The image contains a mathematical problem related to a right-angled triangle $\triangle ABC$ with specific conditions and multiple sub-questions. Here's the summary and approach for each part:

Problem Breakdown:

1. $\triangle ABC$ is a right-angled triangle with:
   • $AB = 2$, $AC = 5$
   • $\angle BAC = 90^\circ$
   • $\angle DAC = 30^\circ$, $AD = DC$
   • $\angle BAD$ is an acute angle.

Questions:

1. The radius of the circumcircle of $\triangle ABC$ is $\sqrt{\text{アイ}/\text{ウ}}$.
2. $\sin \angle ABC = \text{エ}\sqrt{\text{オカ}} / \text{キク}$.
3. $\cos \angle BAD = \text{ケコ}/\text{サ}$.
4. The length of $AD$ is $\text{シ}\sqrt{\text{ス}} / \text{セ}$.
5. The area of $\triangle ABD$ is $\text{ソ} / \text{タ}$.

Solution Steps:

1. Circumradius of $\triangle ABC$:

The circumradius $R$ of a right triangle is given by: $R = \frac{\text{hypotenuse}}{2}$ Since $AC = 5$ is the hypotenuse, the circumradius is: $R = \frac{5}{2}$

2. $\sin \angle ABC$:

Using the given side lengths: $\sin \angle ABC = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{2}{5}$

3. $\cos \angle BAD$:

From the cosine rule in triangle $\triangle BAD$ and given $AD = DC$ and $\angle DAC = 30^\circ$, you can use geometry or trigonometry to find $\cos \angle BAD$. I'll need to explore this further based on the diagram.

4. Length of $AD$:

Using properties of isosceles triangles and trigonometry from the conditions $AD = DC$ and $\angle DAC = 30^\circ$, we can calculate the length of $AD$.

5. Area of $\triangle ABD$:

The area formula for any triangle is: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ You can apply this with the dimensions of $\triangle ABD$ to compute the area.

Do you need a detailed step-by-step explanation for any specific part, or would you like to focus on one of the sub-questions? Feel free to ask!

Follow-up questions:

1. Would you like to see the full process for calculating the circumradius?
2. Should I explore the geometry of $\cos \angle BAD$ further?
3. Would you like to calculate the length of $AD$ in detail?
4. Do you need assistance with calculating the area of $\triangle ABD$?
5. Should I explain how to approach right triangle trigonometry in this case?

Tip:

For right triangles, the circumradius can always be found using half the hypotenuse, a useful shortcut for specific cases like these.