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2022 AMC 12B Problem 5

Problem 5 of 25EasierGeometry

The point (−1,−2)(-1, -2) is rotated 270∘270^\circ counterclockwise about the point (3,1).(3, 1). What are the coordinates of its new position?

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Solution

Relative to the center (3,1),(3, 1), the point is at (−1−3, −2−1)=(−4,−3).(-1 - 3,\, -2 - 1) = (-4, -3). A 270∘270^\circ counterclockwise rotation sends (x,y)(x, y) to (y,−x),(y, -x), so (−4,−3)(-4, -3) becomes (−3,4).(-3, 4). Translating back gives (3−3, 1+4)=(0,5).(3 - 3,\, 1 + 4) = (0, 5). Thus, the correct answer is B.

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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.