Step-by-step explanation:
Finding the Slopes of Parallel and Perpendicular Lines
How do we know if two distinct lines are parallel, perpendicular or neither? To make that determination, we need to review some background knowledge about slope.
Concept 1: When two points are given, the slope of a line can be algebraically solved using the following formula:
Slope Formula
The slope,mm, of a line passing through two arbitrary points \left( {{x_1},{y_1}} \right)(x
1
,y
1
) and \left( {{x_2},{y_2}} \right)(x
2
,y
2
) is calculated as follows…
m = (ysub2-ysub1)/(xsub2-xsub1)
Concept 2: When a linear equation is given, we can find the slope by transforming it into the Slope-Intercept Form. The value of slope will stand out, as it is the coefficient of the linear term (xx-term).
Slope-Intercept Form of a Line
The linear equation written in the form y = mx + by=mx+b is in slope-intercept form where:
m is the slope, and b is the y-intercept
Now, suppose we have two distinct and nonvertical lines, {\ell _1}ℓ
1
and {\ell _2}ℓ
2
in Slope-Intercept Form.
Line 1:
line 1 → y=msub1x+b
Line 2:
line 2 → y = msub2x+b
Parallel Lines: The lines are parallel if their slopes are equal or the same. That means
Equal Slopes:
msub1 = msub2
Graph:
graph showing line 1 and line 2 parallel on the coordinate plane.
Perpendicular Lines: The lines are perpendicular if their slopes are opposite reciprocals of each other. Or, if we multiply their slopes together, we get a product of - \,1−1. These lines intersect at a ninety-degree angle, 90°.
Opposite Reciprocal Slopes:
msub1 = -(1/msub2)
Product of Slopes:
msub1× msub2 = -1
Graph:
graph showing line 1 and line 2 perpendicular to each other on the coordinate plane.
Examples of How to Find the Slopes of Parallel and Perpendicular Lines
Example 1: Line 1 passes through the points \left( {1,3} \right)(1,3) and \left( {4,9} \right)(4,9), while line 2 passes through \left( {2,5} \right)(2,5) and \left( { - \,2, - \,3} \right)(−2,−3). Tell whether these lines are parallel, perpendicular, or neither.
