{"status": "success", "data": {"description_md": "The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\\sqrt{m} + \\sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$<br><center><img class=\"problem-image\" alt='[asy] /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label(\"A\",A,N); label(\"B\",B,W); label(\"C\",C,S); label(\"D\",D,S); label(\"E\",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); [/asy]' class=\"latexcenter\" height=\"252\" src=\"https://latex.artofproblemsolving.com/8/d/3/8d30b5bdb8799955b16c29f895eab1b33ec5d028.png\" width=\"225\"/></center>\n\n$\\textbf{(A)} ~20 \\qquad\\textbf{(B)} ~21 \\qquad\\textbf{(C)} ~22 \\qquad\\textbf{(D)} ~23 \\qquad\\textbf{(E)} ~24$\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 is constructed from  <span class=\"katex--inline\">11</span>  line segments, each of which has length  <span class=\"katex--inline\">2</span> . The area of pentagon  <span class=\"katex--inline\">ABCDE</span>  can be written as  <span class=\"katex--inline\">\\sqrt{m} + \\sqrt{n}</span> , where  <span class=\"katex--inline\">m</span>  and  <span class=\"katex--inline\">n</span>  are positive integers. What is  <span class=\"katex--inline\">m + n ?</span> <br/><center><img class=\"latexcenter\" alt=\"[asy] /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label(&#34;A&#34;,A,N); label(&#34;B&#34;,B,W); label(&#34;C&#34;,C,S); label(&#34;D&#34;,D,S); label(&#34;E&#34;,E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); [/asy]\" height=\"252\" src=\"https://latex.artofproblemsolving.com/8/d/3/8d30b5bdb8799955b16c29f895eab1b33ec5d028.png\" width=\"225\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)} ~20 \\qquad\\textbf{(B)} ~21 \\qquad\\textbf{(C)} ~22 \\qquad\\textbf{(D)} ~23 \\qquad\\textbf{(E)} ~24</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": 3, "problem_name": "2021 AMC 12B Problem 15", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc12B_p16", "prev": "/problem/21_amc12B_p14"}}