document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1012965>Question 1012965</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The area and the perimeter of a parallelogram are 2000 sq. m. and 240 m., respectively. Find the lengths of the short and long diagonals. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #629252 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=629252\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I agree with Theo, and I believe there is some part of the problem that got lost in translation.<BR>\n" );
document.write( "The phrase \"Find the lengths of the short and long diagonals\" tells me that two diagonals of different length are expected.<BR>\n" );
document.write( "That tells me that your parallelogram must be a special kind of parallelogram.<BR>\n" );
document.write( "It is not just any parallelogram, possibly a rectangle or a square,<BR>\n" );
document.write( "because the two diagonals in a rectangle have the same length.<BR>\n" );
document.write( "There is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > solution if it is a rectangle, and it is easy to find.<BR>\n" );
document.write( "There is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > solution if it is a rhombus, and it is relatively easy to find.<BR>\n" );
document.write( "If the there are longer and shorter sides, and acute and obtuse angles,<BR>\n" );
document.write( "the parallelogram is neither a rectangle nor a rhombus,<BR>\n" );
document.write( "and you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > of solutions to the problem.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Here is how your problem should read:<BR>\n" );
document.write( "The area and the perimeter of a parallelogram shown below are 2000 sq. m. and 240 m., respectively. Find the lengths of the short and long diagonals.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C200%2C-60%2C60%2C-20%2C20%2C%0D%0Agreen%28triangle%28-57.42%2C0%2C0%2C-17.42%2C0%2C17.42%29%29%2C%0D%0Agreen%28triangle%28-57.42%2C0%2C57.42%2C0%2C0%2C17.42%29%29%2C%0D%0Agreen%28rectangle%280%2C0%2C2%2C2%29%29%2C%0D%0Aline%28-57.42%2C0%2C0%2C17.42%29%2Cline%2857.42%2C0%2C0%2C17.42%29%2C%0D%0Aline%28-57.42%2C0%2C0%2C-17.42%29%2Cline%2857.42%2C0%2C0%2C-17.42%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > (or maybe the rhombus was tilted so one side would be \"horizontal\").<BR>\n" );
document.write( "The little square in the middle, tells you that the diagonals are perpendicular.<BR>\n" );
document.write( "It also tells you that it is a rhombus, and more importantly,<BR>\n" );
document.write( "it gives you an easy way to calculate the diagonals.<BR>\n" );
document.write( "Let me name a couple variables, and show you how to solve that solvable problem.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C200%2C-60%2C60%2C-20%2C20%2C%0D%0Agreen%28triangle%28-57.42%2C0%2C0%2C-17.42%2C0%2C17.42%29%29%2C%0D%0Agreen%28triangle%28-57.42%2C0%2C57.42%2C0%2C0%2C17.42%29%29%2C%0D%0Agreen%28rectangle%280%2C0%2C2%2C2%29%29%2Clocate%28-28.7%2C8.71%2Cs%29%2C%0D%0Alocate%28-29%2C0%2Cgreen%28a%29%29%2Clocate%2828.5%2C0%2Cgreen%28a%29%29%2C%0D%0Alocate%280.2%2C9.5%2Cgreen%28b%29%29%2Clocate%280.2%2C-8%2Cgreen%28b%29%29%2C%0D%0Arectangle%28-57.42%2C-17.42%2C57.42%2C17.42%29%2C%0D%0Aline%28-57.42%2C0%2C0%2C17.42%29%2Cline%2857.42%2C0%2C0%2C17.42%29%2C%0D%0Aline%28-57.42%2C0%2C0%2C-17.42%29%2Cline%2857.42%2C0%2C0%2C-17.42%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28side=s=sqrt%28a%5E2%2Bb%5E2%29%2Cperimeter=4s=240%2Carea=2ab=2000%29 ALIGN=MIDDLE  ALT=\"system%28side=s=sqrt%28a%5E2%2Bb%5E2%29%2Cperimeter=4s=240%2Carea=2ab=2000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%284sqrt%28a%5E2%2Bb%5E2%29=240%2C2ab=2000%29 ALIGN=MIDDLE  ALT=\"system%284sqrt%28a%5E2%2Bb%5E2%29=240%2C2ab=2000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28sqrt%28a%5E2%2Bb%5E2%29=240%2F4=60%2C2ab=2000%29 ALIGN=MIDDLE  ALT=\"system%28sqrt%28a%5E2%2Bb%5E2%29=240%2F4=60%2C2ab=2000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%5E2%2Bb%5E2=60%5E2=3600%2C2ab=2000%29 ALIGN=MIDDLE  ALT=\"system%28a%5E2%2Bb%5E2=60%5E2=3600%2C2ab=2000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%5E2%2Bb%5E2%2B2ab=3600%2B2000=5600%2Cab=2000%2F2%29 ALIGN=MIDDLE  ALT=\"system%28a%5E2%2Bb%5E2%2B2ab=3600%2B2000=5600%2Cab=2000%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28%28a%2Bb%29%5E2=5600%2Cab=1000%29 ALIGN=MIDDLE  ALT=\"system%28%28a%2Bb%29%5E2=5600%2Cab=1000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2Bb=sqrt%285600%29=20sqrt%2814%29%2Cab=1000%29 ALIGN=MIDDLE  ALT=\"system%28a%2Bb=sqrt%285600%29=20sqrt%2814%29%2Cab=1000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Where do we go from here?<BR>\n" );
document.write( "There is more than one way to solve that system for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but<BR>\n" );
document.write( "we know that the solutions we look for are the solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-%28a%2Bb%29x%2Bab=0 ALIGN=MIDDLE  ALT=\"x%5E2-%28a%2Bb%29x%2Bab=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so we solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-20sqrt%2814%29%2B1000=0 ALIGN=MIDDLE  ALT=\"x%5E2-20sqrt%2814%29%2B1000=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "by using the quadratic formula or by completing the square,<BR>\n" );
document.write( "to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=10sqrt%2814%29+%2B-+20 ALIGN=MIDDLE  ALT=\"x=10sqrt%2814%29+%2B-+20\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the length of the diagonals is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a=2%2810sqrt%2814%29%2B20%29=20sqrt%2814%29-40=about114.833 ALIGN=MIDDLE  ALT=\"2a=2%2810sqrt%2814%29%2B20%29=20sqrt%2814%29-40=about114.833\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the longer diagonal, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b=2%2810sqrt%2814%29-20%29=20sqrt%2814%29-40=about34.833 ALIGN=MIDDLE  ALT=\"2b=2%2810sqrt%2814%29-20%29=20sqrt%2814%29-40=about34.833\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the shorter diagonal. <BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "I was going to do something similar to Theo's tabulation,<BR>\n" );
document.write( "resorting to the Microsoft Excel spreadsheet program<BR>\n" );
document.write( "to create a table showing different parallelograms that met the conditions in the problem,<BR>\n" );
document.write( "and calculating the length of their diagonals,<BR>\n" );
document.write( "but unless it is a rectangle or a rhombus,<BR>\n" );
document.write( "it is not fun.<BR>\n" );
document.write( "For the rectangle found by Theo, both diagonals are the same length, in meters,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28100%5E2%2B20%5E2%29=sqrt%28104000%29=20sqrt%2826%29=about101.98 ALIGN=MIDDLE  ALT=\"sqrt%28100%5E2%2B20%5E2%29=sqrt%28104000%29=20sqrt%2826%29=about101.98\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).<BR>\n" );
document.write( "Just finding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > parallelograms with different lengths diagonals would do.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );