Question

顶点为 $A(0, 2), B(-3, 2)$ ，和 $C(-3, 0)$ 的三角形先关于 $x$ 轴反射，得到的像 $\triangle A'B'C'$ 再绕原点逆时针旋转 $90^{\circ}$ ，得到 $\triangle A''B''C''$ 。下列哪个变换能把 $\triangle A''B''C''$ 变回 $\triangle ABC$ ？

A triangle with vertices $A(0, 2), B(-3, 2),$ and $C(-3, 0)$ is reflected about the $x$ -axis, then the image $\triangle A'B'C'$ is rotated counterclockwise about the origin by $90^{\circ}$ to produce $\triangle A''B''C''.$ Which of the following transformations will return $\triangle A''B''C''$ to $\triangle ABC?$

绕原点逆时针旋转 $90^{\circ}$

counterclockwise rotation about the origin by $90^{\circ}$

绕原点顺时针旋转 $90^{\circ}$

clockwise rotation about the origin by $90^{\circ}$

关于 $x$ 轴反射

reflection about the $x$ -axis

关于直线 $y = x$ 反射

reflection about the line $y = x$

关于 $y$ 轴反射

reflection about the $y$ -axis

答案：D

知识点：变换坐标几何

难度评级：1220

解答：

为判断如何逆转这些变换，可以跟踪一个任意点。

设该点为 $(x, y)$ 。关于 $x$ 轴反射后，它变为 $(x, -y)$ 。

再逆时针旋转后得到 $(y, x)$ 。要把它变回 $(x, y)$ ，唯一的方法是关于 $y = x$ 反射。

所以正确答案是 D。

To figure out how to reverse the transformations, we can analyze a single point and see what happens to it.

Let $(x, y)$ be the point. After being reflected about the $x$ -axis, the point would go to $(x, -y).$

Rotating this counterclockwise would put it at $(y, x).$ The only transformation that puts this back at $(x, y)$ is reflection about $y = x.$

Thus, the correct answer is D .

2016 AMC 10A 第 16 题

答案：D

难度评级：1220

解答：

其他年份的第 16 题