{"status": "success", "data": {"description_md": "In square $ABCD$, points $E$ and $H$ lie on $\\overline{AB}$ and $\\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\\overline{BC}$ and $\\overline{CD}$, respectively, and points $I$ and $J$ lie on $\\overline{EH}$ so that $\\overline{FI} \\perp \\overline{EH}$ and $\\overline{GJ} \\perp \\overline{EH}$. See the figure below. Triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each has area $1.$ What is $FI^2$?\n\n<center>\n<img class=\"problem-image\" height=\"245\" src=\"https://latex.artofproblemsolving.com/4/8/9/4892da26cb6334b359f58d4007f89460883779ab.png\" width=\"252\"/>\n</center>\n\n$\\textbf{(A) } \\frac{7}{3} \\qquad \\textbf{(B) } 8-4\\sqrt2 \\qquad \\textbf{(C) } 1+\\sqrt2 \\qquad \\textbf{(D) } \\frac{7}{4}\\sqrt2 \\qquad \\textbf{(E) } 2\\sqrt2$", "description_html": "<p>In square  <span class=\"katex--inline\">ABCD</span> , points  <span class=\"katex--inline\">E</span>  and  <span class=\"katex--inline\">H</span>  lie on  <span class=\"katex--inline\">\\overline{AB}</span>  and  <span class=\"katex--inline\">\\overline{DA}</span> , respectively, so that  <span class=\"katex--inline\">AE=AH.</span>  Points  <span class=\"katex--inline\">F</span>  and  <span class=\"katex--inline\">G</span>  lie on  <span class=\"katex--inline\">\\overline{BC}</span>  and  <span class=\"katex--inline\">\\overline{CD}</span> , respectively, and points  <span class=\"katex--inline\">I</span>  and  <span class=\"katex--inline\">J</span>  lie on  <span class=\"katex--inline\">\\overline{EH}</span>  so that  <span class=\"katex--inline\">\\overline{FI} \\perp \\overline{EH}</span>  and  <span class=\"katex--inline\">\\overline{GJ} \\perp \\overline{EH}</span> . See the figure below. Triangle  <span class=\"katex--inline\">AEH</span> , quadrilateral  <span class=\"katex--inline\">BFIE</span> , quadrilateral  <span class=\"katex--inline\">DHJG</span> , and pentagon  <span class=\"katex--inline\">FCGJI</span>  each has area  <span class=\"katex--inline\">1.</span>  What is  <span class=\"katex--inline\">FI^2</span> ?</p>\n<center>\n<img class=\"problem-image\" height=\"245\" src=\"https://latex.artofproblemsolving.com/4/8/9/4892da26cb6334b359f58d4007f89460883779ab.png\" width=\"252\"/>\n</center>\n<p> <span class=\"katex--inline\">\\textbf{(A) } \\frac{7}{3} \\qquad \\textbf{(B) } 8-4\\sqrt2 \\qquad \\textbf{(C) } 1+\\sqrt2 \\qquad \\textbf{(D) } \\frac{7}{4}\\sqrt2 \\qquad \\textbf{(E) } 2\\sqrt2</span> </p>\n<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": 4, "problem_name": "2020 AMC 10B Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/20_amc10B_p22", "prev": "/problem/20_amc10B_p20"}}