document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=986115>Question 986115</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A farmer has a field in the shape of right angled triangle with legs (sides other than the hypotenuse) of lengths 16 m and 8 m. She wants to leave a space in the form of a square  of largest area for growing wheat and remaining area for growing vegetables. Find the length of the side of such a square <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #606894 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=606894\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE >\n" );
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document.write( "We are given a right-angled triangle &nbsp;<B>ABC</B>&nbsp; (see the <B>Figure</B>)&nbsp; with the legs &nbsp;|<B>AB</B>| = 8 &nbsp;and &nbsp;|<B>AC</B>| = 16.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\r<BR>\n" );
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document.write( "A square &nbsp;<B>ADEF</B>&nbsp; is inscribed to the triangle &nbsp;<B>ABC</B>&nbsp; in such a way that the triangle and \r<BR>\n" );
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document.write( "the square have the common right angle &nbsp;<I>L</I><B>A</B>.\r<BR>\n" );
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document.write( "We need to find the length of the square side. \r<BR>\n" );
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document.write( "Let <B>x</B> be the square side length. \r<BR>\n" );
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document.write( "Then the length of the segment &nbsp;<B>DB</B>&nbsp; is &nbsp;8 - x&nbsp; and the length of the segment &nbsp;<B>CF</B>&nbsp; is &nbsp;16 - x. \r<BR>\n" );
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document.write( "The triangles &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA ALIGN=MIDDLE  ALT=\"DELTA\"   DISABLED_style= \"max-width:90%; height:auto;\" ><B>DBE</B>&nbsp; and  &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA ALIGN=MIDDLE  ALT=\"DELTA\"   DISABLED_style= \"max-width:90%; height:auto;\" ><B>FEC</B>&nbsp; are right angled triangles. \r<BR>\n" );
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document.write( "They have congruent acute angles &nbsp;<I>L</I><B>FCE</B>&nbsp; and &nbsp;<I>L</I><B>DEB</B>&nbsp; as these angles are the corresponding \r<BR>\n" );
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document.write( "angles at the parallel lines &nbsp;<B>AC</B>&nbsp; and &nbsp;<B>DE</B>&nbsp; and the transverse &nbsp;<B>BC</B>. \r<BR>\n" );
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document.write( "Hence, the triangles &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA ALIGN=MIDDLE  ALT=\"DELTA\"   DISABLED_style= \"max-width:90%; height:auto;\" ><B>DBE</B>&nbsp; and  &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA ALIGN=MIDDLE  ALT=\"DELTA\"   DISABLED_style= \"max-width:90%; height:auto;\" ><B>FEC</B>&nbsp; are similar right-angled triangles. \r<BR>\n" );
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document.write( "Therefore, &nbsp;their legs are proportional: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+200%2C+250%2C+0.5%2C+4.5%2C+0.5%2C+5.5%2C+%0D%0A++++++++++++line%28+1%2C+1%2C+4%2C+1%29%2C+%0D%0A++++++++++++line%28+1%2C+1%2C+1%2C+5%29%2C%0D%0A++++++++++++line%28+4%2C+1%2C+1%2C+5%29%2C+%0D%0A+++++++blue%28line%28+1%2C+2.714%2C+2.714%2C+2.714%29%29%2C+%0D%0A+++++++blue%28line%28+2.714%2C+1%2C+2.714%2C+2.714%29%29%2C+%0D%0A%0D%0A++++++++++++locate+%28+0.9%2C++1%2C++++A%29%2C%0D%0A++++++++++++locate+%28+4.0%2C++1%2C++++B%29%2C%0D%0A++++++++++++locate+%28+0.9%2C++5.3%2C++C%29%2C%0D%0A++++++++++++locate+%28+0.78%2C+2.95%2C+F%29%2C%0D%0A++++++++++++locate+%28+2.8%2C++2.95%2C+E%29%2C%0D%0A++++++++++++locate+%28+2.65%2C+1.0%2C++D%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure</B> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28CF%29%2Fabs%28FE%29 ALIGN=MIDDLE  ALT=\"abs%28CF%29%2Fabs%28FE%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28ED%29%2Fabs%28DB%29 ALIGN=MIDDLE  ALT=\"abs%28ED%29%2Fabs%28DB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;&nbsp;&nbsp;or&nbsp;&nbsp;&nbsp;&nbsp; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816-x%29%2Fx ALIGN=MIDDLE  ALT=\"%2816-x%29%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%288-x%29 ALIGN=MIDDLE  ALT=\"x%2F%288-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(3) \r<BR>\n" );
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document.write( "Now we need to solve the last equation for the unknown &nbsp;<B>x</B>. &nbsp;Simplify the equation &nbsp;(3)&nbsp; step by step: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816-x%29%2A%288-x%29 ALIGN=MIDDLE  ALT=\"%2816-x%29%2A%288-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2A8+-+16x+-+8x+%2B+x%5E2 ALIGN=MIDDLE  ALT=\"16%2A8+-+16x+-+8x+%2B+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=128+-+24x+%2B+x%5E2 ALIGN=MIDDLE  ALT=\"128+-+24x+%2B+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24x ALIGN=MIDDLE  ALT=\"24x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=128 ALIGN=MIDDLE  ALT=\"128\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=128%2F24 ALIGN=MIDDLE  ALT=\"128%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2F3 ALIGN=MIDDLE  ALT=\"16%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<B>Answer</B>. &nbsp;The side measure of the square is of &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; m long. \r<BR>\n" );
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document.write( "For a similar problem see the lesson &nbsp;<A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Triangles/Miscellaneous-problems-on-similar-triangles.lesson>Miscellaneous problems on similar triangles</A>&nbsp; in this site. \r<BR>\n" );
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