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2021 AMC 12B Problem 5

Problem 5 of 25EasierGeometry

The point P(a,b)P(a,b) in the xyxy-plane is first rotated counterclockwise by 90∘90^\circ around the point (1,5)(1,5) and then reflected about the line y=−x.y=-x. The image of PP after these two transformations is at (−6,3).(-6,3). What is b−a?b-a?

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Solution

A 90∘90^\circ counterclockwise rotation about (1,5)(1,5) sends (a,b)(a,b) to (1−(b−5), 5+(a−1))(1-(b-5),\,5+(a-1)) =(6−b, 4+a).=(6-b,\,4+a). Reflecting that about y=−xy=-x (which maps (x,y)(x,y) to (−y,−x)(-y,-x)) gives (−(4+a), −(6−b))(-(4+a),\,-(6-b)) =(−4−a, b−6).=(-4-a,\,b-6). Setting this equal to (−6,3)(-6,3) gives −4−a=−6-4-a=-6 and b−6=3,b-6=3, so a=2a=2 and b=9.b=9. Therefore b−a=9−2=7.b-a=9-2=7. Thus, the correct answer is D.

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Concepts: transformation · work backwards

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