2000 AMC 12 Problem 10

Below is the professionally curated solution for Problem 10 of the 2000 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2000 AMC 12 solutions, or check the answer key.

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Concepts:transformation 3D geometry coordinate geometry

Difficulty rating: 1390

10.

The point $P = (1, 2, 3)$ is reflected in the $xy$ -plane, then its image $Q$ is rotated by $180^\circ$ about the $x$ -axis to produce $R,$ and finally, $R$ is translated by $5$ units in the positive $y$ direction to produce $S.$ What are the coordinates of $S?$

$(1, 7, -3)$

$(-1, 7, -3)$

$(-1, -2, 8)$

$(-1, 3, 3)$

$(1, 3, 3)$

Solution:

Reflecting $(1, 2, 3)$ in the $xy$ -plane gives $Q = (1, 2, -3).$

Rotating $180^\circ$ about the $x$ -axis negates $y$ and $z,$ giving $R = (1, -2, 3).$

Translating $5$ units in the positive $y$ direction gives $S = (1, 3, 3).$

Thus, the correct answer is E.

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