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Segment \( AB \) is on the line \( y - 4 = -5(x - 1) \), and segment \( CD \) is on the line \( y - 4 = \frac{1}{8}(x - 5) \). Which statement proves the relationship of segments \( AB \) and \( CD \)?

A. They are perpendicular because they have slopes that are opposite reciprocals of 5 and \( -\frac{1}{8} \).

B. They are perpendicular because they have slopes that are opposite reciprocals of -5 and \( \frac{1}{8} \).

C. They are parallel because they have the same slope of [tex]\( -\frac{1}{8} \)[/tex].



Answer :

GinnyAnswer
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].

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