document.write( "Question 1208777: If a polygon of n sides has (n/2)(n - 3) diagonals, how many sides will a polygon with 65 diagonals have? Is there a polygon with 80 diagonals?\r
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Algebra.Com's Answer #847224 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "(a) If a polygon of n sides has (n/2)(n - 3) diagonals, how many sides will
\n" ); document.write( "a polygon with 65 diagonals have?
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document.write( "(a) We should determine the number n of sides of a polygon\r\n" );
document.write( "    from equation\r\n" );
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document.write( "        = 65.    (1)\r\n" );
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document.write( "     To solve, multiply both sides by 2 and simplify\r\n" );
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document.write( "        n*(n-3) = 2*65.\r\n" );
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document.write( "        n^2 - 3n - 130 = 0.\r\n" );
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document.write( "     Factor left side\r\n" );
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document.write( "        (n-13)*(n+10) = 0.\r\n" );
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document.write( "     The roots are -10 and 13.  We are looking for positive integer solution, \r\n" );
document.write( "     so n= 13 ideally suits.\r\n" );
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document.write( "     ANSWER.  A polygon with 65 diagonals has 13 sides.\r\n" );
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document.write( "(b)  We should check if the equation\r\n" );
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document.write( "          = 80    (2)\r\n" );
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document.write( "     has a solution in integer positive numbers n >= 3.\r\n" );
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document.write( "     Again, simplify to that as it was done above in part (a).\r\n" );
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document.write( "         n*(n-3) = 2*80\r\n" );
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document.write( "         n^2 - 3n - 160 = 0.\r\n" );
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document.write( "     The discriminant is  d = b^2 - 4ac = (-3)^2 - 4*1*(-160) = 9 + 640 = 649.\r\n" );
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document.write( "     The number 649 is not a perfect square:   = 25.4754784 . . . is an irrational number.\r\n" );
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document.write( "     Thus, equation (2) has only irrational roots and does not have integer solutions.\r\n" );
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document.write( "     From it, we conclude that a polygon with 80 diagonals does not exist.\r\n" );
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