Polygons

The formula for the number of diagonals of a regular polygon with sides is: Given , we substitute into the formula: Solving this quadratic equation: Thus, or . Since must be a positive integer, we have .

The formula for the number of diagonals in a polygon with sides is given by: We are given that the polygon has 275 diagonals, so: Multiply both sides of the equation by 2 to eliminate the fraction: Expanding the equation: Rearranging the terms: Now, solve this quadratic equation using the quadratic formula: Taking the square root of 2209: Thus, the two possible values for are: Since must be a positive integer, we conclude that: Thus, the correct answer is: