document.write( "Question 808682: if the altitudes of a triangle are in ratio 1:2:3 and its perimeter is 36 cm. find its sides \n" ); document.write( "
Algebra.Com's Answer #487188 by Edwin McCravy(20056)\"\" \"About 
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document.write( "Let the sides of the triangle be a, b, and c.\r\n" );
document.write( "The altitudes are given in the ratio of 1:2:3.\r\n" );
document.write( "So we can let x be such that the altitudes have lengths x,2x, and 3x.\r\n" );
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document.write( "Let x be the length of the altitude drawn to side a.\r\n" );
document.write( "Let 2x be the length of the altitude drawn to side b.\r\n" );
document.write( "Let 3x be the length of the altitude drawn to side c.\r\n" );
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document.write( "Now we use the usual formula for the area of a triangle, which is\r\n" );
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document.write( "Area of a triangle = \"1%2F2\"·(any side of triangle)·(altitude drawn to that side)\r\n" );
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document.write( "Therefore\r\n" );
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document.write( "Area of the triangle = \"1%2F2\"a·x = \"1%2F2\"b·2x = \"1%2F2\"·3x\r\n" );
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document.write( "\"1%2F2\"a·x = \"1%2F2\"b·2x = \"1%2F2\"c·3x\r\n" );
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document.write( "Multiply by 2\r\n" );
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document.write( "a·x = 2b·x = c·3x\r\n" );
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document.write( "ax = 2bx = 3cx\r\n" );
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document.write( "Divide by x\r\n" );
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document.write( "a = 2b = 3c\r\n" );
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document.write( "We are given that the perimeter is 36, so\r\n" );
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document.write( "P = a + b + c = 36\r\n" );
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document.write( "So we have the system\r\n" );
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document.write( " a = 2b\r\n" );
document.write( " a = 3c\r\n" );
document.write( " a + b + c = 36\r\n" );
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document.write( "Solve the first eq. for b:  b = \"a%2F2\"\r\n" );
document.write( "Solve the second eq. for c:  c = \"a%2F3\"\r\n" );
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document.write( "  a + \"a%2F2\" + \"a%2F3\" = 36\r\n" );
document.write( " 6a + 3a + 2a = 216\r\n" );
document.write( "          11a = 216\r\n" );
document.write( "            a = \"216%2F11\"\r\n" );
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document.write( "Then b = \"a%2F2\" = \"1%2F2\"a = \"1%2F2\"·\"216%2F11\" = \"108%2F11\"\r\n" );
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document.write( "And c = \"a%2F3\" = \"1%2F3\"a = \"1%2F3\"·\"216%2F11\" = \"72%2F11\"\r\n" );
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document.write( "So the sides would have to be a=\"216%2F11\", b=\"108%2F11\", c=\"72%2F19\"\r\n" );
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document.write( "However, the sum of any two sides of a triangle must be greater than\r\n" );
document.write( "the third side.\r\n" );
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document.write( "So those values are not possible sides for a triangle because b+c < a\r\n" );
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document.write( "So there is no possible solution.  No triangle can have its altitudes in\r\n" );
document.write( "the ratio 1:2:3.  \r\n" );
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document.write( "Edwin
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