If the angles of a triangle are `30^(@) and 45^(@)`, and the included side is `(sqrt3 + 1)` cm, then

Options

A. area of the triangle is `(1)/() (sqrt3 + 1)` sq. units
B. area of the triangle is `(1)/(2) (sqrt3 -1)` sq. units
C. ratio of greater side to smaller side is `(sqrt3 + 1)/(sqrt2)`
D. ratio of greater side to smaller side is `(1)/(4sqrt3)`

Explanation

To solve the problem step by step, we will follow the given information and apply the relevant mathematical concepts.

### Step-by-Step Solution:

1. **Identify the Angles and the Included Side:**
Given angles of the triangle are:
- Angle A = 30°
- Angle B = 45°
- The included side (between angle A and angle B) is \( \sqrt{3} + 1 \) cm.

2. **Calculate the Third Angle:**
Using the property that the sum of angles in a triangle is 180°:
\[ \text{Angle C} = 180° - \text{Angle A} - \text{Angle B} = 180° - 30° - 45° = 105° \]

3. **Apply the Sine Rule:**
The Sine Rule states that:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Here, we have:
- Side a (opposite angle A) is unknown.
- Side b (opposite angle B) is unknown.
- Side c (opposite angle C) is the included side, which is \( \sqrt{3} + 1 \).

4. **Set Up the Equations Using the Sine Rule:**
From the Sine Rule, we can write:
\[ \frac{\sqrt{3} + 1}{\sin 105°} = \frac{b}{\sin 30°} = \frac{c}{\sin 45°} \]
Rearranging gives:
\[ b = \frac{(\sqrt{3} + 1) \cdot \sin 30°}{\sin 105°} \]
\[ c = \frac{(\sqrt{3} + 1) \cdot \sin 45°}{\sin 105°} \]

5. **Calculate the Values of b and c:**
We know:
- \( \sin 30° = \frac{1}{2} \)
- \( \sin 45° = \frac{\sqrt{2}}{2} \)
- \( \sin 105° = \sin(90° + 15°) = \sin 15° = \frac{\sqrt{6} - \sqrt{2}}{4} \)

Substitute these values into the equations:
\[ b = \frac{(\sqrt{3} + 1) \cdot \frac{1}{2}}{\sin 105°} \]
\[ c = \frac{(\sqrt{3} + 1) \cdot \frac{\sqrt{2}}{2}}{\sin 105°} \]

6. **Calculate the Area of the Triangle:**
The area \( A \) of the triangle can be calculated using the formula:
\[ A = \frac{1}{2} \cdot b \cdot c \cdot \sin A \]
Substitute the values of b, c, and \( \sin A \):
\[ A = \frac{1}{2} \cdot \left(\frac{(\sqrt{3} + 1) \cdot \frac{1}{2}}{\sin 105°}\right) \cdot \left(\frac{(\sqrt{3} + 1) \cdot \frac{\sqrt{2}}{2}}{\sin 105°}\right) \cdot \sin 30° \]

7. **Simplify the Area Expression:**
After substituting and simplifying, we will find:
\[ A = \frac{1}{2} \cdot \frac{(\sqrt{3} + 1)^2 \cdot \frac{1}{2} \cdot \frac{\sqrt{2}}{2}}{\sin^2 105°} \]

8. **Final Area Calculation:**
After performing the calculations, we will find the area of the triangle.