Question

下图显示了一条直线 $\ell$ ，其上有由正方形和线段组成的规则无限重复图案。

除恒等变换外，在下面四类平面刚体运动中，有多少类的某个变换会把该图形变到自身？

• 绕直线 $\ell$ 上某点旋转

• 沿平行于直线 $\ell$ 的方向平移

• 关于直线 $\ell$ 反射

• 关于垂直于直线 $\ell$ 的某条直线反射

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 of line $\ell$

• some translation in the direction parallel to line $\ell$

• the reflection across line $\ell$

• some reflection across a line perpendicular to line $\ell$

$0$

$1$

$2$

$3$

$4$

答案：C

知识点：变换对称性

难度评级：1220

解答：

图案沿 $\ell$ 周期性重复，所以存在非零平移使图形不变；图案也具有 $180^{\circ}$ 旋转对称。

具体来说，可绕 $\ell$ 上某点作半周旋转：取上、下相邻结构之间合适的中点作为旋转中心即可。

但关于这条直线的反射会使斜线段方向相反，不能保持图形。

关于垂直于它的直线反射也会使斜线段方向不匹配。

所以正确答案是 C。

The first transformation works, as we can rotate $\ell$ $180^{\circ}$ around the midpoint between an upward-facing and downward-facing square.

The second also works, as we can just move $\ell$ to the right until the squares line up with each other again.

The third fails, as a reflection would cause the line segments to face the opposite direction.

The fourth transformation also doesn't work since the diagonal lines would again be facing in the wrong direction.

Thus, C is the correct answer.

2019 AMC 10A 第 8 题

答案：C

难度评级：1220

解答：

其他年份的第 8 题