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1 Let the sides be n and use the formula: 0.5*(n2-3n) = diagonals

2 So: 0.5*(n2-3n) = 170 => which transposes to: n2-3n-340 = 0

3 Solving the above quadratic equation gives n a positive value of 20

4 So the polygon has 20 sides and (20-2)*180 = 3240 interior angles

5 Each interior angle measures: 3240/20 = 162 degrees

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Q: What are the sizes of each interior angle of a regular polygon that has 170 diagonals showing work?
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How many sides does a regular polygon have when it has 4752 diagonals showing work?

Consider a regular polygon with n sides (and n vertices). Select any vertex. This can be done in n ways. There is no line from that vertex to itself. The lines from the vertex to the immediate neighbour on either side is a side of the polygon and so a diagonal. The lines from that vertex to any one of the remaining n-3 vertices is a diagonal. So, the nuber of ways of selecting the two vertices that deefine a diaginal seem to be n*(n-3). However, this process counts each diagonal twice - once from each end. Therefore a regular polygon with n sides has n*(n-3)/2 diagonals. Now n*(n-3)/2 = 4752 So n*(n-3) = 9504 that is n2 - 3n - 9504 = 0 using the quadratic equation, n = [3 + sqrt(9 + 4*9504)]/2 = 99 sides/vertices. The negative square root in the quadratic formula gives a negative number of sides and that answer can be ignored.


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Consider a regular polygon with n sides (and n vertices). Select any vertex. This can be done in n ways. There is no line from that vertex to itself. The lines from the vertex to the immediate neighbour on either side is a side of the polygon and so a diagonal. The lines from that vertex to any one of the remaining n-3 vertices is a diagonal. So, the nuber of ways of selecting the two vertices that deefine a diaginal seem to be n*(n-3). However, this process counts each diagonal twice - once from each end. Therefore a regular polygon with n sides has n*(n-3)/2 diagonals. Now n*(n-3)/2 = 4752 So n*(n-3) = 9504 that is n2 - 3n - 9504 = 0 using the quadratic equation, n = [3 + sqrt(9 + 4*9504)]/2 = 99 sides/vertices. The negative square root in the quadratic formula gives a negative number of sides and that answer can be ignored.


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