In triangle `A B C ,/_C=(2pi)/3` and `C D` is the internal angle bisector of `/_C ,` meeting the side `A Ba tD` . If Length `C D` is `1,` the H.M. of `aa n db` is equal to: 1 (b) 2 (c) 3 (d) 4

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem step by step, we will use the properties of triangles and the formula for the internal angle bisector.

### Step 1: Identify the given information
We are given:
- In triangle \( ABC \), \( \angle C = \frac{2\pi}{3} \)
- \( CD \) is the internal angle bisector of \( \angle C \)
- Length \( CD = 1 \)

### Step 2: Use the formula for the length of the angle bisector
The formula for the length of the internal angle bisector \( CD \) is given by:

\[ CD = \frac{2ab}{a + b} \cdot \cos\left(\frac{C}{2}\right) \]

### Step 3: Calculate \( \cos\left(\frac{C}{2}\right) \)
Since \( C = \frac{2\pi}{3} \), we find \( \frac{C}{2} \):

\[ \frac{C}{2} = \frac{2\pi}{3} \cdot \frac{1}{2} = \frac{\pi}{3} \]

Now, we can calculate \( \cos\left(\frac{\pi}{3}\right) \):

\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 4: Substitute into the angle bisector formula
Now substituting \( \cos\left(\frac{C}{2}\right) \) into the angle bisector formula:

\[ CD = \frac{2ab}{a + b} \cdot \frac{1}{2} \]

This simplifies to:

\[ CD = \frac{ab}{a + b} \]

### Step 5: Set the equation for \( CD \)
Since we know \( CD = 1 \):

\[ \frac{ab}{a + b} = 1 \]

### Step 6: Cross-multiply to find a relationship between \( a \) and \( b \)
Cross-multiplying gives:

\[ ab = a + b \]

### Step 7: Rearranging the equation
Rearranging the equation, we get:

\[ ab - a - b = 0 \]

### Step 8: Factor the equation
Factoring gives:

\[ (a - 1)(b - 1) = 1 \]

### Step 9: Find the Harmonic Mean (H.M.) of \( a \) and \( b \)
The formula for the harmonic mean (H.M.) of two numbers \( a \) and \( b \) is:

\[ H.M. = \frac{2ab}{a + b} \]

### Step 10: Substitute the value of \( \frac{ab}{a + b} \)
From our earlier result, we know \( \frac{ab}{a + b} = 1 \). Therefore, we can substitute this into the H.M. formula:

\[ H.M. = 2 \cdot 1 = 2 \]

### Final Answer
Thus, the H.M. of \( a \) and \( b \) is \( 2 \).