The vertices of a `triangleOBC` are `O(0,0) , B(-3,-1), C(-1,-3)`. Find the equation of the line parallel to BC and intersecting the sides OB and OC and whose perpendicular distance from the origin is `1/2`.

Answer

To solve the problem, we need to find the equation of a line parallel to the line segment BC of triangle OBC, which intersects the sides OB and OC, and has a perpendicular distance of \( \frac{1}{2} \) from the origin.

### Step-by-Step Solution:

1. **Identify the Coordinates of Points**:
- The vertices of triangle OBC are given as:
- \( O(0, 0) \)
- \( B(-3, -1) \)
- \( C(-1, -3) \)

2. **Find the Slope of Line BC**:
- The slope \( m \) of line BC can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- Here, \( B(-3, -1) \) and \( C(-1, -3) \):
\[ m = \frac{-3 - (-1)}{-1 - (-3)} = \frac{-3 + 1}{-1 + 3} = \frac{-2}{2} = -1 \]

3. **Equation of Line Parallel to BC**:
- Since the slope of line BC is \(-1\), the equation of any line parallel to BC can be written in the form:
\[ y = -x + c \]
- Rearranging gives:
\[ -x - y + c = 0 \]

4. **Calculate the Perpendicular Distance from the Origin**:
- The formula for the perpendicular distance \( d \) from a point \((x_0, y_0)\) to the line \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
- For our line, \( A = -1, B = -1, C = c \), and the point is the origin \((0, 0)\):
\[ d = \frac{|-1(0) - 1(0) + c|}{\sqrt{(-1)^2 + (-1)^2}} = \frac{|c|}{\sqrt{2}} \]
- We know that the perpendicular distance from the origin is \( \frac{1}{2} \):
\[ \frac{|c|}{\sqrt{2}} = \frac{1}{2} \]

5. **Solve for \( c \)**:
- Multiplying both sides by \( \sqrt{2} \):
\[ |c| = \frac{\sqrt{2}}{2} \]
- This gives two possible values for \( c \):
\[ c = \frac{\sqrt{2}}{2} \quad \text{or} \quad c = -\frac{\sqrt{2}}{2} \]

6. **Determine the Correct Value of \( c \)**:
- Since the line must intersect the sides OB and OC, it should have a negative intercept on the y-axis. Thus, we choose:
\[ c = -\frac{\sqrt{2}}{2} \]

7. **Final Equation of the Line**:
- Substituting \( c \) back into the line equation:
\[ -x - y - \frac{\sqrt{2}}{2} = 0 \]
- Rearranging gives:
\[ x + y + \frac{\sqrt{2}}{2} = 0 \]

### Conclusion:
The equation of the required line is:
\[ x + y + \frac{\sqrt{2}}{2} = 0 \]