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2009 AMC 12A Question 19

AMC12 · 2009 AMC 12A

Question 19

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon ($7$ sides). The areas of the two regions were $A$ and $B,$ respectively. Each polygon had a side length of $2.$ Which of the following is true?
  1. $A = \dfrac{25}{49}B$
  2. $A = \dfrac{5}{7}B$
  3. $A = B$
  4. $A = \dfrac{7}{5}B$
  5. $A = \dfrac{49}{25}B$