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The problem statement is in Indonesian and concerns two concentric circles (same center \( O \)) with arcs forming sectors. Here’s a translation and analysis of the given problem:

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**Translation of the Problem:**

*Two circles with the same center \( O \) are arranged as shown in the figure. The length of arcs \( AB + BC = CD + DE = EF + FA \). Given that the radius of the smaller circle is \( 14 \, \text{cm} \) and the area of the shaded region is expressed as \( a - b \sqrt{c} \, \text{cm}^2 \) in its simplest form, determine the value of \( a + b + c \). (Use \( \pi = \frac{22}{7} \)).*

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### Analysis and Solution:

1. **Understanding the Geometry:**
   - We have two circles: a larger one and a smaller one with the same center \( O \).
   - The radius of the smaller circle is \( r = 14 \, \text{cm} \).
   - Arcs \( AB \), \( BC \), \( CD \), \( DE \), \( EF \), and \( FA \) together form sectors of the circles.

2. **Equal Arc Lengths:**
   - The problem states that the lengths of certain arcs are equal: \( AB + BC \), \( CD + DE \), and \( EF + FA \).
   - This suggests that each pair of arcs forms one-third of the circle's circumference.

3. **Finding the Total Area of the Shaded Region:**
   - To find the area of the shaded region between the larger and smaller circles:
     - Calculate the area of each sector formed by the arcs.
     - Subtract the area of the smaller circle from the area of each corresponding larger sector.

4. **Calculating Areas:**
   - Area of the smaller circle: \( A_{\text{small}} = \pi \times 14^2 = \frac{22}{7} \times 196 = 616 \, \text{cm}^2 \).
   - The larger circle's radius isn’t explicitly given, but the relationships of the arc lengths can help determine the corresponding areas.

5. **Expression of the Shaded Area:**
   - The shaded area is given as \( a - b \sqrt{c} \).
   - The problem requires simplifying this to determine the constants \( a \), \( b \), and \( c \).

6. **Determining \( a + b + c \):**
   - After expressing the shaded area, sum up \( a + b + c \).

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This problem involves calculating areas of circle segments and using arc-length relationships. Would you like to proceed with a detailed step-by-step calculation for the solution, including how to deduce the radius of the larger circle or any other part of the problem? 

### Related Questions:
1. What is the formula for the area of a sector of a circle?
2. How do you determine the radius of a larger circle given information about equal arc lengths?
3. How can you simplify expressions involving radicals, like \( a - b \sqrt{c} \)?
4. What is the significance of using \( \pi = \frac{22}{7} \) instead of a more precise value?
5. How do the properties of concentric circles affect the calculation of areas between them?

**Tip:** When dealing with problems involving sectors, make sure to understand the relationship between arc length and central angles to find areas accurately.

Dua buah lingkaran dengan pusat yang sama yaitu titik O disusun seperti gambar berikut. Panjang busur AB + BC = CD + DE = EF + FA. Jika diketahui jari-jari lingkaran kecil 14 cm dan luas arsiran adalah a − b √c cm² dalam bentuk paling sederhana, maka tentukan nilai a + b + c. (π = 22/7).

Solve a geometry problem involving two concentric circles, equal arc lengths, and shaded areas. Determine the values of a, b, and c for the expression of the shaded area a − b √c cm². Suitable for Grades 8-10, this problem involves circle area, arc length, and radical simplification.

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