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This question focuses on geometry involving a right-angled triangle \( \triangle ABC \) with some additional information about angles and lengths. Let's break down and analyze the problem:

### Given:
- \( \triangle ABC \) is a right triangle with \( \angle BAC = 90^\circ \).
- Side lengths: \( AB = 2 \), \( AC = 5 \).
- \( \angle DAC = 30^\circ \), \( AD = DC \).
- \( \angle BAD \) is an acute angle.

The questions ask for several specific calculations, such as the circumradius of \( \triangle ABC \), trigonometric values, and the area of \( \triangle ABD \).

Let's analyze each part:

1. **Circumradius of \( \triangle ABC \)**:
   - The circumradius \( R \) of a right-angled triangle is given by \( R = \frac{c}{2} \), where \( c \) is the hypotenuse.
   - Here, \( AC \) is the hypotenuse of \( \triangle ABC \), so \( R = \frac{5}{2} \).

2. **Sine of \( \angle ABC \)**:
   - \( \sin \angle ABC = \frac{AB}{AC} = \frac{2}{5} \).

3. **Cosine of \( \angle BAD \)**:
   - Since \( \triangle ABD \) involves an acute angle and symmetry conditions like \( AD = DC \), we would need more details about \( D \)'s location, but this angle can be calculated using geometric or trigonometric methods. This part likely involves constructing \( D \) such that additional information on angles can be inferred.

4. **Length of \( AD \)**:
   - This requires some additional trigonometric computation based on the provided \( \angle DAC = 30^\circ \) and the relationship \( AD = DC \). We can use the Law of Sines or other geometric methods to find \( AD \).

5. **Area of \( \triangle ABD \)**:
   - Once we have \( AD \), the area of \( \triangle ABD \) can be calculated using standard area formulas for triangles (e.g., using \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)).

Would you like further breakdowns of each part or more detailed solutions? 

Here are 5 questions you can explore based on this problem:
1. How do you derive the circumradius formula for right triangles?
2. What is the general approach to finding the sine of an angle in a right triangle?
3. How can the Law of Cosines help in solving for angles in triangles like \( \triangle ABD \)?
4. What geometric methods can help find the length of \( AD \) given symmetry conditions?
5. How does the area formula for a triangle change when additional angles are given?

**Tip:** For right-angled triangles, the circumradius is always half the hypotenuse.

In right triangle ABC, ∠BAC = 90°, with AB = 2, AC = 5. Additionally, point D is defined such that ∠DAC = 30° and AD = DC, while ∠BAD is acute. Find the following: (1) The circumradius of triangle ABC. (2) sin(∠ABC). (3) cos(∠BAD). (4) The length of AD. (5) The area of triangle ABD.

Learn how to solve a geometry problem involving a right triangle ABC with given side lengths and angles. The problem covers key concepts such as calculating the circumradius, trigonometric ratios, and finding the area of a triangle. Suitable for high school students in Grades 9-12.

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