1995 AMC 12 第 9 题

Question

图形由一个正方形、它的两条对角线以及连接两组对边中点的线段组成。图中各种大小的三角形总数是

Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is

10

12

14

16

18

答案：D

知识点：geometric counting 对称性

难度评级：1380

小提示：

先按大小分类，再数三角形

Separate the triangles by size before counting

大提示：

先数一个方向的三角形，再利用正方形的旋转对称性

Count one orientation and use the square’s rotational symmetry

解答：

最小的三角形有

8

个，每个由半条边、半条对角线和半条中位线围成。以正方形的一整条边为底、中心为顶点的三角形有

4

个；由对角线截出的半正方形三角形另有

4

个。总数为

8+4+4=16

。因此正确答案是 D。

There are

8

smallest triangles, each bounded by a half-side, a half-diagonal, and a half-midline. There are

4

triangles whose base is a full side and vertex is the center, and

4

half-square triangles cut off by a diagonal. The total is

8+4+4=16.

Thus the correct answer is D.

1995 AMC 12 第 9 题

答案：D

难度评级：1380

小提示：

大提示：

解答：

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