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Difference between revisions of "2001 AMC 10 Problems/Problem 20"

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== Solution 2==
 
== Solution 2==
  
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== Solution 3 (Longer solution-credit: Ileytyn)==
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== Solution 3 (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 <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)
 
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
  
 
== See Also ==
 
== See Also ==

Revision as of 12:13, 4 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 $2000$. 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)

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)

~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

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