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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.

A right-angled triangle ABC is given where AB = 2, AC = 5, ∠BAC = 90°, ∠DAC = 30°, and AD = DC. 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.

Solve a right-angled triangle problem where AB = 2, AC = 5, and ∠BAC = 90°. Learn how to calculate the circumradius, sin(∠ABC), cos(∠BAD), and the area of triangle ABD. This exercise covers essential trigonometry and geometry concepts.

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