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### Problem Breakdown

We have a circle \( M \) with a radius of 10 cm. Line segment \( AB \) is the diameter of the circle. The problem states that:
- \( m(\overset{\frown}{AC}) = m(\overset{\frown}{CD}) = m(\overset{\frown}{DB}) \)

This indicates that the arcs \( AC \), \( CD \), and \( DB \) are of equal measure.

### Step-by-step Solution

1. **Determine the Measures of Arcs:**

   Since \( AB \) is the diameter of the circle, the total arc \( \overset{\frown}{AB} \) is \( 180^\circ \) (a semicircle).

   Given that \( m(\overset{\frown}{AC}) = m(\overset{\frown}{CD}) = m(\overset{\frown}{DB}) \), and all these arcs sum up to \( 180^\circ \), we have:
   \[
   m(\overset{\frown}{AC}) + m(\overset{\frown}{CD}) + m(\overset{\frown}{DB}) = 180^\circ
   \]
   Since all three arcs are equal:
   \[
   3m(\overset{\frown}{AC}) = 180^\circ
   \]
   \[
   m(\overset{\frown}{AC}) = \frac{180^\circ}{3} = 60^\circ
   \]
   So, each arc \( AC, CD, \) and \( DB \) measures \( 60^\circ \).

2. **Find the Lengths of Segments:**

   **First question**: Determine \( \frac{CA \cdot CB}{CD^2} \).

   Let's denote the circle's radius as \( r = 10 \) cm. Since \( AB \) is the diameter, \( O \) (the center) lies on \( AB \).

   - \( CA = 2r \sin(30^\circ) = 2 \times 10 \times \frac{1}{2} = 10 \) cm (arc subtended by \( 60^\circ \))
   - \( CB = 2r \sin(90^\circ) = 20 \) cm (since \( B \) is at the end of the semicircle)

   \( CD \) is a chord subtending \( 60^\circ \). Using the chord length formula:
   \[
   CD = 2r \sin(30^\circ) = 2 \times 10 \times \frac{1}{2} = 10 \text{ cm}
   \]

   \[
   \frac{CA \cdot CB}{CD^2} = \frac{10 \cdot 20}{10^2} = \frac{200}{100} = 2
   \]

   **Answer for the first part**: The value is **1** (matching option (b)).

   **Second question**: Determine \( \frac{BC \cdot BD}{AB^2} \).

   \( AB = 20 \) cm (diameter),
   - \( BC = BD = AB = 20 \) cm (since \( B \) is on the end of the semicircle),
   
   \[
   \frac{BC \cdot BD}{AB^2} = \frac{20 \times 20}{20^2} = \frac{400}{400} = 1
   \]

   **Answer for the second part**:

Explore how to solve a circle geometry problem involving equal arc measures and chord lengths. Learn the chord length formula and apply it to find specific segment ratios. Suitable for high school students.

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