document.write( "Question 912543: In a polygon, no three diagonals are concurrent. If the total number of point of intersection of diagonals interior to the polygon is 70, then the number of diagonals of the polygon is: \n" ); document.write( "

Algebra.Com's Answer #553930 by Edwin McCravy(20059) $\"\"$ $\"About$

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document.write( "I'm going to assume that your polygon is convex, not concave.\r\n" );
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document.write( "Every convex quadrilateral has 2 diagonals which \r\n" );
document.write( "intersect in 1 interior point.  Every combination of 4 vertices\r\n" );
document.write( "determines a convex quadrilateral, contributing one diagonal\r\n" );
document.write( "intersection point.\r\n" );
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document.write( "So the formula for the number of such points, since none are\r\n" );
document.write( "concurrent, is the number of combinations of four vertices.\r\n" );
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document.write( "So the equation is\r\n" );
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document.write( "C(n,4) = 70\r\n" );
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document.write( "Using synthetic division, we can factor the polynomial \r\n" );
document.write( "completely as\r\n" );
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document.write( "Only the solution n=8 is applicable.\r\n" );
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document.write( "Answer: The polygon has 8 sides.\r\n" );
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document.write( "Edwin

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