{"status": "success", "data": {"description_md": "The figure below shows line $\\ell$ with a regular, infinite, recurring pattern of squares and line segments.<br><center><img class=\"problem-image\" alt=\"[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 &lt;= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i &lt;= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0)); [/asy]\" class=\"latexcenter\" height=\"72\" src=\"https://latex.artofproblemsolving.com/c/6/6/c66986a019a481aa0dad6c2a690111ffd5a52014.png\" width=\"502\"/></center><br>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?<br>*some rotation around a point of line $\\ell$<br>*some translation in the direction parallel to line $\\ell$<br>*the reflection across line $\\ell$<br>*some reflection across a line perpendicular to line $\\ell$\n\n$\\textbf{(A) } 0 \\qquad\\textbf{(B) } 1 \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } 3 \\qquad\\textbf{(E) } 4$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>The figure below shows line  <span class=\"katex--inline\">\\ell</span>  with a regular, infinite, recurring pattern of squares and line segments.<br/><center><img class=\"latexcenter\" alt=\"[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 &lt;= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i &lt;= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0)); [/asy]\" height=\"72\" src=\"https://latex.artofproblemsolving.com/c/6/6/c66986a019a481aa0dad6c2a690111ffd5a52014.png\" width=\"502\"/></center><br/>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?<br/>*some rotation around a point of line  <span class=\"katex--inline\">\\ell</span> <br/>*some translation in the direction parallel to line  <span class=\"katex--inline\">\\ell</span> <br/>*the reflection across line  <span class=\"katex--inline\">\\ell</span> <br/>*some reflection across a line perpendicular to line  <span class=\"katex--inline\">\\ell</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 0 \\qquad\\textbf{(B) } 1 \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } 3 \\qquad\\textbf{(E) } 4</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 2, "problem_name": "2019 AMC 12A Problem 6", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12A_p07", "prev": "/problem/19_amc12A_p05"}}