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 counterclockwise rotation around the origin by $reflection about the y = x$ , so the answer is $\boxed{\textbf{(D)}}$ .

2016 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2016 AMC 10A Problems/Problem 16

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