document.write( "Question 1079865: The angles of polygon are in arithmetic expression 172°,168° and 164°. How many sides does the polygon have? \n" ); document.write( "

Algebra.Com's Answer #694103 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "The angles of polygon are in arithmetic expression 172°,168° and 164°. How many sides does the polygon have?
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\n" ); document.write( "\n" ); document.write( "Your formulation is not perfect, unfortunately.\r
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\n" ); document.write( "\n" ); document.write( "The correct formulation is this: \r
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document.write( "     The angles of polygon are in arithmetic progression 172°,168°, 164° and so on . . . . \r\n" );
document.write( "     How many sides does the polygon have?\r\n" );
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document.write( "The corresponding sequence of exterior angles is 8°, 12°, 16°  and so on . . . \r\n" );
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document.write( "It is an arithmetic progression with the first term of 8 and the common difference of 4. \r\n" );
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document.write( "The sum of exterior angles of any (convex) polygon is 360°.\r\n" );
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document.write( "So, you need to find \"n\", the number of sides/vertices, from the condition \r\n" );
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document.write( " = 360°, where  is the sum of the first n terms of this AP.\r\n" );
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document.write( "You can use the formula for  = ,\r\n" );
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document.write( "which gives you an equation\r\n" );
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document.write( " = 360,   or, which is the same\r\n" );
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document.write( "(8 + 2*(n-1))*n = 360.\r\n" );
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document.write( "It reduces to a quadratic equation\r\n" );
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document.write( "2n^2 + 6n - 360 = 0,   which is equivalent to\r\n" );
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document.write( " = 0. \r\n" );
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document.write( "It can be solved by factoring\r\n" );
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document.write( "(n-12)*(n+15) = 0,\r\n" );
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document.write( "which gives you only one positive solution n = 12.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. n= 12. The polygon has 12 sides.\r
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