document.write( "Question 1132966: The perimeter of a right triangle with side lengths that are integers, and having the same area as a rectangle with dimensions 36 cm by 45 cm, is in cm...? \n" ); document.write( "

Algebra.Com's Answer #750145 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "The area of the rectangle is equal to 36*45 = 1620 cm^2.\r\n" );
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document.write( "It is ,  where \"a\" and \"b\" are the legs of a triangle;\r\n" );
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document.write( "hence, a*b = 3240 cm^2.\r\n" );
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document.write( "Thus our task is to find right-angled triangles with integer sides \"a\" and \"b\" such that ab = 3240 and \r\n" );
document.write( "the hypotenuse    is an integer number, too. \r\n" );
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document.write( "The reasonable way to organize the search is to try all factors \"a\" and \"b\" of the number 3240 and check \r\n" );
document.write( "for each pair (a,b) whether the hypotenuse   is integer.\r\n" );
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document.write( "3240 = 324*10 = 18^2*10 = 2^3*3^4*5.\r\n" );
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document.write( "It is easy to perform such search in Excel : see the Table below.\r\n" );
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document.write( "a    b= 3240/a  sqrt(a^2+b^2)\r\n" );
document.write( "-----------------------------\r\n" );
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document.write( "1	3240	3240.00015432\r\n" );
document.write( "2	1620	1620.00123457\r\n" );
document.write( "4	810	810.00987648\r\n" );
document.write( "8	405	405.07900464\r\n" );
document.write( "3	1080	1080.00416666\r\n" );
document.write( "6	540	540.03333230\r\n" );
document.write( "12	270	270.26653511\r\n" );
document.write( "24	135	137.11673858\r\n" );
document.write( "9	360	360.11248243\r\n" );
document.write( "18	180	180.89776118\r\n" );
document.write( "36	90	96.93296653\r\n" );
document.write( "108	30	112.08925015\r\n" );
document.write( "27	120	123.00000000     <<<---===\r\n" );
document.write( "54	60	80.72174428\r\n" );
document.write( "108	30	112.08925015\r\n" );
document.write( "324	10	324.15428425\r\n" );
document.write( "5	648	648.01928984\r\n" );
document.write( "10	324	324.15428425\r\n" );
document.write( "20	162	163.22989922\r\n" );
document.write( "40	81	90.33825325\r\n" );
document.write( "15	216	216.52020691\r\n" );
document.write( "30	108	112.08925015\r\n" );
document.write( "60	54	80.72174428\r\n" );
document.write( "120	27	123.00000000     <<<---===\r\n" );
document.write( "45	72	84.90583019\r\n" );
document.write( "90	36	96.93296653\r\n" );
document.write( "180	18	180.89776118\r\n" );
document.write( "540	6	540.03333230\r\n" );
document.write( "135	24	137.11673858\r\n" );
document.write( "270	12	270.26653511\r\n" );
document.write( "540	6	540.03333230\r\n" );
document.write( "1620	2	1620.00123457\r\n" );
document.write( "3240	1	3240.00015432\r\n" );
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document.write( "From the table, there is, actually, only ONE such a triangle with the legs 27 and 120 centimeters and the hypotenuse of 123 centimeters.\r\n" );
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document.write( "Its perimeter is  27 + 120 + 127 = 270 centimeters.      ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved. // The primitive Pythagorean triple for this triangle is (9, 40, 41).\r
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