2021 AMC 12B Spring Problem 5

Attempt Problem 5 of the 2021 AMC 12B Spring below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Spring solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

$1$

$3$

$5$

$7$

$9$

Answer: D

Concepts:transformation work backwards

Difficulty rating: 1330

Solution:

A $90^\circ$ counterclockwise rotation about $(1,5)$ sends $(a,b)$ to $(1-(b-5),\,5+(a-1))$ $=(6-b,\,4+a).$

Reflecting that about $y=-x$ (which maps $(x,y)$ to $(-y,-x)$ ) gives $(-(4+a),\,-(6-b))$ $=(-4-a,\,b-6).$

Setting this equal to $(-6,3)$ gives $-4-a=-6$ and $b-6=3,$ so $a=2$ and $b=9.$

Therefore $b-a=9-2=7.$

Thus, the correct answer is D.

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