document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=282344>Question 282344</A>: <I><SPAN STYLE=\"word-break: break-all;\"> which ways can I make a triangle with 25 centimeters of perimeter? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #204999 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=204999\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The answer is that, if you allow the length of a side to be any real number, then the number of ways is infinite.\r<BR>\n" );
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document.write( "One example will show you why:\r<BR>\n" );
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document.write( "Let's assume the length of the largest side of the triangle is 12 cm.\r<BR>\n" );
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document.write( "This means that the sum of the lengths of the two other sides has to be equal to 13 cm.\r<BR>\n" );
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document.write( "Let's assume length of one of the smaller sides is 8 cm.\r<BR>\n" );
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document.write( "This means the length of the other of the smaller sides is 5 cm.\r<BR>\n" );
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document.write( "The sum of the lengths of all 3 sides of this triangle is 12 + 8 + 5 = 25 cm.\r<BR>\n" );
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document.write( "Now let's take another possibility:\r<BR>\n" );
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document.write( "The length of one of the smaller sides is 7 cm and the length of the other of the smaller side is 6 cm.\r<BR>\n" );
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document.write( "The sum of the lengths of all 3 sides of this triangle is 12 + 7 + 6 = 25 cm.\r<BR>\n" );
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document.write( "We have at least two ways we can make a triangle with a perimeter of 25 cm.\r<BR>\n" );
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document.write( "In between these two triangles there is an infinite number of ways.\r<BR>\n" );
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document.write( "One of the smaller sides can vary between 7 and 8 cm in length while the other of the smaller sides can vary between 5 and 6 cm in length.\r<BR>\n" );
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document.write( "Some examples would be triangle with the following lengths:\r<BR>\n" );
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document.write( "12, 8, 5<BR>\n" );
document.write( "12, 7.9, 5.1<BR>\n" );
document.write( "12, 7.8, 5.2<BR>\n" );
document.write( "12, 7.7, 5.3<BR>\n" );
document.write( "12, 7.6, 5.4<BR>\n" );
document.write( "12, 7.5, 5.5<BR>\n" );
document.write( "etc.\r<BR>\n" );
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document.write( "Between a triangle having sides of 12, 8, 5 and a triangle having sides of 12, 7, 6, there are an infinite number of possible points in between.\r<BR>\n" );
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document.write( "You can divide 1 cm into 2, 4, 6, 8, 16, 32, 64, 128, 512, ...... x number of pieces, where there is no limit to how large x can grow.\r<BR>\n" );
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document.write( "Bottom line is there are an infinite number of ways if you assume that the length of each side can be any real number.\r<BR>\n" );
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document.write( "If you restrict the length of each side to be something like an integer, then there are countable number of ways in which you can get a triangle to have a perimeter of 25 cm.\r<BR>\n" );
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document.write( "Assume the same 12 cm for the largest side of the triangle.\r<BR>\n" );
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document.write( "The sum of the lengths of the other 2 sides of the triangle would have to be 13 cm in order for the perimeter to be equal to 25 cm.\r<BR>\n" );
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document.write( "This can happen in the following ways:\r<BR>\n" );
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document.write( "12,12,1 <BR>\n" );
document.write( "12,11,2<BR>\n" );
document.write( "12,10,3<BR>\n" );
document.write( "12,9,4<BR>\n" );
document.write( "12,8,5<BR>\n" );
document.write( "12,7,6\r<BR>\n" );
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document.write( "Assume the largest side is 11 and you get:\r<BR>\n" );
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document.write( "11,11,3<BR>\n" );
document.write( "11,10,4<BR>\n" );
document.write( "11,9,5<BR>\n" );
document.write( "11,8,6<BR>\n" );
document.write( "11,7,7\r<BR>\n" );
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document.write( "Assume the largest side is 10 and you get:\r<BR>\n" );
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document.write( "10,10,5<BR>\n" );
document.write( "10,9,6<BR>\n" );
document.write( "10,8,7\r<BR>\n" );
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document.write( "Assume the largest side is 9 and you get:\r<BR>\n" );
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document.write( "9,9,7<BR>\n" );
document.write( "9,8,8\r<BR>\n" );
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document.write( "Assume the largest side is 8 and you get:\r<BR>\n" );
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document.write( "8,8,9 which can't be, because this would make 8 NOT he largest side of the triangle, so we stop at the largest side of the triangle having a minimum length of 9.\r<BR>\n" );
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document.write( "As you can see, now the number of ways are countable.   \r<BR>\n" );
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document.write( "This is not the case when the length of the side of a triangle can be any real number.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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