AMC12 2019 B

AMC12 2019 B · Q3

AMC12 2019 B · Q3. It mainly tests Coordinate geometry, Transformations.

Which one of the following rigid transformations (isometries) maps the line segment $\overline{AB}$ onto the line segment $\overline{A'B'}$ so that the image of $A(-2,1)$ is $A'(2,-1)$ and the image of $B(-1,4)$ is $B'(1,-4)$?

下列哪一个刚性变换（等距变换）将线段 $\overline{AB}$ 映射到线段 $\overline{A'B'}$，使得点 $A(-2,1)$ 的像为 $A'(2,-1)$，点 $B(-1,4)$ 的像为 $B'(1,-4)$？

(A) reflection in the $y$-axis $y$轴的反射

(B) counterclockwise rotation around the origin by 90° 绕原点逆时针旋转90°

(D) reflection in the $x$-axis $x$轴的反射

(E) clockwise rotation around the origin by 180° 绕原点顺时针旋转180°

Answer

Correct choice: (E)

正确答案：（E）

Solution

Rotation around the origin by 180° can be described by the rule $(x, y) \rightarrow (-x, -y)$, so (E) is the correct answer. The observation that $\overline{AB}$ is in the second quadrant and $\overline{A'B'}$ is in the fourth quadrant excludes answer options (A), (B), and (D), because these transformations map $\overline{AB}$ into a neighboring quadrant. The translation described in (C) maps points in $\overline{AB}$ onto points in $\overline{A'B'}$, but the image is not oriented properly (for instance, $B$ is mapped to $A'$).

绕原点旋转 180° 可由规则 $(x, y) \rightarrow (-x, -y)$ 描述，故 (E) 是正确答案。注意到 $\overline{AB}$ 在第二象限而 $\overline{A'B'}$ 在第四象限，排除选项 (A)、(B) 和 (D)，因为这些变换将 $\overline{AB}$ 映射到相邻象限。选项 (C) 描述的平移将 $\overline{AB}$ 中的点映射到 $\overline{A'B'}$ 中的点，但像的方向不正确（例如，$B$ 映射到 $A'$）。

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