Pentagonal Recursion

Geometry Level 5

The diagonals of a regular pentagon are drawn, yielding a smaller pentagon within. The ratio of the side of the larger pentagon to the side of the smaller pentagon has the form \( \frac {a + \sqrt{b}}{c} \), where \(a, b\) and \(c\) are nonnegative integers and \(a\) and \(c\) are coprime. What is \(a+b+c\)?

Details and assumptions

\(b\) can be a multiple of a square number, \(a\) can be 0, and \(c\) can be 1. If you calculate the ratio to be \( 2\sqrt{3} = \frac { 0 + \sqrt{12} } {1} \), then your answer should be \( 0 + 12 + 1 = 13\).

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