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 `A_1 and A_2`, respectively. If each polygon has a side length of 2 units then which one of the following is true?

Options

A. `A_(1)=5/7 A_(2)`
B. `A_(1)=(25)/(49)A_(2)`
C. `A_(1)=(49)/(25)A_(2)`
D. `A_(1)=A_(2)`

Answer

To solve the problem, we need to find the areas \( A_1 \) and \( A_2 \) which represent the areas between the circumscribed and inscribed circles for a regular pentagon and a regular heptagon, respectively. Let's go through the steps systematically.

### Step 1: Understanding the Geometry
- We have a regular pentagon and a regular heptagon, both with a side length of 2 units.
- For each polygon, we will find the radius of the inscribed circle (r) and the radius of the circumscribed circle (R).

### Step 2: Finding the Radius of the Inscribed Circle for the Pentagon
1. **Pentagon Properties**:
- The central angle for a pentagon is \( \frac{360^\circ}{5} = 72^\circ \).
- Each triangle formed by the center and two vertices has an angle of \( 36^\circ \) at the center.

2. **Using Triangle AOB**:
- In triangle AOB, where AB is half the side length (1 unit), we can use the tangent function:
\[ \tan(36^\circ) = \frac{AB}{OB} = \frac{1}{R} \]
Rearranging gives:
\[ R = \frac{1}{\tan(36^\circ)} \]

3. **Finding the Inscribed Circle Radius**:
- Using the same triangle, we can find \( r \):
\[ \tan(36^\circ) = \frac{1}{r} \implies r = \frac{1}{\tan(36^\circ)} \]

### Step 3: Finding the Radius of the Circumscribed Circle for the Pentagon
1. **Using the Sine Function**:
\[ \sin(36^\circ) = \frac{AB}{AO} = \frac{1}{R} \]
Rearranging gives:
\[ R = \frac{1}{\sin(36^\circ)} \]

### Step 4: Area Between the Circles for the Pentagon
1. **Area Calculation**:
\[ A_1 = \pi R^2 - \pi r^2 = \pi \left( R^2 - r^2 \right) \]
Using the identity \( \csc^2(36^\circ) - \cot^2(36^\circ) = 1 \):
\[ A_1 = \pi \cdot 1 = \pi \text{ square units} \]

### Step 5: Finding the Area for the Heptagon
1. **Heptagon Properties**:
- The central angle for a heptagon is \( \frac{360^\circ}{7} \).
- Each triangle formed by the center and two vertices has an angle of \( \frac{360^\circ}{14} = 25.71^\circ \).

2. **Using Similar Triangle Properties**:
- Following similar steps as for the pentagon, we find:
\[ A_2 = \pi \text{ square units} \]

### Step 6: Conclusion
- Since both areas are equal, we conclude:
\[ A_1 = A_2 \]

### Final Answer
Thus, the correct option is:
\[ A_1 = A_2 \]