2019 AMC 12B Problem 3

Below is the professionally curated solution for Problem 3 of the 2019 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 12B solutions, or check the answer key.

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Concepts:transformation coordinate geometry

Difficulty rating: 1080

Which one 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 $B'(1,-4)?$

reflection in the $y$ -axis

counterclockwise rotation around the origin by $90^\circ$

translation by $3$ units to the right and $5$ units down

reflection in the $x$ -axis

clockwise rotation about the origin by $180^\circ$

Solution:

Each point maps by $(x,y)\to(-x,-y):$ indeed $(-2,1)\to(2,-1)$ and $(-1,4)\to(1,-4).$

The map $(x,y)\to(-x,-y)$ is a $180^\circ$ rotation about the origin.

Thus, E is the correct answer.

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