document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=473138>Question 473138</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A garden shaped like a right trianglr has one leg 6 meters less that the length of the other. The hypotenuse is 6 meters less than 2 times the length of the shorter leg.  What are the lengths of the sides? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #324428 by </B><A HREF=/tutors/aboutme.mpl?userid=karaoz><B>karaoz(32)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=karaoz><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=karaoz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=324428\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br> There are three unknown values:<BR>\n" );
document.write( "a = shorter leg<BR>\n" );
document.write( "b = longer leg<BR>\n" );
document.write( "c = hypotenuse<BR>\n" );
document.write( "<br>Translation of the relevant information into algebraic statements:<BR>\n" );
document.write( "b = a + 6<BR>\n" );
document.write( "c = 2a - 6<BR>\n" );
document.write( "a = ?<BR>\n" );
document.write( "b = ?<BR>\n" );
document.write( "<br>So, there are three variables and only two equations.<BR>\n" );
document.write( "To be able to solve the system we need one more equation.<BR>\n" );
document.write( "This equation will come from the fact that the triangle is a right triangle.<BR>\n" );
document.write( "For any right triangle, it is true that a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> and this will be our third equation.<BR>\n" );
document.write( "Hence the system to solve is:<br><BR>\n" );
document.write( "b = a + 6<BR>\n" );
document.write( "c = 2a - 6<BR>\n" );
document.write( "a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup><BR>\n" );
document.write( "<br>We can quickly reduce the system by substituting b and c in the third equation by the expressions on the right hand sides of the first two equations.<BR>\n" );
document.write( "This will give us:<br><BR>\n" );
document.write( "a<sup>2</sup> + (a + 6)<sup>2</sup> = (2a - 6)<sup>2</sup>,<BR>\n" );
document.write( "which is one equation with one unknown.<BR>\n" );
document.write( "<br>After squaring the expressions in brackets, we have<BR>\n" );
document.write( "a<sup>2</sup> + a<sup>2</sup> + 12a + 36 = 4a<sup>2</sup> -24a + 36<BR>\n" );
document.write( "<br>After simplifying, the equation becomes:<BR>\n" );
document.write( "2a<sup>2</sup> - 36a = 0, which is equivalent to:<BR>\n" );
document.write( "2a(a - 18) = 0<BR>\n" );
document.write( "<br>There are two solutions to this equation: a = 0 and a = 18.<BR>\n" );
document.write( "If a = 0 then b = 6 and c = -6, which is clearly not a real triangle even though the values satisfy all the three equations.<BR>\n" );
document.write( "So, we will disregard this solution and keep only a = 18.<BR>\n" );
document.write( "<br>When a = 18 then b = 24 and c = 30.<BR>\n" );
document.write( "So the answer to the question is:<BR>\n" );
document.write( "<i>The lengths of the sides are 18 meters and 24 meters.</i>\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );