To determine whether the two lines given are parallel, perpendicular, or neither, we need to calculate the slopes of each line.

Step 1: Calculate the Slope of the First Line

The first line goes through the points $(-5, 10)$ and $(-1, -6)$.

The slope $m$ is calculated using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

For the points $(-5, 10)$ and $(-1, -6)$: $m_1 = \frac{-6 - 10}{-1 + 5} = \frac{-16}{4} = -4$

So, the slope of the first line is $-4$.

Step 2: Calculate the Slope of the Second Line

The second line goes through the points $(6, 7)$ and $(22, 11)$.

Using the same formula: $m_2 = \frac{11 - 7}{22 - 6} = \frac{4}{16} = \frac{1}{4}$

So, the slope of the second line is $\frac{1}{4}$.

Step 3: Determine the Relationship Between the Slopes

1. Parallel lines have equal slopes.
2. Perpendicular lines have slopes that are negative reciprocals of each other.

In this case:

• The slope of the first line is $-4$.
• The slope of the second line is $\frac{1}{4}$.

Since $-4$ is the negative reciprocal of $\frac{1}{4}$, the two lines are perpendicular.

Answer:

The correct choice is:

• Perpendicular

Would you like further details on how slopes determine the relationship between lines?

Here are five related questions to explore this concept further:

1. What is the general condition for two lines to be parallel?
2. How would we calculate the angle between two lines given their slopes?
3. Can two lines be neither parallel nor perpendicular? Under what conditions?
4. How do we determine if two lines are collinear?
5. What happens to the slope if the line is vertical or horizontal?

Tip: When finding the relationship between lines, always check if the slopes are negative reciprocals for perpendicularity or equal for parallelism.