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This image contains a math problem written in Persian. Here's the translated version:

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**Problem 8**:
The endpoints of line segment \(AB\) are \( A(3, 1) \) and \( B(-1, 2) \).

**a)** Write the equation of a line that is perpendicular to the line segment \(AB\) and passes through point \(A\).

**b)** Write the equation of a line that is parallel to the line segment \(AB\) and passes through the origin.

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Let me solve these problems.

### Part (a): Line Perpendicular to \(AB\) through \(A(3, 1)\)

1. **Slope of Line Segment \(AB\)**:
   The slope \(m_{AB}\) of line segment \(AB\) is given by:

   \[
   m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 1}{-1 - 3} = \frac{1}{-4} = -\frac{1}{4}
   \]

2. **Slope of Perpendicular Line**:
   The slope \(m_{\perp}\) of the line perpendicular to \(AB\) is the negative reciprocal of \(m_{AB}\):

   \[
   m_{\perp} = 4
   \]

3. **Equation of Perpendicular Line**:
   Using the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \), where the point is \(A(3, 1)\) and the slope is \(4\):

   \[
   y - 1 = 4(x - 3)
   \]

   Simplifying this:

   \[
   y - 1 = 4x - 12
   \]
   \[
   y = 4x - 11
   \]

Thus, the equation of the line perpendicular to \(AB\) passing through point \(A\) is \( y = 4x - 11 \).

### Part (b): Line Parallel to \(AB\) through the Origin

1. **Slope of Parallel Line**:
   The slope of a line parallel to \(AB\) is the same as the slope of \(AB\). Thus, the slope is \( m_{AB} = -\frac{1}{4} \).

2. **Equation of Parallel Line through the Origin**:
   The equation of a line with slope \( m = -\frac{1}{4} \) passing through the origin \( (0, 0) \) is:

   \[
   y = -\frac{1}{4}x
   \]

Thus, the equation of the line parallel to \(AB\) passing through the origin is \( y = -\frac{1}{4}x \).

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Would you like more details on these steps or further clarification?

Here are 5 follow-up questions based on this topic:

1. How would you find the distance between points \(A(3, 1)\) and \(B(-1, 2)\)?
2. What is the geometric interpretation of a perpendicular line in the coordinate plane?
3. How can we derive the midpoint of the line segment \(AB\)?
4. How would the equation of a line change if it were parallel to \(AB\) but passed through a different point instead of the origin?
5. Can you find the point where the perpendicular line intersects the line passing through \(B\)?

**Tip**: When finding the equation of a perpendicular line, remember that the slopes are negative reciprocals of each other.

The endpoints of line segment AB are A(3, 1) and B(-1, 2). (a) Write the equation of a line that is perpendicular to the line segment AB and passes through point A. (b) Write the equation of a line that is parallel to the line segment AB and passes through the origin.

Find the equations of lines perpendicular and parallel to line segment AB where A(3, 1) and B(-1, 2). This problem involves the concept of slopes, linear equations, and the relationship between parallel and perpendicular lines. Suitable for Grades 9-11.

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