Which statement is true about the graphs of the two lines [tex]y=-\frac{4}{5} x+2[/tex] and [tex]y=-\frac{5}{4} x-\frac{1}{2}[/tex]?
The lines are perpendicular to each other because [tex]-\frac{4}{5}[/tex] and [tex]-\frac{5}{4}[/tex] are opposite reciprocals of each other.
The lines are perpendicular to each other because 2 and [tex]-\frac{1}{2}[/tex] are opposite reclprocats of each other.
The lines are neither parallel nor perpendicular to each other because [tex]-\frac{4}{5}[/tex] and [tex]-\frac{5}{4}[/tex] are not opposite reciprocals of each other.
The lines are neither parallel nor perpendicular to each other because 2 and [tex]-\frac{1}{2}[/tex] are not opposite reciprocals of each other.
Answer (2)
The slopes of the lines are m 1 = − 5 4 and m 2 = − 4 5 .
The lines are not parallel because m 1 = m 2 .
The lines are not perpendicular because m 1 × m 2 = 1 = − 1 .
Therefore, the lines are neither parallel nor perpendicular. The correct answer is: The lines are neither parallel nor perpendicular to each other because − 5 4 and − 4 5 are not opposite reciprocals of each other.
Explanation
Understanding the Problem We are given two lines: y = − 5 4 x + 2 and y = − 4 5 x − 2 1 . We need to determine the relationship between these lines. Specifically, we want to know if they are parallel, perpendicular, or neither.
Identifying Slopes and Intercepts The slope of the first line is m 1 = − 5 4 , and the slope of the second line is m 2 = − 4 5 . The y-intercept of the first line is b 1 = 2 , and the y-intercept of the second line is b 2 = − 2 1 .
Checking for Parallel Lines Two lines are parallel if their slopes are equal. In this case, m 1 = − 5 4 and m 2 = − 4 5 . Since − 5 4 = − 4 5 , the lines are not parallel.
Checking for Perpendicular Lines Two lines are perpendicular if the product of their slopes is -1. Let's check if m 1 × m 2 = − 1 : m 1 × m 2 = ( − 5 4 ) × ( − 4 5 ) = 20 20 = 1 Since the product of the slopes is 1, not -1, the lines are not perpendicular.
Conclusion Since the lines are neither parallel nor perpendicular, the correct statement is: The lines are neither parallel nor perpendicular to each other because − 5 4 and − 4 5 are not opposite reciprocals of each other.
Examples
Understanding the relationships between lines is crucial in various fields. For instance, architects use parallel and perpendicular lines to design buildings and ensure structural stability. City planners use these concepts to design road layouts, ensuring efficient traffic flow and minimizing accidents. In computer graphics, understanding line relationships is essential for creating realistic images and animations.
The correct answer is that the lines are neither parallel nor perpendicular to each other because their slopes − 5 4 and − 4 5 are not opposite reciprocals. This can be verified since their product is 1 and not -1. Thus, the two lines do not exhibit a perpendicular relationship.
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