AMC10 2001 A

AMC10 2001 A · Q20

AMC10 2001 A · Q20. It mainly tests Exponents & radicals, Polygons.

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?

通过从边长为2000的正方形的每个角切掉一个等腰直角三角形，形成一个正八边形。正八边形的每边长是多少？

(A) $\frac{1}{3}(2000)$ $\frac{1}{3}(2000)$

(B) $2000(\sqrt{2} - 1)$ $2000(\sqrt{2} - 1)$

(D) 1000 1000

(E) $1000\sqrt{2}$ $1000\sqrt{2}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) Let $x$ represent the length of each side of the octagon, which is also the length of the hypotenuse of each of the right triangles. Each leg of the right triangles has length $\frac{x\sqrt{2}}{2}$, so $$2\cdot\frac{x\sqrt{2}}{2}+x=2000,\text{ and }x=\frac{2000}{\sqrt{2}+1}=2000(\sqrt{2}-1).$$

（B）设 $x$ 表示八边形每条边的长度，这也等于每个直角三角形斜边的长度。每个直角三角形的直角边长度为 $\frac{x\sqrt{2}}{2}$，所以 $$2\cdot\frac{x\sqrt{2}}{2}+x=2000,\text{ 且 }x=\frac{2000}{\sqrt{2}+1}=2000(\sqrt{2}-1).$$

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