The angles A, B, C of a triangle are in the ratio 2:5:5. What is the value of tan B tan C?

Options

A. `4+sqrt3`
B. `4+2sqrt3`
C. `7+4sqrt3`
D. `3+3sqrt3`

Answer

To solve the problem where the angles A, B, and C of a triangle are in the ratio 2:5:5, and we need to find the value of \( \tan B \tan C \), we can follow these steps:

### Step 1: Define the angles in terms of a variable
Given the ratio of the angles A, B, and C as 2:5:5, we can express them as:
- \( A = 2x \)
- \( B = 5x \)
- \( C = 5x \)

### Step 2: Use the triangle angle sum property
We know that the sum of the angles in a triangle is \( 180^\circ \). Therefore, we can write the equation:
\[ A + B + C = 180^\circ \]
Substituting the expressions for A, B, and C:
\[ 2x + 5x + 5x = 180^\circ \]
This simplifies to:
\[ 12x = 180^\circ \]

### Step 3: Solve for x
To find the value of \( x \), we divide both sides by 12:
\[ x = \frac{180^\circ}{12} = 15^\circ \]

### Step 4: Calculate the angles A, B, and C
Now, substituting \( x \) back into the expressions for the angles:
- \( A = 2x = 2 \times 15^\circ = 30^\circ \)
- \( B = 5x = 5 \times 15^\circ = 75^\circ \)
- \( C = 5x = 5 \times 15^\circ = 75^\circ \)

### Step 5: Find \( \tan B \) and \( \tan C \)
Since both \( B \) and \( C \) are \( 75^\circ \):
\[ \tan B = \tan 75^\circ \]
\[ \tan C = \tan 75^\circ \]

### Step 6: Calculate \( \tan B \tan C \)
Thus, we need to find:
\[ \tan B \tan C = \tan 75^\circ \tan 75^\circ = \tan^2 75^\circ \]

### Step 7: Use the tangent addition formula
To find \( \tan 75^\circ \), we can use the tangent addition formula:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
where \( A = 45^\circ \) and \( B = 30^\circ \):
\[ \tan 75^\circ = \tan(45^\circ + 30^\circ) = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} \]
Substituting known values:
\[ \tan 45^\circ = 1, \quad \tan 30^\circ = \frac{1}{\sqrt{3}} \]
So,
\[ \tan 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} = \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} \]

### Step 8: Simplify the expression
Finding a common denominator for the numerator and denominator:
\[ = \frac{\frac{\sqrt{3} + 1}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]
Now, rationalizing the denominator:
\[ = \frac{(\sqrt{3} + 1)(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3} \]

### Step 9: Calculate \( \tan^2 75^\circ \)
Now, we find:
\[ \tan^2 75^\circ = (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

### Final Answer
Thus, the value of \( \tan B \tan C \) is:
\[ \boxed{7 + 4\sqrt{3}} \]