Two plane mirrors makes an angle of `120^(@)` with each other. The distance between the two images of a point source formed in them is 20 cm . Determine the distance from the light source of the point where the mirrors touch, the light source lies on the bisector of the angle formed by the mirrors.

Explanation

To solve the problem, we need to determine the distance from the light source to the point where the two mirrors touch, given that the distance between the two images formed by the mirrors is 20 cm and the angle between the mirrors is 120 degrees.

### Step-by-step Solution:

1. **Understanding the Geometry**:
- We have two plane mirrors making an angle of \(120^\circ\) with each other.
- The light source lies on the bisector of this angle, which means it is at an angle of \(60^\circ\) from each mirror (since \(120^\circ / 2 = 60^\circ\)).

2. **Setting Up the Diagram**:
- Let the point source be at point \(O\) and the point where the mirrors touch be point \(A\).
- The images formed by the mirrors will be denoted as \(I_1\) and \(I_2\).
- The distance between the two images \(I_1\) and \(I_2\) is given as \(20 \, \text{cm}\).

3. **Identifying the Angles**:
- The angle between the line \(OI_1\) and the line \(OA\) is \(60^\circ\) (the angle bisector).
- The angle between the line \(OI_2\) and the line \(OA\) is also \(60^\circ\).

4. **Using Trigonometry**:
- The distance from the light source \(O\) to the point where the mirrors touch \(A\) is denoted as \(D\).
- The distance from \(O\) to \(I_1\) and \(I_2\) can be expressed using trigonometric functions:
\[ \text{Distance from } O \text{ to } I_1 = D \cos(60^\circ) = D \cdot \frac{1}{2} \]
\[ \text{Distance from } O \text{ to } I_2 = D \cos(60^\circ) = D \cdot \frac{1}{2} \]

5. **Calculating the Total Distance**:
- The total distance between the two images \(I_1\) and \(I_2\) can be expressed as:
\[ \text{Distance } I_1 I_2 = 2 \cdot D \cos(60^\circ) = 2 \cdot D \cdot \frac{1}{2} = D \]
- We know from the problem statement that this distance is \(20 \, \text{cm}\):
\[ D = 20 \, \text{cm} \]

6. **Finding the Distance from the Light Source**:
- Since we have established that the distance \(D\) is equal to \(20 \, \text{cm}\), we can now find the distance from the light source \(O\) to the point where the mirrors touch \(A\):
\[ D = \frac{20}{\cos(30^\circ)} \]
- The value of \(\cos(30^\circ)\) is \(\frac{\sqrt{3}}{2}\):
\[ D = \frac{20}{\frac{\sqrt{3}}{2}} = \frac{20 \cdot 2}{\sqrt{3}} = \frac{40}{\sqrt{3}} \approx 23.09 \, \text{cm} \]

### Final Answer:
The distance from the light source to the point where the mirrors touch is approximately \(23.09 \, \text{cm}\).