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Let's go through the questions on the worksheet step by step.

### Part 1: Determine the Slope of Each Line Named Below
You need to find the slopes of lines \(a\), \(b\), \(c\), and \(d\) on the provided graph.

The slope of a line is calculated as:
\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.

Let's identify the points for each line and calculate the slopes:

1. **Line \(a\):**
   - Points: \( (0,3) \) and \( (2,2) \)
   - Slope \(m = \frac{2 - 3}{2 - 0} = \frac{-1}{2} = -\frac{1}{2} \)

2. **Line \(b\):**
   - Points: \( (-4,2) \) and \( (2,2) \)
   - Slope \(m = \frac{2 - 2}{2 - (-4)} = \frac{0}{6} = 0 \) (This is a horizontal line.)

3. **Line \(c\):**
   - Points: \( (-2,-3) \) and \( (2,2) \)
   - Slope \(m = \frac{2 - (-3)}{2 - (-2)} = \frac{5}{4} = \frac{5}{4} \)

4. **Line \(d\):**
   - Points: \( (-5,0) \) and \( (-1,-1) \)
   - Slope \(m = \frac{-1 - 0}{-1 - (-5)} = \frac{-1}{4} = -\frac{1}{4} \)

### Part 2: Any Line Parallel or Perpendicular
5. **Any line parallel to \(b\)**: 
   - The slope will be the same as \(b\), so the slope is 0.

6. **Any line perpendicular to \(d\)**:
   - The slope will be the negative reciprocal of \(d\), so the slope is \(4\).

### Part 3: Graph the Line
7. **Graph the line with slope \(-3\) passing through \(P(1, -2)\)**:
   - Use the point-slope form \(y - y_1 = m(x - x_1)\) to graph this line.

8. **Graph the line passing through \(P(-1, 2)\) and perpendicular to the line determined by \(A(1, 4)\) and \(B(2, 7)\)**:
   - First, calculate the slope of the line \(AB\):
     \[
     \text{slope of AB} = \frac{7 - 4}{2 - 1} = 3
     \]
   - The slope of the perpendicular line will be the negative reciprocal, \(-\frac{1}{3}\).

### Part 4: Determine the Value of \(r\)
9. **Find \(r\) such that the slope between \((5, 3)\) and \((r, 6)\) is 1**:
   - Set up the equation for the slope and solve for \(r\):
     \[
     1 = \frac{6 - 3}{r - 5} \Rightarrow r - 5 = 3 \Rightarrow r = 8
     \]

10. **Find \(r\) such that the slope between \((2, r)\) and \((-2, 6)\) is \(\frac{1}{2}\)**:
    - Set up the equation for the slope and solve for \(r\):
      \[
      \frac{1}{2} = \frac{6 - r}{-2 - 2} \Rightarrow -2 = 6 - r \Rightarrow r = 8
      \]

Let me know if you want detailed solutions or have any questions! Here are five related questions to expand on this content:

1. What is the significance of parallel and perpendicular slopes in geometry?
2. How can you find the equation of a line given only the slope and one point?
3. Why is it important to understand the concept of negative reciprocals when dealing with perpendicular lines?
4. How do you determine if two lines are parallel or perpendicular just by looking at their equations?
5. How can you graph a line if you are only given two points?

**Tip:** When determining the slope of a line, always carefully subtract the coordinates to avoid sign errors.

Determine the slope of each line, find the equations for parallel and perpendicular lines, and solve for points given specific slopes.

Learn how to determine the slope of lines, graph lines through points with a given slope, and find equations for parallel and perpendicular lines. This tutorial covers key concepts in coordinate geometry, including the slope formula, point-slope form, and solving for unknowns.

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