2019 AMC 10A 第 8 题

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

小提示：

沿

\ell

方向的平移会保留重复图案。

Translations along

\ell

preserve the repeating pattern

大提示：

检查反射是否会把斜线段方向反过来。

Check whether reflections reverse the slanted segments

解答：

第一种变换可行：可以绕

\ell

上某点旋转

180^{\circ}

，旋转中心取在一个朝上的正方形与一个朝下的正方形之间的中点。

第二种变换也可行：只要把图形沿

\ell

向右平移，直到各正方形重新对齐即可。

第三种不行，因为这样的反射会使线段方向相反。

第四种变换同样不行，因为斜线段又会指向错误的方向。

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

小提示：

大提示：

解答：

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