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. $(1,3,3) \Rightarrow \text {(E) }$

See Also

2000 AMC 12 (ProblemsAnswer KeyResources)
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. AMC logo.png

Invalid username
Login to AoPS