With the usual notation , in `Delta ABC` if `angle A+ angle B= 120^@` ,a=sqrt(3)+1 " and " b= sqrt(3)-1` then the ratio `angle A : angle B ` is :

Options

A. `7:1`
B. `5:3`
C. `9:7`
D. `3:1`

Answer

To solve the problem step by step, we will use the information provided and apply the properties of triangles and trigonometric identities.

### Step 1: Understand the given information
We have a triangle ABC where:
- \( \angle A + \angle B = 120^\circ \)
- Side \( a = \sqrt{3} + 1 \)
- Side \( b = \sqrt{3} - 1 \)

### Step 2: Find angle C
Using the property that the sum of angles in a triangle is \( 180^\circ \):
\[ \angle C = 180^\circ - (\angle A + \angle B) = 180^\circ - 120^\circ = 60^\circ \]

### Step 3: Use the tangent half-angle formula
We can apply the tangent half-angle formula:
\[ \tan\left(\frac{\angle A - \angle B}{2}\right) = \frac{a - b}{a + b} \cdot \cot\left(\frac{\angle C}{2}\right) \]

### Step 4: Calculate \( a - b \) and \( a + b \)
Substituting the values of \( a \) and \( b \):
\[ a - b = (\sqrt{3} + 1) - (\sqrt{3} - 1) = 2 \]
\[ a + b = (\sqrt{3} + 1) + (\sqrt{3} - 1) = 2\sqrt{3} \]

### Step 5: Calculate \( \cot\left(\frac{\angle C}{2}\right) \)
Since \( \angle C = 60^\circ \):
\[ \frac{\angle C}{2} = 30^\circ \quad \text{and} \quad \cot(30^\circ) = \sqrt{3} \]

### Step 6: Substitute into the tangent formula
Now substituting into the formula:
\[ \tan\left(\frac{\angle A - \angle B}{2}\right) = \frac{2}{2\sqrt{3}} \cdot \sqrt{3} = 1 \]

### Step 7: Solve for \( \angle A - \angle B \)
Since \( \tan\left(\frac{\angle A - \angle B}{2}\right) = 1 \):
\[ \frac{\angle A - \angle B}{2} = 45^\circ \quad \Rightarrow \quad \angle A - \angle B = 90^\circ \]

### Step 8: Set up the equations
Now we have two equations:
1. \( \angle A + \angle B = 120^\circ \)
2. \( \angle A - \angle B = 90^\circ \)

### Step 9: Solve the equations
Adding the two equations:
\[ (\angle A + \angle B) + (\angle A - \angle B) = 120^\circ + 90^\circ \]
\[ 2\angle A = 210^\circ \quad \Rightarrow \quad \angle A = 105^\circ \]

Now substituting back to find \( \angle B \):
\[ \angle B = 120^\circ - \angle A = 120^\circ - 105^\circ = 15^\circ \]

### Step 10: Find the ratio \( \angle A : \angle B \)
Now we can find the ratio:
\[ \angle A : \angle B = 105^\circ : 15^\circ = 7 : 1 \]

Thus, the final answer is:
\[ \text{The ratio } \angle A : \angle B = 7 : 1 \]

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