2001 AMC 10 Problems/Problem 20
Contents
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
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 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)
See Also
2001 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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