate to
, ( since the octagon is regular, meaning all of the side's lengths are congruent) giving us the following equation:
. By isolating the variable and simplifying the right side, we get the following:
. Dividing both sides by
, we arrive with
, now, to find the length of the side of the octagon, we can plug in
and use the equation
,
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
, which is the same as
, factoring out a
, we derive the following:
, by rationalizing the denominator of
, we get
, after expanding, finally, we get
!(not a factorial symbol, just an exclamation point)
~Ileytyn
See Also
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 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.