# 2019 AMC 12B Problems/Problem 3

## Problem

Which 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 $B'(1, -4)$? $\textbf{(A) }$ reflection in the $y$-axis $\textbf{(B) }$ counterclockwise rotation around the origin by $90^{\circ}$ $\textbf{(C) }$ translation by 3 units to the right and 5 units down $\textbf{(D) }$ reflection in the $x$-axis $\textbf{(E) }$ clockwise rotation about the origin by $180^{\circ}$

## Solution

We can simply graph the points, or use coordinate geometry, to realize that both $A'$ and $B'$ are, respectively, obtained by rotating $A$ and $B$ by $180^{\circ}$ about the origin. Hence the rotation by $180^{\circ}$ about the origin maps the line segment $\overline{AB}$ to the line segment $\overline{A'B'}$, so the answer is $\boxed{(\text{E})}$.

~Dodgers66

## Video Solution 1

~Education, the Study of Everything

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