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To solve this problem, let's analyze the key information given and use geometry concepts to find the area of triangle \( ABC \).

### Step-by-Step Solution

1. **Identify Given Information**:
   - The circumradius \( R \) of triangle \( ABC \) is \( 5 \, \text{cm} \).
   - \( D \) is the intersection of the angle bisector of \( \angle A \) with the circumcircle of \( ABC \).
   - \( \angle ADB \) is a right angle, indicating that \( D \) lies on the circumcircle of \( \triangle ABC \), making \( \triangle ABC \) a right triangle.
   - The ratio of sides \( AC : CB = 3 : 4 \).

2. **Determine the Side Lengths Using the Ratio**:
   Since \( \triangle ABC \) is a right triangle and follows a \( 3:4:5 \) ratio:
   - Let \( AC = 3x \) and \( CB = 4x \).
   - By the Pythagorean theorem for right triangles, the hypotenuse \( AB = 5x \).

3. **Utilize the Circumradius**:
   - In a right triangle, the hypotenuse is equal to the diameter of the circumcircle.
   - Therefore, \( AB = 2R = 2 \times 5 = 10 \, \text{cm} \).
   - Since \( AB = 5x \), we can equate and solve for \( x \):
     \[
     5x = 10 \Rightarrow x = 2
     \]

4. **Calculate the Side Lengths**:
   - \( AC = 3x = 3 \times 2 = 6 \, \text{cm} \)
   - \( CB = 4x = 4 \times 2 = 8 \, \text{cm} \)
   - \( AB = 10 \, \text{cm} \) (as verified).

5. **Calculate the Area of Triangle \( ABC \)**:
   - The area \( A \) of a right triangle is given by:
     \[
     A = \frac{1}{2} \times AC \times CB
     \]
   - Substitute \( AC = 6 \) and \( CB = 8 \):
     \[
     A = \frac{1}{2} \times 6 \times 8 = \frac{1}{2} \times 48 = 24 \, \text{cm}^2
     \]

### Final Answer
The area of triangle \( ABC \) is \( 24 \, \text{cm}^2 \).

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Would you like more details on any specific step? Here are some additional questions for further exploration:

1. How can the circumradius help in identifying the type of triangle?
2. What other methods could we use to find the area of triangle \( ABC \)?
3. How would the area change if the circumradius were different?
4. What properties of right triangles with a \( 3:4:5 \) ratio make calculations easier?
5. How is the circumcircle related to the hypotenuse in a right triangle?

**Tip:** In right triangles, if you know the circumradius, you can directly find the hypotenuse, as it's twice the circumradius.

In the figure, the circumradius of triangle ABC is 5 cm. Let D be the intersection of the angle bisector of angle A with the circumcircle of triangle ABC. It’s known that angle ADB is a right angle. In the triangle, the ratio of the sides AC:CB = 3:4. Then find the area of triangle ABC.

Solve for the area of triangle ABC with a circumradius of 5 cm and sides in the ratio 3:4. Using geometry and trigonometry, this solution applies the Pythagorean Theorem and properties of circumcircles. Suitable for high school students in Grades 9-12.

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