Let's break down the given problem step-by-step to find the slopes of the lines and then determine their relationships (parallel, perpendicular, or neither).

Part (a) - Find the slope of each line:

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Slope of Line 1 (passes through $(-4, -3)$ and $(8, 0)$):

$m_1 = \frac{0 - (-3)}{8 - (-4)} = \frac{0 + 3}{8 + 4} = \frac{3}{12} = \frac{1}{4}$

Slope of Line 2 (passes through $(-1, 6)$ and $(0, 2)$):

$m_2 = \frac{2 - 6}{0 - (-1)} = \frac{-4}{0 + 1} = -4$

Slope of Line 3 (passes through $(6, -7)$ and $(-3, -4)$):

$m_3 = \frac{-4 - (-7)}{-3 - 6} = \frac{-4 + 7}{-3 - 6} = \frac{3}{-9} = -\frac{1}{3}$

Part (b) - Determine whether the lines are parallel, perpendicular, or neither:

• Parallel lines have equal slopes.
• Perpendicular lines have slopes that are negative reciprocals of each other, i.e., $m_1 \times m_2 = -1$.
• If neither of these conditions is met, the lines are neither parallel nor perpendicular.

Line 1 and Line 2:

• Slope of Line 1: $\frac{1}{4}$
• Slope of Line 2: $-4$
• Check if they are negative reciprocals: $\frac{1}{4} \times (-4) = -1$.

Thus, Line 1 and Line 2 are perpendicular.

Line 1 and Line 3:

• Slope of Line 1: $\frac{1}{4}$
• Slope of Line 3: $-\frac{1}{3}$
• They are not equal, and $\frac{1}{4} \times \left(-\frac{1}{3}\right) \neq -1$.

Thus, Line 1 and Line 3 are neither parallel nor perpendicular.

Line 2 and Line 3:

• Slope of Line 2: $-4$
• Slope of Line 3: $-\frac{1}{3}$
• They are not equal, and $-4 \times \left(-\frac{1}{3}\right) \neq -1$.

Thus, Line 2 and Line 3 are neither parallel nor perpendicular.

Final Answers:

• Slope of Line 1 = $\frac{1}{4}$
• Slope of Line 2 = $-4$
• Slope of Line 3 = $-\frac{1}{3}$

For each pair:

• Line 1 and Line 2: Perpendicular
• Line 1 and Line 3: Neither
• Line 2 and Line 3: Neither

Would you like more details or have any questions?

Here are five related questions you might find helpful:

1. What does it mean for two lines to have slopes that are negative reciprocals?
2. How do we calculate the slope of vertical or horizontal lines?
3. Can we have more than two lines being perpendicular to one another?
4. What is the significance of parallel and perpendicular lines in coordinate geometry?
5. How can we determine the equation of a line given its slope and one point?

Tip: When comparing slopes, always check for both equality (parallel) and negative reciprocal (perpendicular) relationships to avoid errors.