Question

Use the slopes to determine whether the lines are parallel, perpendicular, or neither.

Lines: 5, 4, 3, 2, 1, -5, -4, 3, -2, -1, 1, 2, K, -4, -5
Parallel Lines: _
Perpendicular Lines: _

a) ( {1}/{4} ) and ( {1}/{4} )
b) ( -{4}/{1} ) and ( {4}/{1} )
c) ( -{4}/{1} ) and ( -{4}/{1} )
d) ( {1}/{4} ) and ( {4}/{1} )

Answer 1

Parallel Lines: b) ( -{4}/{1} ) and ( {4}/{1} )

Perpendicular Lines: a) ( {1}/{4} ) and ( {1}/{4} )

Step-by-step explanation:

In mathematics, the relationship between the slopes of two lines helps determine whether they are parallel, perpendicular, or neither. The slope of a line is typically represented as "m" in the equation y = mx + b, where "m" is the slope.

a) The slopes of both lines in option a) are {1}/{4}. Since they are equal, the lines are parallel.

b) In option b), the slopes are -{4}/{1} and {4}/{1}. These slopes are negative reciprocals of each other ({4}/{1} is the reciprocal of -{4}/{1}), meeting the criteria for perpendicular lines.

c) The slopes in option c) are both -{4}/{1}, indicating that the lines are parallel.

d) The slopes in option d) are {1}/{4} and {4}/{1}. Since they are neither equal nor negative reciprocals, the lines are neither parallel nor perpendicular.

In conclusion, option b represents lines that are parallel, and option a represents lines that are perpendicular.

Answer 2

Parallel Lines: b) ( -{4}/{1} ) and ( {4}/{1} )

Perpendicular Lines: a) ( {1}/{4} ) and ( {1}/{4} )

Step-by-step explanation:

The lines in option (b) have slopes that are negative reciprocals of each other, indicating perpendicularity. On the other hand, the lines in option (a) have the same slope, signifying parallelism.

In option (b), the slopes are -4 and 4. The negative reciprocal of -4 is 1/4, confirming perpendicularity. For option (a), both lines have a slope of 1/4, indicating parallelism since they share the same slope.

Understanding the concept of slopes is crucial in determining the relationship between lines. When two lines are parallel, their slopes are equal. When two lines are perpendicular, the product of their slopes is -1. These principles help establish the nature of the relationship between the given lines in each option.

In conclusion, examining the slopes of the lines in each option allows us to categorize them as either parallel or perpendicular based on the defined relationships between their slopes.