AMC12 2019 A

AMC12 2019 A · Q6

AMC12 2019 A · Q6. It mainly tests Transformations, Symmetry.

The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments. How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself? • some rotation around a point on line $\ell$ • some translation in the direction parallel to line $\ell$ • the reflection across line $\ell$ • some reflection across a line perpendicular to line $\ell$

下面的图形显示直线 $\ell$ 上有一个规则的、无限的、重复的正方形和线段图案。在这个图形所在的平面中，以下四种刚体运动变换（除了恒等变换外），有多少种会将这个图形变换成自身？ • 绕直线 $\ell$ 上某点的某些旋转 • 平行于直线 $\ell$ 方向的某些平移 • 关于直线 $\ell$ 的反射 • 关于垂直于直线 $\ell$ 的直线的某些反射

(A) 0 0

(B) 1 1

(D) 3 3

(E) 4 4

Answer

Correct choice: (C)

正确答案：（C）

Solution

A translation in the direction parallel to line $\ell$ by an amount equal to the distance between the left sides of successive squares above the line (or any integer multiple thereof), will take the figure to itself. The translation vector could be PQ in the figure below. In addition, a rotation of 180° around any point on line $\ell$ that is halfway between the bases on $\ell$ of a square above the line and a nearest square below the line, such as point R in the figure, will also take the figure to itself. Either of the given reflections, however, will result in a figure in which the “tails” attached to the squares above the line are on the left side of the squares instead of the right side. Therefore 2 of the listed non-identity transformations will transform this figure into itself.

平行于直线 $\ell$ 方向的平移，平移量等于线上面连续正方形左侧之间的距离（或其任何整数倍），会将图形映射到自身。平移向量可以是图中的 PQ。此外，绕直线 $\ell$ 上任意一点的 180° 旋转，该点位于线上面一个正方形底边与最近的线下面正方形底边中间的位置（如图中的点 R），也会将图形映射到自身。然而，给定的两种反射任一种都会导致线上面正方形附着的“尾巴”在正方形的左侧而不是右侧。因此，列出的非恒等变换中有 2 种会将此图形变换成自身。

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