2007 AMC 8 第 12 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
12.
一个单位六角星由一个边长为 的正六边形和它的 个等边三角形延伸部分组成，如图所示。延伸部分的面积与原正六边形面积之比是多少？
A unit hexagram is composed of a regular hexagon of side length and its equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?
小提示：
将正六边形分成六个全等等边三角形
Divide the regular hexagon into six congruent equilateral triangles.
大提示：
每个外侧三角形都与六个内部三角形之一共边
Each exterior triangle shares a side with one of the six interior triangles.
解答：
可以把六边形分成 个全等等边三角形，如下图。
因为每个内部三角形都与一个外侧三角形共边，所以所有这些三角形全等。因此面积比为 。
所以正确答案是 A。
Note that we can split the hexagon into congruent equilateral triangles as follows.
Since each of them share an edge with an exterior triangle, all the triangles are congruent. Therefore, the ratio of areas is
Thus, A is the correct answer.
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