Question

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 $\triangle ABC?$

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

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

reflection about the $x$ -axis

reflection about the line $y = x$

reflection about the $y$ -axis

Solution:

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 Problem 16

Solution:

Problem 16 in Other Years