2016 AMC 10A Problems/Problem 16

Problem

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 $\triangle ABC$?

$\textbf{(A)}$ counterclockwise rotation about the origin by $90^{\circ}$.

$\textbf{(B)}$ clockwise rotation about the origin by $90^{\circ}$.

$\textbf{(C)}$ reflection about the $x$-axis

$\textbf{(D)}$ reflection about the line $y = x$

$\textbf{(E)}$ reflection about the $y$-axis.

Solution

Consider a point $(x, y)$. Reflecting it about the $x$-axis will map it to $(x, -y)$, and rotating it counterclockwise about the origin by $90^{\circ}$ will map it to $(y, x)$. The operation that undoes this is a reflection about the $y = x$, so the answer is $\boxed{\textbf{(D)}}$.

Video Solution

https://www.youtube.com/watch?v=VHXDQz5BG8U

Video Solution 2

https://youtu.be/9AqQiA0aE24

~IceMatrix

See Also

2016 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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All AMC 10 Problems and Solutions

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