2000 AMC 12 Problems/Problem 10

Problem

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 $S$ ?

$\text {(A) } (1,7, - 3) \qquad \text {(B) } ( - 1,7, - 3) \qquad \text {(C) } ( - 1, - 2,8) \qquad \text {(D) } ( - 1,3,3) \qquad \text {(E) } (1,3,3)$

Solution

Step 1: Reflect in the xy plane. That is the same as creating the additive inverse of z and sticking it into the z coordinate. $(1,2,-3)$

Step 2: Rotate around x-axis 180 degrees. That is the same as taking the additive inverse of y and sticking it in the y coordinate, then taking the additive inverse of z and sticking it into the z coordinate. $(1, -2, 3)$

Step 3: Translate 5 units in positive-y direction. That is the same as adding 5 to y and sticking it in the y coordinate. $(1,3,3) \Rightarrow \text {(E) }$

2000 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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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2000 AMC 12 Problems/Problem 10

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