In a triangle ABC, let BC = 1, AC=2 and measure of `/_ C` is `30^@`. Which of the following statement(s) is (are) correct?

Options

A. (a) `2 sin A =sin B`
B. (b) Length of side AB equals `5-2 sqrt3`
C. (c) measure of `angle A` is less than `30^(@)`
D. (d) Circumradius of `/_\ ABC` is equal to length of side AB

Answer

To solve the problem, we need to analyze the triangle ABC with the given parameters: BC = 1, AC = 2, and angle C = 30 degrees. We will evaluate each statement one by one.

### Step-by-Step Solution:

1. **Understanding the Triangle and Given Values:**
- We have triangle ABC with:
- Side BC = 1
- Side AC = 2
- Angle C = 30 degrees

2. **Using the Law of Sines:**
- According to the Law of Sines, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
- Here, \( a = BC = 1 \), \( b = AC = 2 \), and \( c = AB \).

3. **Finding Side AB (c):**
- We can use the cosine rule to find side AB (c):
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos C \]
- Substituting the known values:
\[ c^2 = 1^2 + 2^2 - 2 \cdot 1 \cdot 2 \cdot \cos(30^\circ) \]
- Since \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \):
\[ c^2 = 1 + 4 - 2 \cdot 1 \cdot 2 \cdot \frac{\sqrt{3}}{2} \]
\[ c^2 = 5 - 2\sqrt{3} \]
- Thus, \( c = \sqrt{5 - 2\sqrt{3}} \).

4. **Evaluating Each Statement:**
- **Statement A:** \( 2 \sin A = \sin B \)
- Using the Law of Sines, we can verify this statement.
- **Statement B:** Length of side AB equals \( 5 - 2\sqrt{3} \)
- We found \( c = \sqrt{5 - 2\sqrt{3}} \), so this statement is incorrect.
- **Statement C:** Measure of angle A is less than 30 degrees
- Since side AB (c) is greater than BC, angle A must be less than angle C (30 degrees). Thus, this statement is correct.
- **Statement D:** Circumradius of triangle ABC equals length of side AB
- From the Law of Sines, we can show that \( R = \frac{c}{2\sin C} \) and substituting gives us \( R = \sqrt{5 - 2\sqrt{3}} \). Thus, this statement is correct.

5. **Conclusion:**
- The correct statements are A, C, and D.

### Final Answer:
- The correct statements are A, C, and D.