In AABC, angle A, B and C are in the ratio 1:2:3, then which of the following is (are) correct? (All symbol used have usual meaning in a triangle.)

Options

A. Circum-radius of `DeltaABC=c`
B. `a:b:c=1:sqrt3:2`
C. Permimeter of `DeltaABC=3+sqrt3`
D. Area of `Delta ABC=(sqrt3)/(8) c ^(2)`

Answer

To solve the problem, we need to find the angles of triangle ABC given that they are in the ratio 1:2:3. We will then analyze the options provided to determine which are correct.

### Step-by-Step Solution:

1. **Understanding the Angles**:
Let the angles of triangle ABC be:
- Angle A = 1x
- Angle B = 2x
- Angle C = 3x

According to the angle sum property of triangles, we know that:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]
Therefore,
\[ 1x + 2x + 3x = 180^\circ \]
Simplifying this gives:
\[ 6x = 180^\circ \implies x = 30^\circ \]

2. **Finding the Angles**:
Now substituting the value of x:
- Angle A = 1x = 30°
- Angle B = 2x = 60°
- Angle C = 3x = 90°

Thus, we have:
\[ \text{Angle A} = 30^\circ, \quad \text{Angle B} = 60^\circ, \quad \text{Angle C} = 90^\circ \]
This shows that triangle ABC is a right-angled triangle.

3. **Circumradius of the Triangle**:
For a right-angled triangle, the circumradius (R) is given by:
\[ R = \frac{C}{2} \]
where C is the hypotenuse. Here, since C is the side opposite the right angle, we can say:
\[ R = C \]
Thus, option 1 is correct.

4. **Finding the Sides**:
Using the sine rule:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
Substituting the values:
\[ \frac{a}{\frac{1}{2}} = \frac{b}{\frac{\sqrt{3}}{2}} = \frac{c}{1} = 2R \]
This gives us:
\[ a = R, \quad b = \sqrt{3}R, \quad c = 2R \]
Therefore, the ratio of the sides is:
\[ a : b : c = 1 : \sqrt{3} : 2 \]
Thus, option 2 is also correct.

5. **Calculating the Perimeter**:
The perimeter (P) of triangle ABC is given by:
\[ P = a + b + c = R + \sqrt{3}R + 2R = (1 + \sqrt{3} + 2)R = (3 + \sqrt{3})R \]
Since R = C, we can express the perimeter as:
\[ P = (3 + \sqrt{3})C \]
Thus, option 3 is correct.

6. **Calculating the Area**:
The area (A) of triangle ABC can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take:
\[ A = \frac{1}{2} \times a \times b = \frac{1}{2} \times R \times \sqrt{3}R = \frac{\sqrt{3}}{2}R^2 \]
Substituting R = C/2 gives:
\[ A = \frac{\sqrt{3}}{2} \left(\frac{C}{2}\right)^2 = \frac{\sqrt{3}}{2} \cdot \frac{C^2}{4} = \frac{\sqrt{3}}{8}C^2 \]
Thus, option 4 is incorrect as it states the area is \(\frac{\sqrt{3}}{8}C^2\).

### Conclusion:
The correct options are:
- Option 1: Circumradius of triangle ABC is equal to C (Correct)
- Option 2: a : b : c = 1 : \(\sqrt{3}\) : 2 (Correct)
- Option 3: Perimeter of triangle ABC is equal to \(3 + \sqrt{3}\) (Correct)
- Option 4: Area of triangle ABC is equal to \(\frac{\sqrt{3}}{8}C^2\) (Incorrect)