Which one of the following rigid transformations (isometries) maps the line segment AB onto the line segment A'B' so that the image of A(-2,1) is A'(2,-1) and the image of B(-1,4) is B'(1,-4)?

(E) clockwise rotation about the origin by 180°

2019 AMC 12B Problem 3

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Which one of the following rigid transformations (isometries) maps the line segment

\overline{AB}

onto the line segment

\overline{A'B'}

so that the image of

A(-2,1)

A'(2,-1)

and the image of

B(-1,4)

B'(1,-4)?

reflection in the

y

-axis

counterclockwise rotation around the origin by

90^\circ

translation by

3

units to the right and

5

units down

reflection in the

x

-axis

clockwise rotation about the origin by

180^\circ

Track the coordinate change:

(-2,1)\to(2,-1)

and

(-1,4)\to(1,-4)

Both points follow the rule

(x,y)\to(-x,-y)

Each point maps by

(x,y)\to(-x,-y):

indeed

(-2,1)\to(2,-1)

and

(-1,4)\to(1,-4).

The map

(x,y)\to(-x,-y)

is a

180^\circ

rotation about the origin.

Thus, E is the correct answer.