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.