Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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].
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].
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.