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. Replace $z$ with its additive inverse: $(1,2,-3)$

Step 2: Rotate around $x$ -axis 180 degrees. Replace $y$ and $z$ with their respective additive inverses. $(1, -2, 3)$

Step 3: Translate $5$ units in positive- $y$ direction. Replace $y$ with $y+5$ . $(1,3,3) \Rightarrow \text {(E) }$

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
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

2000 AMC 12 Problems/Problem 10

Problem

Solution

See Also