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2022 AMC 12B Problem 5

Below is the professionally curated solution for Problem 5 of the 2022 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2022 AMC 12B solutions, or check the answer key.

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Concepts:transformationcoordinate geometry

Difficulty rating: 1200

5.

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?

(−3,−4)(-3, -4)

(0,5)(0, 5)

(2,−1)(2, -1)

(4,3)(4, 3)

(6,−3)(6, -3)

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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