2019 AMC 12A Problem 6

Question

The figure below shows line

\ell

with a regular, infinite, recurring pattern of squares and line segments.

How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?

• some rotation around a point of line

\ell

• some translation in the direction parallel to line

\ell

• the reflection across line

\ell

• some reflection across a line perpendicular to line

\ell

0

1

2

3

4

Answer: C

Concepts:transformation symmetry

Difficulty rating: 1310

Small hint:

A translation by one period clearly works

Big hint:

Test a

180^\circ

rotation about a point on

\ell,

and check that neither reflection maps the diagonal segments correctly

Solution:

A translation by one full period maps the figure to itself, so translation works.

A

180^\circ

rotation about a suitable point on

\ell

sends each square above the line to the square below it, with the diagonal segments matching, so this rotation works.

Reflection across

\ell

sends the top-right diagonals to top-right diagonals below the line, but the actual below-line diagonals point to the bottom-left, so it fails. A reflection across a perpendicular line fails for the same reason. Only

2

of the four transformations work.

Thus, the correct answer is C.