document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=495663>Question 495663</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the least possible diagonal of a rectangle with a perimeter of 100 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #336238 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=336238\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If a and b are the side lengths with a+b = 50, we can find the minimum possible value of a diagonal several ways:\r<BR>\n" );
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document.write( "If you know power means, we have <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sqrt{\frac{a^2+b^2}{2}} \ge \frac{a+b}{2} \Rightarrow \sqrt{\frac{a^2+b^2}{2}} \ge 25 \Rightarrow \sqrt{a^2+b^2} \ge 25\sqrt{2}\">, equality occurs when a=b (square).<BR>\n" );
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document.write( "The Cauchy-Schwarz inequality states that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a^2+b^2)(1+1) \ge (a+b)^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a^2+b^2)(2) \ge 50^2 \Rightarrow \sqrt{a^2+b^2} \ge 25\sqrt{2}\">\r<BR>\n" );
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document.write( "We can say that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a+b)^2 = a^2 + b^2 + 2ab = d^2 + 2ab\">, the LHS is constant so we can maximize 2ab (twice the area of the rectangle) to minimize the diagonal d. We can show several ways the optimal case is a=b.<BR>\n" );
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document.write( "All of these solutions yield <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 25\sqrt{2}\"> as the minimal possible diagonal. </SPAN>\n" );
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