For the following exercises, use the descriptions of each...

For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?
Line $1 :$ Passes through $(0,6)$ and $(3,-24)$
Line 2 : Passes through $(-1,19)$ and $(8,-71)$

Key Concepts

Parallel Lines

Parallel lines are lines in the same plane that never intersect, and their slopes are equal. This equality of slopes is a key criterion for identifying parallelism in coordinate geometry, ensuring that the lines maintain a consistent distance apart.

Slope

The slope of a line is a measure of its steepness, defined as the ratio between the vertical change (rise) and the horizontal change (run) between any two points on the line. It is a foundational concept in coordinate geometry that quantifies how much one variable changes in relation to another.

Computing Slope from Two Points

To determine the slope from two points, you subtract the y-coordinates to get the rise and the x-coordinates to get the run, then divide the rise by the run. This calculation provides a constant value for the slope of a straight line and is critical for further analysis of linear relationships.

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle. In coordinate geometry, the slopes of perpendicular lines are negative reciprocals of one another, meaning that when multiplied together, the result is -1. This criterion is used to verify whether two lines meet at a 90-degree angle.

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