document.write( "Question 1142419: The number of sides of two polygons differ by 4 and the number of diagonals differ by 30. Find the sides, diagonals and the sum of the interior angles. \n" ); document.write( "

Algebra.Com's Answer #763138 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "The number of diagonals for any n-sided convex polygon is  .\r\n" );
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document.write( "So, if \"m\" and \"n\" are the numbers of sides in our polygons, then the problem gives you two equations \r\n" );
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document.write( "    n - m = 4                     (1)\r\n" );
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document.write( "     -  = 30       (2)\r\n" );
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document.write( "Step 1.  Multiply equation (2) by 2 (both sides).  You will get\r\n" );
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document.write( "    n*(n-3) - m*(m-3) = 60.      (3)\r\n" );
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document.write( "Step 2.  From equation (1) express n = m + 4 and substitute it into equation (3), replacing and excluding m.  You will get\r\n" );
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document.write( "    (m+4)*(m+1) - m*(m-3) = 60.\r\n" );
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document.write( "Simplify it\r\n" );
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document.write( "    m^2 + 4m + m + 4 - m^2 + 3m = 60\r\n" );
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document.write( "    8m                          = 56\r\n" );
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document.write( "     m                          = 56/8 = 7.\r\n" );
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document.write( "ANSWER.  The polygons have 7 and 11 sides.\r\n" );
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document.write( "CHECK.    -  =  -  = 11*4 - 7*2 = 44 - 14 = 30.    ! Correct !\r\n" );
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