document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=444115>Question 444115</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many different isosceles triangles are possible if the sides must have whole number lengths and the perimeter must be ninety-three inches? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #306239 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20055)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=306239\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> How many different isosceles triangles are possible if the sides must have whole number lengths and the perimeter must be ninety-three inches?<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"><font face = \"consolas\" color = \"indigo\" size = 4><b>\r\n" );
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document.write( "In any triangle whose sides have measures a,b, and c,\r\n" );
document.write( "the triangular inequality tells us:\r\n" );
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document.write( "a + b > c\r\n" );
document.write( "a + c > b\r\n" );
document.write( "b + c > a\r\n" );
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document.write( "Since the triangles here have two sides with the same\r\n" );
document.write( "measure, we will suppose that a = b and the above becomes\r\n" );
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document.write( "a + a > c\r\n" );
document.write( "a + c > a\r\n" );
document.write( "a + c > a\r\n" );
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document.write( " or\r\n" );
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document.write( "2a > c\r\n" );
document.write( " c > 0\r\n" );
document.write( " c > 0\r\n" );
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document.write( "So we have \r\n" );
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document.write( "1)    2a > c > 0\r\n" );
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document.write( "Since the perimeter is 93,\r\n" );
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document.write( "we have\r\n" );
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document.write( "   a + a + c = 93\r\n" );
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document.write( "2)    2a + c = 93\r\n" );
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document.write( "Solve 2) for c, \r\n" );
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document.write( "c = 93 - 2a\r\n" );
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document.write( "Substitute in 1)\r\n" );
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document.write( "1)    2a > 93 - 2a\r\n" );
document.write( "      4a > 93\r\n" );
document.write( "       a > 23.25\r\n" );
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document.write( "and since a is an integer\r\n" );
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document.write( "       a > 23 \r\n" );
document.write( "        \r\n" );
document.write( "Solve 2) for 2a\r\n" );
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document.write( "      2a = 93 - c  \r\n" );
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document.write( "Substitute 93 - c for 2a in 1)\r\n" );
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document.write( " 93 - c > c \r\n" );
document.write( "     93 > 2c\r\n" );
document.write( "   46.5 > c\r\n" );
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document.write( "and since c is an integer,\r\n" );
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document.write( "     47 > c\r\n" );
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document.write( "or    c < 47 \r\n" );
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document.write( " \r\n" );
document.write( "So we have\r\n" );
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document.write( "     a > 23\r\n" );
document.write( "     c < 47\r\n" );
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document.write( "Let p be the positive integer amount that a is greater than 23\r\n" );
document.write( "Let q be the positive integer amount that a is less than 47\r\n" );
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document.write( "So \r\n" );
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document.write( "     a = 23 + p \r\n" );
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document.write( "and\r\n" );
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document.write( "     c = 47 - q\r\n" );
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document.write( "Substituting those into 2)\r\n" );
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document.write( "    2a + c = 93\r\n" );
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document.write( " 2(23 + p) + (47 - q) = 93\r\n" );
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document.write( "     46 + 2p + 47 - q = 93\r\n" );
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document.write( "          93 + 2p - q = 93\r\n" );
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document.write( "               2p - q = 0\r\n" );
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document.write( "                   2p = q\r\n" );
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document.write( " So now we have\r\n" );
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document.write( "     a = 23+p \r\n" );
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document.write( "and\r\n" );
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document.write( "     c = 47-2p\r\n" );
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document.write( "Since c > 0\r\n" );
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document.write( " 47 - 2p > 0\r\n" );
document.write( "     -2p > -47\r\n" );
document.write( "       p < 23.5\r\n" );
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document.write( "and since p is an integer\r\n" );
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document.write( "       p < 24\r\n" );
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document.write( "And p > 0 so\r\n" );
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document.write( "   0 < p < 24\r\n" );
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document.write( "So there are 23 values p can take on, any of the integers\r\n" );
document.write( "which are greater than 0 and less than 24.  That's 1 through\r\n" );
document.write( "23, so there are 23 possible isoceles triangles with integer\r\n" );
document.write( "sides that have perimeter 93.\r\n" );
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document.write( "Here they all are:\r\n" );
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document.write( " p   a = 23+p   b = a   c = 47-2p\r\n" );
document.write( "----------------------------------\r\n" );
document.write( " 1.     24       24        45\r\n" );
document.write( " 2.     25       25        43\r\n" );
document.write( " 3.     26       26        41\r\n" );
document.write( " 4.     27       27        39\r\n" );
document.write( " 5.     28       28        37\r\n" );
document.write( " 6.     29       29        35\r\n" );
document.write( " 7.     30       30        33\r\n" );
document.write( " 8.     31       31        31\r\n" );
document.write( " 9.     32       32        29\r\n" );
document.write( "10.     33       33        27\r\n" );
document.write( "11.     34       34        25\r\n" );
document.write( "12.     35       35        23\r\n" );
document.write( "13.     36       36        21\r\n" );
document.write( "14.     37       37        19\r\n" );
document.write( "15.     38       38        17\r\n" );
document.write( "16.     39       39        15\r\n" );
document.write( "17.     40       40        13\r\n" );
document.write( "18.     41       41        11\r\n" );
document.write( "19.     42       42        9\r\n" );
document.write( "20.     43       43        7\r\n" );
document.write( "21.     44       44        5\r\n" );
document.write( "22.     45       45        3\r\n" );
document.write( "23.     46       46        1\r\n" );
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document.write( "Edwin</PRE><BR>\n" );
document.write( "     <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );