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To solve this problem, let's analyze the information provided in the image:

The problem gives the following information about an isosceles trapezoid:
- Bases \( a = 21 \) cm and \( b = 9 \) cm.
- Height \( h = 8 \) cm.

We need to find the radius \( R \) of the circumscribed circle around this trapezoid.

### Step-by-step Solution:

1. **Midline of the Trapezoid**: The midline (or median) \( m \) of a trapezoid is the average of the lengths of the two bases:
   \[
   m = \frac{a + b}{2} = \frac{21 + 9}{2} = 15 \text{ cm}
   \]

2. **Finding the Length of the Legs**: Let’s denote the legs of the isosceles trapezoid as \( c \). To find \( c \), we use the Pythagorean theorem. Since the trapezoid is isosceles, the perpendiculars dropped from the ends of the smaller base \( b \) to the larger base \( a \) divide \( a \) into three segments: \( x \), \( b \), and \( x \) where \( x = \frac{a - b}{2} \).

   \[
   x = \frac{21 - 9}{2} = 6 \text{ cm}
   \]

   Now, using the Pythagorean theorem in one of the right triangles formed:
   \[
   c^2 = x^2 + h^2 = 6^2 + 8^2 = 36 + 64 = 100
   \]
   \[
   c = \sqrt{100} = 10 \text{ cm}
   \]

3. **Radius of the Circumscribed Circle**: For an isosceles trapezoid, the radius \( R \) of the circumscribed circle can be calculated using the formula:
   \[
   R = \frac{\sqrt{(a + b)^2 + 4h^2}}{2}
   \]

   Plugging in the values:
   \[
   R = \frac{\sqrt{(21 + 9)^2 + 4 \cdot 8^2}}{2} = \frac{\sqrt{30^2 + 4 \cdot 64}}{2}
   \]
   \[
   R = \frac{\sqrt{900 + 256}}{2} = \frac{\sqrt{1156}}{2}
   \]
   \[
   R = \frac{34}{2} = 17 \text{ cm}
   \]

So, the radius \( R \) of the circumscribed circle around the trapezoid is **17 cm**.

Would you like more details on any step, or do you have any questions?

Here are some related questions:
1. How is the median of a trapezoid used in other geometric calculations?
2. What are the properties of an isosceles trapezoid?
3. How do you calculate the area of an isosceles trapezoid?
4. What is the significance of the circumscribed circle in geometry?
5. How does the Pythagorean theorem apply to trapezoids in general?

**Tip:** Knowing the properties of specific geometric shapes can help simplify complex problems.

Learn how to find the radius of the circumscribed circle around an isosceles trapezoid with a detailed step-by-step solution. This problem covers key geometric concepts, including the properties of isosceles trapezoids and the formula for the radius of the circumscribed circle. Suitable for high school students.

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