2019 AMC 10A Problem 8

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: 1220

Small hint:

Translations along

\ell

preserve the repeating pattern

Big hint:

Check whether reflections reverse the slanted segments

Solution:

The first transformation works, as we can rotate

\ell

180^{\circ}

around the midpoint between an upward-facing and downward-facing square.

The second also works, as we can just move

\ell

to the right until the squares line up with each other again.

The third fails, as a reflection would cause the line segments to face the opposite direction.

The fourth transformation also doesn’t work since the diagonal lines would again be facing in the wrong direction.

Thus, C is the correct answer.