2023 AMC 10A Problem 19

Problem 19 of 25HarderGeometry

The line segment formed by

A(1, 2)

and

B(3, 3)

is rotated to the line segment formed by

A'(3, 1)

and

B'(4, 3)

about the point

P(r, s).

What is

|r - s|?

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Solution

A rotation keeps its center equidistant from each point and its image. So

P

is equidistant from

A

and

A',

and from

B

and

B',

which puts it at the intersection of two perpendicular bisectors. The bisector of

BB'

from

(3,3)

(4,3)

x = 3.5.

The bisector of

AA'

from

(1,2)

(3,1)

2x - y = 2.5.

Then

y = 2(3.5) - 2.5 = 4.5,

P = (3.5, 4.5)

and

|r - s| = |3.5 - 4.5| = 1.

Thus, E is the correct answer.