Symmetry
Source: 2019 AMC 10A #8 / AMC 12A #6
February 8, 2019
AMCAMC 10symmetryAMC 122019 AMC 12A2019 AMC 10A2019 AMC
Problem Statement
The figure below shows line with a regular, infinite, recurring pattern of squares and line segments.[asy]
size(300);
defaultpen(linewidth(0.8));
real r = 0.35;
path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r);
path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r);
for(int i=0;i <= 4;i=i+1)
{
draw(shift((4*i,0)) * P);
draw(shift((4*i,0)) * Q);
}
for(int i=1;i <= 4;i=i+1)
{
draw(shift((4*i-2,0)) * Pp);
draw(shift((4*i-1,0)) * Qp);
}
draw((-1,0)--(18.5,0),Arrows(TeXHead));
[/asy]
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
[*] some translation in the direction parallel to line
[*] the reflection across line
[*] some reflection across a line perpendicular to line