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The problem asks for the shaded area between two concentric circles, given that the chord \( AB \) of the larger circle has a length of \( \frac{14}{\sqrt{\pi}} \).

### Step 1: Analyze the Given Information
- The two circles are concentric, meaning they share the same center.
- \( AB \) is a chord of the larger circle, and its length is \( \frac{14}{\sqrt{\pi}} \).
- We need to find the area of the shaded region, which is the area between the outer circle and the inner circle (an annular region).

### Step 2: Relate the Chord to the Radius of the Large Circle
The length of a chord in a circle is related to the radius \( R \) and the perpendicular distance \( d \) from the center to the chord by the formula:
\[
L = 2\sqrt{R^2 - d^2}
\]
where \( L \) is the length of the chord.

Since the chord length \( AB = \frac{14}{\sqrt{\pi}} \), we can use this information to solve for \( R \) (the radius of the outer circle). However, to simplify the problem, let's assume \( AB \) passes through the center of the circle, making it a diameter of the larger circle. In this case:
\[
AB = 2R
\]
Thus,
\[
R = \frac{\frac{14}{\sqrt{\pi}}}{2} = \frac{7}{\sqrt{\pi}}.
\]

### Step 3: Calculate the Area of the Larger Circle
The area of a circle is given by:
\[
\text{Area} = \pi R^2.
\]
Substituting \( R = \frac{7}{\sqrt{\pi}} \):
\[
\text{Area of the larger circle} = \pi \left( \frac{7}{\sqrt{\pi}} \right)^2 = \pi \times \frac{49}{\pi} = 49.
\]

### Step 4: Calculate the Area of the Smaller Circle
We don't have the direct radius of the smaller circle, but the problem suggests we need only the difference between the areas of the two circles, which defines the shaded region. Let's assume the radius of the smaller circle is \( r \).

### Step 5: Find the Area of the Shaded Region
The area of the shaded region (the annulus) is the difference between the area of the larger circle and the area of the smaller circle:
\[
\text{Area of the shaded region} = \pi R^2 - \pi r^2 = \pi(R^2 - r^2).
\]
We already know \( R = \frac{7}{\sqrt{\pi}} \). To complete the calculation, we need additional information about the smaller circle's radius \( r \), which isn't provided explicitly. However, if this were given, the remaining area calculation would follow the same steps.

Would you like to clarify the information about the smaller circle, or proceed with assumptions to estimate \( r \)? Feel free to ask further!

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Here are 5 related questions:
1. How do we determine the radius of a circle from the length of its chord?
2. What are the properties of concentric circles?
3. How can you find the area of an annular region without knowing the radius of both circles?
4. What is the relationship between a chord and the radius of a circle?
5. How can the Pythagorean theorem help in calculating the distance from the center of the circle to a chord?

**Tip:** In geometry problems involving circles, always check if the chord is a diameter—it simplifies calculations by directly relating the chord to the radius!

En la siguiente figura los dos círculos tienen el mismo centro y la cuerda AB mide \( \frac{14}{\sqrt{\pi}} \). ¿Cuál es el área de la región sombreada?

This problem involves calculating the shaded region between two concentric circles where the length of the chord AB is given. Learn how to use geometric formulas and concepts, including the area of circles and chord length, to solve this problem. Suitable for students in Grades 9-12.

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