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 $b - a ?$

$1$

$3$

$5$

$7$

$9$

Answer: D

Concepts:transformation coordinate geometry work backwards

Difficulty rating: 1220

Video solution:

Click to load, then click again to play

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

Answer: D

Difficulty rating: 1220

Video solution:

Written solution:

Problem 9 in Other Years