Question

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 $b - a ?$

$1$

$3$

$5$

$7$

$9$

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Written solution:

Work backward. Reflecting $(-6,3)$ across $y=-x$ gives $(-3,6)$ .

Now undo the $90^\circ$ counterclockwise rotation by rotating $(-3,6)$ clockwise about $(1,5)$ . Relative to $(1,5)$ , the point is $(-4,1)$ . A clockwise quarter-turn sends this to $(1,4)$ , and translating back gives $(2,9)$ .

Thus $a=2$ , $b=9$ , and $b-a=7$ .

Thus, the answer is D .

2021 AMC 10B Spring Problem 9

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Problem 9 in Other Years