AMC10 2019 A

AMC10 2019 A · Q8

AMC10 2019 A · Q8. 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 $\ell$

下图显示直线 $\ell$ 上有一个规则、无限、周期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映射到自身？ * 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

Answer (C): 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 $\overline{PQ}$ in the figure below. In addition, a rotation of $180^\circ$ 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.

答案（C）：沿着与直线$\ell$平行的方向进行平移，平移量等于直线上方相邻两个正方形左边界之间的距离（或其任意整数倍），会使该图形与自身重合。平移向量可以是下图中的$\overline{PQ}$。此外，绕直线$\ell$上的任意一点作$180^\circ$旋转；该点位于直线上方某个正方形在$\ell$上的底边与直线下方最近的一个正方形在$\ell$上的底边的中点处（如图中的点$R$），也会使图形与自身重合。然而，题目给出的任一反射都会得到一个图形：直线上方正方形所连接的“尾巴”会出现在正方形的左侧而不是右侧。因此，在所列的非恒等变换中，有2个会把该图形变换到其自身。

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