¶ 2019 AMC 10A Problem 8
Problem:
The figure below shows line 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 on line
. some translation in the direction parallel to line
. the reflection across line
. some reflection across a line perpendicular to line
Answer Choices:
A.
B.
C.
D.
E.
Solution:
A translation in the direction parallel to line by an amount equal to the distance between the left sides of successive squares above the line (or any integer multiple thereof), will take the figure to itself. The translation vector could be in the figure below. In addition, a rotation of around any point on line that is halfway between the bases on of a square above the line and a nearest square below the line, such as point in the figure, will also take the figure to itself. Either of the given reflections, however, will result in a figure in which the "tails" attached to the squares above the line are on the left side of the squares instead of the right side. Therefore of the listed non-identity transformations will transform this figure into itself.
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