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To determine the slope of a line perpendicular to the given line segment \( \overline{AB} \), we need to:

1. **Calculate the slope of \( \overline{AB} \)**:
   - The coordinates for points \( A \) and \( B \) appear to be approximately:
     - \( A(-3, -3) \)
     - \( B(1, 2) \)
   - Using the slope formula:
     \[
     \text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-3)}{1 - (-3)} = \frac{2 + 3}{1 + 3} = \frac{5}{4}
     \]

2. **Find the slope of the perpendicular line**:
   - The slope of a line perpendicular to another is the negative reciprocal. So, the perpendicular slope is:
     \[
     -\frac{4}{5}
     \]

3. **Identify the correct answer**:
   - The slope that is perpendicular to \( \overline{AB} \) is \( -\frac{4}{5} \), but since this specific option isn't listed, let's verify if the question might imply a different value or a calculation is missed.

Let me know if this needs further clarification or if there are questions about other possible interpretations of the options. Here are five follow-up questions for exploration:

1. How do we determine the midpoint of \( \overline{AB} \) given points \( A \) and \( B \)?
2. Why do perpendicular slopes involve a negative reciprocal relationship?
3. What would be the equation of a line passing through point \( A \) and perpendicular to \( \overline{AB} \)?
4. How can we verify if two lines are perpendicular solely by looking at their slopes?
5. What is the process of finding the distance between points \( A \) and \( B \)?

**Tip:** Remember that the negative reciprocal of a slope is the key to identifying perpendicular lines on a coordinate plane.

Determine the slope of the line that is perpendicular to this line segment.

Explore how to determine the slope of a line perpendicular to a given line segment using coordinate geometry. Learn key concepts such as calculating slope and understanding perpendicular slopes with negative reciprocal relationships. Ideal for high school students studying geometry.

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