{"status": "success", "data": {"description_md": "Let $ABCD$ be a [[rhombus]] with $AC = 16$ and $BD = 30$. Let $N$ be a point on $\\overline{AB}$, and let $P$ and $Q$ be the feet of the perpendiculars from $N$ to $\\overline{AC}$ and $\\overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $PQ$?\n  \n![[asy] size(200); defaultpen(0.6); pair O = (15*15/17,8*15/17), C = (17,0), D = (0,0), P = (25.6,19.2), Q = (25.6, 18.5); pair A = 2*O-C, B = 2*O-D; pair P = (A+O)/2, Q=(B+O)/2, N=(A+B)/2; draw(A--B--C--D--cycle); draw(A--O--B--O--C--O--D); draw(P--N--Q); label(\"\\(A\\)\",A,WNW); label(\"\\(B\\)\",B,ESE); label(\"\\(C\\)\",C,ESE); label(\"\\(D\\)\",D,SW); label(\"\\(P\\)\",P,SSW); label(\"\\(Q\\)\",Q,SSE); label(\"\\(N\\)\",N,NNE); [/asy]](https://latex.artofproblemsolving.com/c/2/2/c22d4d20155faaf8bd0cf51f26a5795d14b32322.png)\n\n$\\mathrm{(A)}\\ 6.5\\qquad\\mathrm{(B)}\\ 6.75 \\qquad\\mathrm{(C)}\\ 7\\qquad\\mathrm{(D)}\\ 7.25\\qquad\\mathrm{(E)}\\ 7.5$\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>Let <span class=\"katex--inline\">ABCD</span> be a [[rhombus]] with <span class=\"katex--inline\">AC = 16</span> and <span class=\"katex--inline\">BD = 30</span>. Let <span class=\"katex--inline\">N</span> be a point on <span class=\"katex--inline\">\\overline{AB}</span>, and let <span class=\"katex--inline\">P</span> and <span class=\"katex--inline\">Q</span> be the feet of the perpendiculars from <span class=\"katex--inline\">N</span> to <span class=\"katex--inline\">\\overline{AC}</span> and <span class=\"katex--inline\">\\overline{BD}</span>, respectively. Which of the following is closest to the minimum possible value of <span class=\"katex--inline\">PQ</span>?</p>&#10;<p><img src=\"https://latex.artofproblemsolving.com/c/2/2/c22d4d20155faaf8bd0cf51f26a5795d14b32322.png\" alt=\"[asy] size(200); defaultpen(0.6); pair O = (15*15/17,8*15/17), C = (17,0), D = (0,0), P = (25.6,19.2), Q = (25.6, 18.5); pair A = 2*O-C, B = 2*O-D; pair P = (A+O)/2, Q=(B+O)/2, N=(A+B)/2; draw(A--B--C--D--cycle); draw(A--O--B--O--C--O--D); draw(P--N--Q); label(&#34;(A)&#34;,A,WNW); label(&#34;(B)&#34;,B,ESE); label(&#34;(C)&#34;,C,ESE); label(&#34;(D)&#34;,D,SW); label(&#34;(P)&#34;,P,SSW); label(&#34;(Q)&#34;,Q,SSE); label(&#34;(N)&#34;,N,NNE); [/asy]\"/></p>&#10;<p><span class=\"katex--inline\">\\mathrm{(A)}\\ 6.5\\qquad\\mathrm{(B)}\\ 6.75 \\qquad\\mathrm{(C)}\\ 7\\qquad\\mathrm{(D)}\\ 7.25\\qquad\\mathrm{(E)}\\ 7.5</span></p>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2003 AMC 12B Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/03_amc12B_p23", "prev": "/problem/03_amc12B_p21"}}