if the sides of a triangle are in the ratio `2:sqrt6 : sqrt3 + 1, `then the largest ange of the trangle will be (1) 60 (2) 72 (3) 75 (4) 90

Options

A. `60^(@)`
B. `72^(@)`
C. `75^(@)`
D. `90^(@)`

Answer

To solve the problem, we need to find the largest angle of a triangle whose sides are in the ratio \(2 : \sqrt{6} : \sqrt{3} + 1\). We will follow these steps:

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

where \(k\) is a positive constant.

### Step 2: Determine the largest side
To find the largest angle, we first need to determine which side is the largest. We compare \(2k\), \(\sqrt{6}k\), and \((\sqrt{3} + 1)k\).

Calculating the approximate values:
- \(2k\) is \(2\)
- \(\sqrt{6} \approx 2.45\), so \(\sqrt{6}k \approx 2.45k\)
- \(\sqrt{3} \approx 1.73\), hence \(\sqrt{3} + 1 \approx 2.73\), so \((\sqrt{3} + 1)k \approx 2.73k\)

Thus, \((\sqrt{3} + 1)k\) is the largest side.

### Step 3: Use the Cosine Rule
The angle opposite the largest side \(c\) (which is \((\sqrt{3} + 1)k\)) will be the largest angle \(C\). We will use the cosine rule to find \(C\):

\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

Substituting the values of \(a\), \(b\), and \(c\):

\[ \cos C = \frac{(2k)^2 + (\sqrt{6}k)^2 - ((\sqrt{3} + 1)k)^2}{2 \cdot 2k \cdot \sqrt{6}k} \]

### Step 4: Simplify the expression
Calculating the squares:
- \(a^2 = (2k)^2 = 4k^2\)
- \(b^2 = (\sqrt{6}k)^2 = 6k^2\)
- \(c^2 = ((\sqrt{3} + 1)k)^2 = (\sqrt{3})^2 + 2\cdot\sqrt{3}\cdot1 + 1^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}\)

Now substituting back into the cosine formula:

\[ \cos C = \frac{4k^2 + 6k^2 - (4 + 2\sqrt{3})k^2}{4\sqrt{6}k^2} \]

This simplifies to:

\[ \cos C = \frac{10k^2 - (4 + 2\sqrt{3})k^2}{4\sqrt{6}k^2} = \frac{(10 - 4 - 2\sqrt{3})k^2}{4\sqrt{6}k^2} = \frac{6 - 2\sqrt{3}}{4\sqrt{6}} \]

### Step 5: Further simplify
Factoring out 2 from the numerator:

\[ \cos C = \frac{2(3 - \sqrt{3})}{4\sqrt{6}} = \frac{3 - \sqrt{3}}{2\sqrt{6}} \]

### Step 6: Recognize the angle
We need to recognize that \(\frac{3 - \sqrt{3}}{2\sqrt{6}}\) corresponds to \(\cos 75^\circ\). Therefore, we conclude that:

\[ C = 75^\circ \]

### Final Answer
The largest angle of the triangle is \(75^\circ\).

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