document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=163403>Question 163403</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangle has a diagonal of length 20 ft. and a perimeter or 56 ft. Find the dimensions of the rectangle. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #120388 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=120388\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> diagonal 20 feet.<BR>\n" );
document.write( "perimeter 56 feet.<BR>\n" );
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document.write( "diagonal of the rectangle is the hypotenuse of the 2 triangles created by it.<BR>\n" );
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document.write( "let x = length of the rectangle<BR>\n" );
document.write( "let y = width of the rectangle.<BR>\n" );
document.write( "x + y are also legs of the right triangles created.<BR>\n" );
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document.write( "formula for perimeter of the rectangle is<BR>\n" );
document.write( "2*x + 2*y = 56<BR>\n" );
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document.write( "formula for legs of the right triangle is<BR>\n" );
document.write( "x^2 + y^2 = 20^2<BR>\n" );
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document.write( "since 2*x + 2*y = 56, then<BR>\n" );
document.write( "2*y = 56 - 2*x<BR>\n" );
document.write( "which becomes<BR>\n" );
document.write( "y = 28 - x<BR>\n" );
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document.write( "substituting for y in the second equation gets<BR>\n" );
document.write( "x^2 + (28-x)^ = 20^2<BR>\n" );
document.write( "which expands to become<BR>\n" );
document.write( "x^2 + 784 - 56*x + x^2 = 400<BR>\n" );
document.write( "subtracting 400 from both sides of the equation and combining like terms gets<BR>\n" );
document.write( "2*x^2 - 56*x + 384 = 0<BR>\n" );
document.write( "dividing both sides of the equation by 2 gets<BR>\n" );
document.write( "x^2 - 28*x + 192 = 0<BR>\n" );
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document.write( "solving this equation (i used the quadratic formula because i wasn't able to see the correct factors) yields<BR>\n" );
document.write( "x = 16 or x = 12<BR>\n" );
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document.write( "both values are good by substituting in the original equation.<BR>\n" );
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document.write( "if x = 12, then <BR>\n" );
document.write( "2*x + 2*y = 56 becomes 24+2*y=56 becomes 2*y = 32 becomes y = 16<BR>\n" );
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document.write( "if x = 16, then<BR>\n" );
document.write( "2*x + 2*y = 56 becomes 32+2*y=56 becomes 2*y = 24 becomes y = 12<BR>\n" );
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document.write( "if x = 12, y = 16<BR>\n" );
document.write( "if x = 16, y = 12<BR>\n" );
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document.write( "substituting either one in the equation for the legs of the triangle gets<BR>\n" );
document.write( "x^2 + y^2 = 400 becomes (12)^2 + (16)^2 = 400 becomes 144 + 256 = 400 becomes 400 = 400.<BR>\n" );
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document.write( "x = 12 and y = 16, or x = 16 and y = 12 works for both the perimeter and for the legs of the triangle so these values are good.<BR>\n" );
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document.write( "answer is<BR>\n" );
document.write( "x = 12 or 16<BR>\n" );
document.write( "y = 16 or 12<BR>\n" );
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document.write( "the rectangle is 12 by 16, or 16 by 12, whichever way you want to look at it.<BR>\n" );
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