SOLUTION: 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 polygo

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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:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
I'm going to assume that your polygon is convex, not concave.

Every convex quadrilateral has 2 diagonals which 
intersect in 1 interior point.  Every combination of 4 vertices
determines a convex quadrilateral, contributing one diagonal
intersection point.

So the formula for the number of such points, since none are
concurrent, is the number of combinations of four vertices.

So the equation is

C(n,4) = 70

n%28n-1%29%28n-2%29%28n-3%29%2F%284%2A3%2A2%2A1%29=70

%28n%5E4-6n%5E3%2B11n%5E2-6n%29%2F24=70

n%5E4-6n%5E3%2B11n%5E2-6n=1680

n%5E4-6n%5E3%2B11n%5E2-6n-1680=0

Using synthetic division, we can factor the polynomial 
completely as

%28n-8%29%28n%2B5%29%28n%5E2-3n%2B42%29+=+0

Only the solution n=8 is applicable.

Answer: The polygon has 8 sides.

Edwin