Which statement is true about the graphs of the two lines [tex]y=-8 x-\frac{5}{4}[/tex] and [tex]y=\frac{1}{8} x+\frac{4}{5}[/tex] ?
A. The lines are perpendicular to each other because -8 and [tex]\frac{1}{8}[/tex] are opposite reciprocals of each other.
B. The lines are perpendicular to each other because [tex]-\frac{5}{4}[/tex] and [tex]\frac{4}{5}[/tex] are opposite reciprocals of each other.
C. The lines are neither parallel nor perpendicular to each other because -8 and [tex]\frac{1}{8}[/tex] are not opposite reciprocals of each other.
D. The lines are neither parallel nor perpendicular to each other because [tex]-\frac{5}{4}[/tex] and [tex]\frac{4}{5}[/tex] are not opposite reciprocals of each other.
Answer (2)
Identify the slopes of the two lines: m 1 = − 8 and m 2 = 8 1 .
Check if the lines are parallel: Since m 1 = m 2 , the lines are not parallel.
Check if the lines are perpendicular: Calculate the product of the slopes: ( − 8 ) × ( 8 1 ) = − 1 .
Conclude that the lines are perpendicular because the product of their slopes is -1, and the final answer is: The lines are perpendicular to each other because -8 and 8 1 are opposite reciprocals of each other.
Explanation
Analyzing the Problem We are given two lines: y = − 8 x − 4 5 and y = 8 1 x + 5 4 . We need to determine the relationship between these lines. Specifically, we want to know if they are parallel, perpendicular, or neither. The key to determining this lies in examining the slopes of the lines.
Identifying the Slopes The slope of the first line, y = − 8 x − 4 5 , is m 1 = − 8 . The slope of the second line, y = 8 1 x + 5 4 , is m 2 = 8 1 .
Checking for Parallel Lines Two lines are parallel if their slopes are equal. In this case, m 1 = − 8 and m 2 = 8 1 . Since − 8 = 8 1 , the lines are not parallel.
Checking for Perpendicular Lines Two lines are perpendicular if the product of their slopes is -1. Let's calculate the product of the slopes: m 1 × m 2 = ( − 8 ) × ( 8 1 ) = − 1 .
Determining the Relationship Since the product of the slopes is -1, the lines are perpendicular. The slopes − 8 and 8 1 are also opposite reciprocals of each other, which confirms that the lines are perpendicular.
Final Answer Therefore, the lines are perpendicular to each other because -8 and 8 1 are opposite reciprocals of each other.
Examples
Understanding the relationship between lines is crucial in various real-world applications. For instance, architects use perpendicular lines to design stable and balanced structures, ensuring walls meet floors at right angles. Similarly, in navigation, understanding perpendicular and parallel lines helps in plotting courses and determining directions. In computer graphics, these concepts are fundamental for creating and manipulating objects in a virtual space. Knowing how to analyze slopes and their relationships allows for precise and efficient problem-solving in these diverse fields.
The two lines given are perpendicular to each other because their slopes (-8 and 8 1 ) are opposite reciprocals. Thus, their product equals -1, confirming their perpendicularity. The correct answer is A.
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