`P` is a point on the line `y+2x=1,` and `Qa n dR` two points on the line `3y+6x=6` such that triangle `P Q R` is an equilateral triangle. The length of the side of the triangle is `2/(sqrt(5))` (b) `3/(sqrt(5))` (c) `4/(sqrt(5))` (d) none of these

Options

A. `2//sqrt(15)`
B. `3//sqrt(5)`
C. `4//sqrt(5)`
D. none of these

Explanation

To solve the problem, we need to find the length of the side of the equilateral triangle \( PQR \) where \( P \) is on the line \( y + 2x = 1 \) and \( Q \) and \( R \) are on the line \( 3y + 6x = 6 \).

### Step-by-Step Solution:

1. **Identify the lines**:
- The first line is given by the equation \( y + 2x = 1 \).
- The second line can be simplified from \( 3y + 6x = 6 \) to \( y + 2x = 2 \) by dividing the entire equation by 3.

2. **Determine the nature of the lines**:
- The equations \( y + 2x = 1 \) and \( y + 2x = 2 \) represent two parallel lines since they have the same slope (the coefficient of \( x \) is the same).

3. **Calculate the distance between the two parallel lines**:
- The formula for the distance \( d \) between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
- Here, \( C_1 = 1 \) and \( C_2 = 2 \), \( A = 2 \), and \( B = 1 \).
- Thus, the distance \( d \) is:
\[ d = \frac{|2 - 1|}{\sqrt{2^2 + 1^2}} = \frac{1}{\sqrt{5}} \]

4. **Relate the distance to the side of the equilateral triangle**:
- In an equilateral triangle, the height \( h \) can be related to the side length \( s \) using the formula:
\[ h = \frac{\sqrt{3}}{2} s \]
- The height \( h \) also corresponds to the distance between the two parallel lines, which we calculated as \( \frac{1}{\sqrt{5}} \).

5. **Set up the equation**:
- Equating the two expressions for height gives:
\[ \frac{1}{\sqrt{5}} = \frac{\sqrt{3}}{2} s \]

6. **Solve for \( s \)**:
- Rearranging the equation:
\[ s = \frac{2}{\sqrt{3}} \cdot \frac{1}{\sqrt{5}} = \frac{2}{\sqrt{15}} \]

7. **Match with the options**:
- The calculated side length \( s = \frac{2}{\sqrt{15}} \) does not match any of the provided options (a) \( \frac{2}{\sqrt{5}} \), (b) \( \frac{3}{\sqrt{5}} \), (c) \( \frac{4}{\sqrt{5}} \), or (d) none of these.

### Conclusion:
The length of the side of the triangle \( PQR \) is \( \frac{2}{\sqrt{15}} \), which corresponds to option (d) none of these.