A circle is inscribed inside a regular pentagon and another circle is circumscribed about this pentagon. Similarly, a circle is inscribed in a regular heptagon and another circumscribed about the heptagon. The area of the regions between the two circles in two cases are A₁
and A₂, respectively. If each polygon has a side length of 2 units, then which one of the following is true ?

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$A_{1} = \frac{49}{25} A_{2}$

$A_{1} = \frac{25}{49} A_{2}$

$A_{1} = A_{2}$

$A_{1} = \frac{5}{7} A_{2}$

answer is D.

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Detailed Solution

$\begin{array}{l} In first case r = \cot \frac{π}{5}; R = cosec \frac{π}{5} \\ In second case r_{1} = \cot \frac{π}{7}; R_{1} = cosec \frac{π}{7} \\ 1_{1} = π (R^{2} - r^{2}) = π ({cosec}^{3} \frac{π}{5} - \cot^{2} \frac{π}{5}) = π \end{array}$

$\begin{array}{l} M A_{2} = π (R_{1}^{2} - r_{1}^{2}) \\ = π ({cosec}^{2} \frac{π}{7} - \cot^{2} \frac{π}{7}) = π \\ \Rightarrow A_{1} = A_{2} \end{array}$