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2019 AMC 12B Problem 3

Problem 3 of 25EasierGeometry

Which one of the following rigid transformations (isometries) maps the line segment AB‾\overline{AB} onto the line segment A′B′‾\overline{A'B'} so that the image of A(−2,1)A(-2,1) is A′(2,−1)A'(2,-1) and the image of B(−1,4)B(-1,4) is B′(1,−4)?B'(1,-4)?

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Solution

Each point maps by (x,y)→(−x,−y):(x,y)\to(-x,-y): indeed (−2,1)→(2,−1)(-2,1)\to(2,-1) and (−1,4)→(1,−4).(-1,4)\to(1,-4). The map (x,y)→(−x,−y)(x,y)\to(-x,-y) is a 180∘180^\circ rotation about the origin. Thus, E is the correct answer.

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Concepts: transformation · coordinate geometry

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