The image shows a math problem involving finding the slopes of lines formed by pairs of points and classifying the lines as parallel, perpendicular, or neither. Here's a step-by-step explanation:

Problem Breakdown

The points provided are:

• $A(3, 13)$
• $B(4, -15)$
• $C(-2, 5)$
• $D(1, -7)$

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

Steps:

1. Find the slope of $\overrightarrow{AB}$: Using points $A(3, 13)$ and $B(4, -15)$: $m_{AB} = \frac{-15 - 13}{4 - 3} = \frac{-28}{1} = -28$

2. Find the slope of $\overrightarrow{CD}$: Using points $C(-2, 5)$ and $D(1, -7)$: $m_{CD} = \frac{-7 - 5}{1 - (-2)} = \frac{-12}{3} = -4$

3. Classify the lines:

   • Two lines are parallel if their slopes are equal.
   • Two lines are perpendicular if the product of their slopes is $-1$.
   • Otherwise, they are classified as neither.

   The slopes are: $m_{AB} = -28, \quad m_{CD} = -4$ Since $-28 \neq -4$ and $(-28) \times (-4) \neq -1$, the lines are neither parallel nor perpendicular.

Final Answer:

• Slopes: $m_{AB} = -28$, $m_{CD} = -4$
• Classification: Neither

Would you like further clarifications or help with related problems? Here are 5 questions to expand your understanding:

1. How are perpendicular slopes calculated for any two lines?
2. Can two vertical lines ever be classified as perpendicular?
3. What happens when two lines have a slope of 0?
4. How can you determine if a line passes through a specific point?
5. What are the characteristics of parallel lines in three-dimensional space?

Tip: Always simplify your slopes to their lowest terms before comparing them.