1992 AMC 12 第 9 题

Question

五个边长均为

2\sqrt3

的等边三角形排列在同一直线的一侧，且每个三角形都有一条边在该直线上。沿着这条直线，一个三角形底边的中点是下一个三角形的一个顶点。这五个三角形区域的并集所覆盖的平面区域面积为

Five equilateral triangles, each with side

2\sqrt3,

are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is

10

12

15

10\sqrt3

12\sqrt3

答案：E

知识点：equilateral triangle area 容斥原理 overlapping regions

难度评级：1780

小提示：

先把五个大三角形的面积相加，再减去重叠部分

Add the areas of the five large triangles, then subtract their overlaps

大提示：

每一对相邻三角形的重叠部分都是边长为原来一半的等边三角形

Each adjacent overlap is an equilateral triangle with half the original side length

解答：

每个大三角形的面积为

\frac{\sqrt3}{4}(2\sqrt3)^2=3\sqrt3\text{。}

四个重叠部分都是边长为

\sqrt3

的等边三角形，面积均为

\frac{3\sqrt3}{4}

。不存在三个三角形共同重叠的区域，所以并集的面积为

5(3\sqrt3)-4\left(\frac{3\sqrt3}{4}\right)=12\sqrt3\text{。}

因此正确答案是 E。

Each large triangle has area

\frac{\sqrt3}{4}(2\sqrt3)^2=3\sqrt3.

Each of the four overlaps is an equilateral triangle of side

\sqrt3,

hence area

\frac{3\sqrt3}{4}.

There are no triple overlaps, so the union has area

5(3\sqrt3)-4\left(\frac{3\sqrt3}{4}\right)=12\sqrt3.

Thus the correct answer is E.

1992 AMC 12 第 9 题

答案：E

难度评级：1780

小提示：

大提示：

解答：

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