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2000 AMC 12 Problem 10

Problem 10 of 25EasierGeometry

The point P=(1,2,3)P = (1, 2, 3) is reflected in the xyxy-plane, then its image QQ is rotated by 180∘180^\circ about the xx-axis to produce R,R, and finally, RR is translated by 55 units in the positive yy direction to produce S.S. What are the coordinates of S?S?

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Solution

Reflecting (1,2,3)(1, 2, 3) in the xyxy-plane gives Q=(1,2,−3).Q = (1, 2, -3). Rotating 180∘180^\circ about the xx-axis negates yy and z,z, giving R=(1,−2,3).R = (1, -2, 3). Translating 55 units in the positive yy direction gives S=(1,3,3).S = (1, 3, 3). Thus, the correct answer is E.

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Concepts: transformation · 3D geometry · coordinate geometry

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.