2016 AMC 10A Problem 16

Problem 16 of 25IntermediateGeometry

A triangle with vertices

A(0, 2),

B(-3, 2),

and

C(-3, 0)

is reflected about the

x

-axis, then the image

\triangle A'B'C'

is rotated counterclockwise about the origin by

90^{\circ}

to produce

\triangle A''B''C''.

Which of the following transformations will return

\triangle A''B''C''

\triangle ABC?

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Solution

To figure out how to reverse the transformations, we can analyze a single point and see what happens to it. Let

(x, y)

be the point. After being reflected about the

x

-axis, the point would go to

(x, -y).

Rotating this counterclockwise would put it at

(y, x).

The only transformation that puts this back at

(x, y)

is reflection about

y = x.

Thus, the correct answer is D .