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### Part 1: Perpendicular Line to the Given Line
The goal is to find the equation of a line that passes through point **A** and is **perpendicular** to the given line on the graph.

1. **Determine the slope of the given line:**
   - From the graph, the given line goes through the points \((-3, -6)\) and \((0, -4)\). 
   - Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
     \[
     m = \frac{-4 - (-6)}{0 - (-3)} = \frac{-4 + 6}{3} = \frac{2}{3}.
     \]
   - So, the slope of the given line is \(\frac{2}{3}\).

2. **Find the slope of the perpendicular line:**
   - The slope of a perpendicular line is the **negative reciprocal** of the slope of the original line.
     \[
     m_{\text{perpendicular}} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2}.
     \]

3. **Write the equation of the perpendicular line:**
   - We need to write the equation in point-slope form: \(y - y_1 = m(x - x_1)\).
   - The coordinates of point A are \((3, -5)\).
   - Plug in the slope \(m = -\frac{3}{2}\) and point \(A(3, -5)\):
     \[
     y - (-5) = -\frac{3}{2}(x - 3).
     \]
   - Simplifying this, we get:
     \[
     y + 5 = -\frac{3}{2}(x - 3).
     \]
   - This matches option **(C)**.

### Part 2: Equation of the Line through Point A with a Slope of Zero
- A line with a slope of zero is a **horizontal line**.
- The equation of a horizontal line is \(y = c\), where \(c\) is the y-coordinate of the point it passes through.
- Point A has coordinates \((3, -5)\), so the equation is:
  \[
  y = -5.
  \]

### Part 3: Equation of the Line through Point A with an Undefined Slope
- A line with an undefined slope is a **vertical line**.
- The equation of a vertical line is \(x = c\), where \(c\) is the x-coordinate of the point it passes through.
- For point A \((3, -5)\), the equation is:
  \[
  x = 3.
  \]

### Summary:
1. The answer to the perpendicular line equation is **(C)**.
2. The equation of the line through point A with a slope of zero is \(y = -5\).
3. The equation of the line through point A with an undefined slope is \(x = 3\).

Would you like more details on any of these steps?

### Related Questions:
1. How do you find the equation of a line given two points?
2. What is the slope of a vertical line, and why is it undefined?
3. Can you explain why the product of slopes of perpendicular lines is \(-1\)?
4. What happens to the graph of a line as its slope increases?
5. How can you graph a line given its equation in point-slope form?

**Tip**: The negative reciprocal of a slope is essential for perpendicular lines. Always remember this relationship when dealing with perpendicularity!

Which of the following best represents the equation of the line that goes through point A and is perpendicular to the line, and additional questions about the slope of lines passing through point A?

Solve the problem of finding the equation of a line that goes through point A and is perpendicular to the given line, with questions about lines with zero and undefined slopes. Learn key concepts like slope relationships and point-slope form, suitable for students in Grades 8-10.

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