document.write( "Question 535964: Sam is studying isosceles triangles. He wants to know how many isosceles triangles exist with a perimeter of 99 inches and side lengths are positive whole numbers. What are the lengths of the sides of these triangles? How do you know when you've found them all? \n" ); document.write( "
Algebra.Com's Answer #352058 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Let the two equal sides have length a=b and the base have length c\r\n" );
document.write( "\r\n" );
document.write( "The perimeter = a+b+c = a+a+c = 2a+c = 99\r\n" );
document.write( "\r\n" );
document.write( "By the triangle inequality we always have a + c > a true so the only\r\n" );
document.write( "triangular inequality we have to be concerned about is\r\n" );
document.write( "\r\n" );
document.write( "a + a > c \r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "2a > c\r\n" );
document.write( "\r\n" );
document.write( "and\r\n" );
document.write( "\r\n" );
document.write( "c > 0\r\n" );
document.write( "\r\n" );
document.write( "So putting everything we have together:\r\n" );
document.write( "\r\n" );
document.write( "(1)     2a+c = 99\r\n" );
document.write( "(2)       2a > c\r\n" );
document.write( "(3)        c > 0\r\n" );
document.write( "\r\n" );
document.write( "Solve (1) for c\r\n" );
document.write( "\r\n" );
document.write( "(1)     2a+c = 99\r\n" );
document.write( "(4)        c = 99 - 2a  \r\n" );
document.write( "\r\n" );
document.write( "Substitute (4) into (2)\r\n" );
document.write( "\r\n" );
document.write( "(2)       2a > c\r\n" );
document.write( "          2a > 99 - 2a\r\n" );
document.write( "          4a > 99\r\n" );
document.write( "(5)        a > 24.75\r\n" );
document.write( "\r\n" );
document.write( "Substitute (4) into (3)\r\n" );
document.write( "\r\n" );
document.write( "(3)        c > 0\r\n" );
document.write( "     99 - 2a > 0\r\n" );
document.write( "         -2a > -99\r\n" );
document.write( "(6)        a < 49.5\r\n" );
document.write( "\r\n" );
document.write( "Putting (5) and (6) together\r\n" );
document.write( "\r\n" );
document.write( "        24.75 < a < 49.75\r\n" );
document.write( "\r\n" );
document.write( "So a can be any integer from 25 to 49, inclusive.\r\n" );
document.write( "\r\n" );
document.write( "There are 49 integers from 1 thru 49.\r\n" );
document.write( "We subtract the number of integers 34 or less, and there are 24\r\n" );
document.write( "of them, so the number of triangles satisfying the given\r\n" );
document.write( "conditions is 49 - 24 or 25.\r\n" );
document.write( "\r\n" );
document.write( "That's the answer, 25.\r\n" );
document.write( "\r\n" );
document.write( "Here they all are, where \"a\" can be any integer from 25 \r\n" );
document.write( "through 49, inclusive.\r\n" );
document.write( "\r\n" );
document.write( "     a  b  c\r\n" );
document.write( " 1.  25 25 49\r\n" );
document.write( " 2.  26 26 47\r\n" );
document.write( " 3.  27 27 45\r\n" );
document.write( " 4.  28 28 43\r\n" );
document.write( " 5.  29 29 41\r\n" );
document.write( " 6.  30 30 39\r\n" );
document.write( " 7.  31 31 37\r\n" );
document.write( " 8.  32 32 35\r\n" );
document.write( " 9.  33 33 33\r\n" );
document.write( "10.  34 34 31\r\n" );
document.write( "11.  35 35 29\r\n" );
document.write( "12.  36 36 27\r\n" );
document.write( "13.  37 37 25\r\n" );
document.write( "14.  38 38 23\r\n" );
document.write( "15.  39 39 21\r\n" );
document.write( "16.  40 40 19\r\n" );
document.write( "17.  41 41 17\r\n" );
document.write( "18.  42 42 15\r\n" );
document.write( "19.  43 43 13\r\n" );
document.write( "20.  44 44 11\r\n" );
document.write( "21.  45 45 9\r\n" );
document.write( "22.  46 46 7\r\n" );
document.write( "23.  47 47 5\r\n" );
document.write( "24.  48 48 3\r\n" );
document.write( "25.  49 49 1\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );