The problem involves finding the slopes ($m$) of the lines $\overline{AB}$ and $\overline{CD}$, and determining the type of lines they represent. Let's solve step-by-step:

1. Find the slope formula:
   The slope $m$ of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

2. Calculate $m(\overline{AB})$ for points $A(-3, 8)$ and $B(3, 2)$:
   $m(\overline{AB}) = \frac{2 - 8}{3 - (-3)} = \frac{-6}{6} = -1$

3. Calculate $m(\overline{CD})$ for points $C(7, 1)$ and $D(5, -1)$:
   $m(\overline{CD}) = \frac{-1 - 1}{5 - 7} = \frac{-2}{-2} = 1$

4. Determine the relationship between the lines:

   • The slope of $\overline{AB}$ is $-1$.
   • The slope of $\overline{CD}$ is $1$.
     Since the slopes are negative reciprocals ($-1 \times 1 = -1$), the lines are perpendicular.

Final Answer:

Let me know if you'd like further explanation!

Here are 5 related questions to consider:

1. How do we determine if two lines are parallel?
2. How can we find the equation of a line given two points?
3. What is the geometric significance of perpendicular slopes?
4. Can two lines with undefined slopes be perpendicular?
5. How do negative reciprocals relate to rotations in geometry?

Tip: Always simplify fractions for slopes and check the sign carefully to avoid errors in classification!