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

Undo the transformations: reflecting

(-6,3)

about

y=-x

reverses the last step

Reflecting

(x,y)

about

y=-x

gives

(-y,-x);

90^\circ

rotation about

(1,5)

sends

(a,b)

(6-b,\,4+a)

90^\circ

counterclockwise rotation about

(1,5)

sends

(a,b)

(1-(b-5),\,5+(a-1))

=(6-b,\,4+a).

Reflecting that about

y=-x

(which maps

(x,y)

(-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,

a=2

and

b=9.

Therefore

b-a=9-2=7.

Thus, the correct answer is D.

Problem 5 in other years