To determine the relationship between the segments \( AB \) and \( CD \) from their respective equations, we need to analyze the slopes of their lines.
Step-by-Step Solution:
1. Identify the slope of line \( AB \):
- Given the equation of line \( AB \): \( y - 4 = -5(x - 1) \).
- Rearrange to the slope-intercept form \( y = mx + b \):
[tex]\[
y - 4 = -5x + 5 \implies y = -5x + 9
\][/tex]
- Therefore, the slope of line \( AB \) is \( -5 \).
2. Identify the slope of line \( CD \):
- Given the equation of line \( CD \): \( y - 4 = \frac{1}{8}(x - 5) \).
- Rearrange to the slope-intercept form \( y = mx + b \):
[tex]\[
y - 4 = \frac{1}{8}x - \frac{5}{8} \implies y = \frac{1}{8}x + \frac{27}{8}
\][/tex]
- Therefore, the slope of line \( CD \) is \( \frac{1}{8} \).
3. Determine the relationship between the slopes:
- Two lines are perpendicular if the product of their slopes is \( -1 \). Check for slopes \( -5 \) and \( \frac{1}{8} \):
[tex]\[
(-5) \times \left(\frac{1}{8}\right) = -\frac{5}{8}, \quad \text{which is not equal to} \, -1.
\][/tex]
- Therefore, they are not perpendicular by this condition.
- Two lines are parallel if they have the same slope. Here, the slopes are \( -5 \) and \( \frac{1}{8} \), which are not equal.
Given this analysis:
The correct statement that proves the relationship of segments \( AB \) and \( CD \) is:
They are perpendicular because they have slopes that are opposite reciprocals of -5 and [tex]\(\frac{1}{8}\)[/tex].
