2009 AMC 10A 第 21 题

Question

许多哥特式大教堂的窗户中，有一些部分含有一圈全等小圆，并由一个大圆外接。在图中，小圆的数量为四个。四个小圆的面积之和与大圆面积的比是多少？

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?

3 - 2\sqrt{2}

2 - \sqrt{2}

4(3 - 2\sqrt{2})

\dfrac{1}{2}(3 - \sqrt{2})

2\sqrt{2} - 2

答案：C

知识点：相切圆面积比分母有理化

难度评级：1690

小提示：

设每个小圆半径为

1

；它们的圆心构成边长为

2

的正方形。

Let each small circle have radius

1;

their centers form a square of side

2

大提示：

大圆的直径等于该正方形的对角线加上两个小圆半径。

The large circle’s diameter is the square’s diagonal plus two small radii

解答：

设每个小圆半径为

1

。它们的圆心构成边长为

2

的正方形，该正方形对角线为

2\sqrt2

。

大圆的直径为

2 + 2\sqrt2

，所以半径为

1 + \sqrt2

。

所求比值为

\begin{aligned} \dfrac{4 \cdot \pi (1)^2}{\pi (1 + \sqrt2)^2} &= \dfrac{4}{3 + 2\sqrt2} \\ &= 4(3 - 2\sqrt2) \end{aligned}\text{。}

所以正确答案是 C。

Let each small circle have radius

1.

Their centers form a square of side

2,

whose diagonal is

2\sqrt2.

The large circle’s diameter is

2 + 2\sqrt2,

so its radius is

1 + \sqrt2.

The desired ratio is

\begin{aligned} \dfrac{4 \cdot \pi (1)^2}{\pi (1 + \sqrt2)^2} &= \dfrac{4}{3 + 2\sqrt2} \\ &= 4(3 - 2\sqrt2). \end{aligned}

Thus, the correct answer is C.

2009 AMC 10A 第 21 题

答案：C

难度评级：1690

小提示：

大提示：

解答：

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