AMC10 2009 A

AMC10 2009 A · Q21

AMC10 2009 A · Q21. It mainly tests Circle theorems, Area & perimeter.

Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle. In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?

许多哥特式大教堂的窗户中有部分包含一个由全等圆组成的环，这些圆被一个更大的圆外切。在图示中，小圆的数量是四个。小圆面积之和与大圆面积的比率为多少？

(A) $3 - 2\sqrt{2}$ $3 - 2\sqrt{2}$

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

(D) $\frac{1}{2}(3 - \sqrt{2})$ $\frac{1}{2}(3 - \sqrt{2})$

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

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): It may be assumed that the smaller circles each have radius 1. Their centers form a square with side length 2 and diagonal length $2\sqrt{2}$. Thus the diameter of the large circle is $2+2\sqrt{2}$, so its area is $(1+\sqrt{2})^2\pi=(3+2\sqrt{2})\pi$. The desired ratio is $$ \frac{4\pi}{(3+2\sqrt{2})\pi}=4(3-2\sqrt{2}). $$

答案（C）：可以假设每个小圆的半径为 1。它们的圆心构成一个边长为 2、对角线长为 $2\sqrt{2}$ 的正方形。因此大圆的直径为 $2+2\sqrt{2}$，其面积为 $(1+\sqrt{2})^2\pi=(3+2\sqrt{2})\pi$。所求比值为 $$ \frac{4\pi}{(3+2\sqrt{2})\pi}=4(3-2\sqrt{2}). $$

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