document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=460423>Question 460423</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are how many triangles with perimeter 20 and integer sides such that altitudes ratio is 1:2:3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #315803 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=315803\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> None.\r<BR>\n" );
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document.write( "Let a, b, and c be the lengths of the sides of the triangle.\r<BR>\n" );
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document.write( "Then by the given, a + b + c = 20.\r<BR>\n" );
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document.write( "Now the altitudes being in the ratio 1:2:3 means that if x is the shortest altitude , then the longer altitude is 2x, and the longest altitude is 3x.  \r<BR>\n" );
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document.write( "Without loss of generality, let x be the altitude for a, 2x be the altitude for b, and 3x be the altitude for c.\r<BR>\n" );
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document.write( "Then by using the area formula, we get <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ax%29%2F2+=+%282bx%29%2F2+=+%283cx%29%2F2 ALIGN=MIDDLE  ALT=\"%28ax%29%2F2+=+%282bx%29%2F2+=+%283cx%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <==> a = 2b = 3c.<BR>\n" );
document.write( "==> c = a/3 and b = a/2.\r<BR>\n" );
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document.write( "==> a + a/2 + a/3 = 20<BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2811a%29%2F6+=+20 ALIGN=MIDDLE  ALT=\"%2811a%29%2F6+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+120%2F11 ALIGN=MIDDLE  ALT=\"a+=+120%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since a is not an integer, neither will b and c be.   </SPAN>\n" );
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