If the sides of a triangle are in the ratio `2 : sqrt(6) : 1 + sqrt(3)`, then what is the smallest angle of the triangle ?

Options

A. `75^(@)`
B. `60^(@)`
C. `45^(@)`
D. `30^(@)`

Explanation

To find the smallest angle of the triangle with sides in the ratio \(2 : \sqrt{6} : 1 + \sqrt{3}\), we will follow these steps:

### Step 1: Assign the sides of the triangle
Let the sides of the triangle be:
- \( a = 1 + \sqrt{3} \)
- \( b = 2 \)
- \( c = \sqrt{6} \)

### Step 2: Identify the smallest side
To find the smallest angle, we first need to determine which side is the smallest. We compare the lengths:
- \( a = 1 + \sqrt{3} \approx 1 + 1.732 = 2.732 \)
- \( b = 2 \)
- \( c = \sqrt{6} \approx 2.449 \)

From this, we see that \( b = 2 \) is the smallest side.

### Step 3: Use the Cosine Rule
The angle opposite the smallest side \( b \) is \( B \). We will use the cosine rule to find \( \cos B \):
\[ \cos B = \frac{a^2 + c^2 - b^2}{2ac} \]

### Step 4: Calculate \( a^2 \), \( b^2 \), and \( c^2 \)
Now, we calculate:
- \( a^2 = (1 + \sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3} \)
- \( b^2 = 2^2 = 4 \)
- \( c^2 = (\sqrt{6})^2 = 6 \)

### Step 5: Substitute into the cosine formula
Now substitute these values into the cosine formula:
\[ \cos B = \frac{(4 + 2\sqrt{3}) + 6 - 4}{2(1 + \sqrt{3})(\sqrt{6})} \]
\[ \cos B = \frac{6 + 2\sqrt{3}}{2(1 + \sqrt{3})\sqrt{6}} \]

### Step 6: Simplify the expression
Now simplify the denominator:
\[ 2(1 + \sqrt{3})\sqrt{6} = 2\sqrt{6} + 2\sqrt{18} = 2\sqrt{6} + 6\sqrt{2} \]
Thus,
\[ \cos B = \frac{6 + 2\sqrt{3}}{2\sqrt{6} + 6\sqrt{2}} \]

### Step 7: Calculate \( B \)
To find \( B \), we can use a calculator or trigonometric tables to find the angle corresponding to \( \cos B \).

### Step 8: Determine the smallest angle
After calculating, we find that:
\[ B \approx 45^\circ \]

### Conclusion
The smallest angle of the triangle is \( 45^\circ \).

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