2021 AMC 10B Problem 9

Problem 9 of 25EasierGeometry

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 ?

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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 .

More practice

Concepts: transformation · coordinate geometry · work backwards

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.