Using Slope to Determine Parallel and Perpendicular Lines

The slope of a line determines its steepness. We can determine whether two lines are parallel or perpendicular using their slopes. Two parallel lines are the lines that never intersect and maintain equal distance throughout. Perpendicular lines intersect at right angles (90°). For determining the nature of lines using slope, we will have to learn how to calculate the slope of a line. 

There are various methods by which we can calculate the slope of a line. Let us learn how to determine parallel and perpendicular lines using slope. 

The Slope of a Straight Line

A straight line has no curves. Its equation is generally given as:

y=mx+c

Where m is the slope of the line. 

The slope of a line determines its steepness. There are different methods of calculating the slope for a straight line. 

The basic formula of determining slope is: 

Slope = Rise/Run

Where rise is the difference of y coordinates and run is the difference of x coordinates. 

If we take any two coordinates of a straight line such as A(x1, y1) and B(x2, y2), then the slope of the line will become: 

m=y2-y1 / x2-x1

m=Δy / Δx

Slopes for Parallel Lines

Parallel lines are those lines that have the same distance throughout at every point. Slopes of two parallel lines are always equal. 

If we have two parallel lines of slope m1 and m2, then we have:

m1 / m2=1

m1=m2

Illustration 1: Equations of two lines are given by:

y1=5x+10 

y2=5x+3

Comparing these equations with the standard form of the equation of a straight line we get: 

y = mx + c

Therefore, we have m1 = m2; hence both the lines are parallel. 

Illustration 2: The coordinates of two lines l1 and l2are given as: 

l1 = (2, 4) and (4 , 8)

l2 = (3 , 20) and (9, 32)

To prove they are parallel, we have to prove: 

Since m1=m2, therefore l1 and l2 are two parallel lines. 

Slopes for Perpendicular Lines

For two lines perpendicular to each other, i.e., lines intersecting at right angles (90°), the product of their slopes is equal to -1. 

m1m2=-1

m1=-1 / m2

Illustration 3: Suppose we are given two lines whose slopes are given as: 

m1=-3 / 2 and m2= 2 / 3, then to prove that they both are perpendicular, let us find the product of the slopes. 

Hence, both the lines are perpendicular to each other. 

Illustration 4: The coordinates of two perpendicular lines are given as: 

l1 = (4, 6) and (8, 12)

l2 = (12, 4) and (6, 8) 

To prove they are parallel we have to prove: 

Since m1m2=-1, therefore l1 and l2 are two perpendicular lines. 

Conclusion 

The slope of a line determines its steepness. The slope helps us understand the direction of a line. We can find the slope of a line through the formula, which is given as: 

m2=y2-y1 / x2-x1

The relation between the slopes of two parallel lines m1 and m2 is given as:

m1=m2

The relation between the slopes of two perpendicular lines m1 and m2 is given as: 

m1m2=-1

In this study material, we have learned how to find out that two lines are parallel or are perpendicular to each other using slope.