Question 178210
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Each exterior angle measures 25°

The sum of the exterior angles of any polygon is 360°.  Therefore if a regular polygon has exterior angles of x°, then


*[tex \Large {360 \over x}] must be an integer.


*[tex \Large {360 \over 25} = 14.4]  Not an integer, therefore there is no regular polygon with exterior angles measuring 25°


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The number of diagonals of a polygon is given by


*[tex \Large {n(n - 3) \over 2}]


You are given that there are 4860 diagonals, so


*[tex \Large {n(n - 3) \over 2} = 4860]


Multiply by 2, remove parentheses, and put the equation in standard quadratic form:


*[tex \Large n^2 - 3n - 9720 = 0]


Applying the quadratic formula:


*[tex \Large n = \frac {-(-3) \pm sqrt{(-3)^2 - 4(1)(-9720)}}{2(1)} = \frac {3 \pm sqrt {38889}}{2} = \frac {3 \pm 3 sqrt{4321}}{2}]


But one of these root is a negative number and both of them are irrational.  Therefore there is no regular polygon with 4860 diagonals.

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