Let's solve the problem.

We are given two lines defined by the points $A(1, 5)$, $B(4, 4)$, $C(9, 10)$, and $D(-6, -5)$, and we are tasked with determining whether these lines $AB$ and $CD$ are parallel, perpendicular, or neither.

Step 1: Find the slope of line $AB$

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

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

For line $AB$, we have points $A(1, 5)$ and $B(4, 4)$. Using the slope formula:

$m_{AB} = \frac{4 - 5}{4 - 1} = \frac{-1}{3}$

So, the slope of line $AB$ is $m_{AB} = -\frac{1}{3}$.

Step 2: Find the slope of line $CD$

For line $CD$, we have points $C(9, 10)$ and $D(-6, -5)$. Using the slope formula:

$m_{CD} = \frac{-5 - 10}{-6 - 9} = \frac{-15}{-15} = 1$

So, the slope of line $CD$ is $m_{CD} = 1$.

Step 3: Determine the relationship between the slopes

• If the slopes are equal ($m_1 = m_2$), the lines are parallel.
• If the product of the slopes is $-1$ ($m_1 \times m_2 = -1$), the lines are perpendicular.
• Otherwise, the lines are neither.

Let's check the product of the slopes:

$m_{AB} \times m_{CD} = -\frac{1}{3} \times 1 = -\frac{1}{3}$

Since the product is not $-1$ and the slopes are not equal, the lines $AB$ and $CD$ are neither parallel nor perpendicular.

Would you like further clarification or details on any step?

Here are 5 related questions to deepen your understanding:

1. How do we know if two lines are parallel in general?
2. What is the geometric interpretation of the slope of a line?
3. How can you determine if two lines are perpendicular using slopes?
4. Can two lines be neither parallel nor perpendicular and still intersect?
5. How would the slopes change if one line was vertical or horizontal?

Tip: Always remember that parallel lines have equal slopes, and perpendicular lines have slopes whose product is $-1$.