{"status": "success", "data": {"description_md": "Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\\circ}, 180^{\\circ},$ and $270^{\\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)\n\n$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 17 \\qquad \\textbf{(D) } 20 \\qquad \\textbf{(E) } 25$", "description_html": "<p>Let <span class=\"katex--inline\">T</span> be the triangle in the coordinate plane with vertices <span class=\"katex--inline\">(0,0), (4,0),</span> and <span class=\"katex--inline\">(0,3).</span> Consider the following five isometries (rigid transformations) of the plane: rotations of <span class=\"katex--inline\">90^{\\circ}, 180^{\\circ},</span> and <span class=\"katex--inline\">270^{\\circ}</span> counterclockwise around the origin, reflection across the <span class=\"katex--inline\">x</span>-axis, and reflection across the <span class=\"katex--inline\">y</span>-axis. How many of the <span class=\"katex--inline\">125</span> sequences of three of these transformations (not necessarily distinct) will return <span class=\"katex--inline\">T</span> to its original position? (For example, a <span class=\"katex--inline\">180^{\\circ}</span> rotation, followed by a reflection across the <span class=\"katex--inline\">x</span>-axis, followed by a reflection across the <span class=\"katex--inline\">y</span>-axis will return <span class=\"katex--inline\">T</span> to its original position, but a <span class=\"katex--inline\">90^{\\circ}</span> rotation, followed by a reflection across the <span class=\"katex--inline\">x</span>-axis, followed by another reflection across the <span class=\"katex--inline\">x</span>-axis will not return <span class=\"katex--inline\">T</span> to its original position.)</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A) } 12 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 17 \\qquad \\textbf{(D) } 20 \\qquad \\textbf{(E) } 25</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2020 AMC 10A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/20_amc10A_p24", "prev": "/problem/20_amc10A_p22"}}