Parallel lines and perpendicular lines

In this parallel and perpendicular lines study material, we will see what makes the parallel lines and perpendicular lines different and how to calculate the equations.

Two lines lying on the same plane represent parallel lines when they never cross. Two parallel lines have the same slope. The only distinction between them is their y-intercept. If we were to shift one line vertically towards the y-intercept of the other line, they would then become one line.

Parallel lines:

f (x) = -2x + 6 or f (x) = -2x – 4

Not parallel lines:

f (x) = 3x + 2 or f (x) = 2x + 1

We can tell by these equations and comparing the slopes if they are in parallel. If slopes are identical and the y-intercepts differ, the lines are parallel. If the slopes differ and the y-intercepts are different, the lines are not parallel.

Unlike parallel lines, perpendicular lines do intersect. Their intersection creates a right or a 90-degree angle.

Perpendicular lines are not able to share the same slope. They have slopes that differ; they are distinct from each other in a particular way. A line’s slope will be the reverse of the opposite line. The product of an integer and its reciprocal is 1. If m1 and m2 are negative reciprocals of each other, they are multiplied to give -1.

m₁ x m₂= −1

To determine the reciprocal of a number, you can simply divide 1 by the number. The reciprocal of 8 is 1/8 while the reciprocal of ⅛ is 8. To determine the negative reciprocal, first find the reciprocal then alter the sign.

Similar to parallel lines as mentioned in this parallel and perpendicular lines study material, we can establish the perpendicularity of two lines by looking at their slopes. Each line’s slope is negative reciprocal to the other. Therefore, the lines appear to be perpendicular.

f(x) = 14(x) + 2, negative reciprocal of 14 is -4.

f(x) = -4x + 3, negative reciprocal of -4 is 14.

The product of the slopes is –1.

-4^1/4 = -1

Writing Equations on Parallel Lines

If we have the equation for the line, we could utilise the information we have about slope to formulate the equation for any line that is perpendicular or parallel to it.

Let’s say we have the following equation:

f (x) = 3x + 1

We are aware that the slope of the line is 3. We also know that the y-intercept is (0, 1). Any other line that has a slope of three will run parallel with f(x). The lines resulting from all of the functions listed below are parallel to f(x).

g (x) = 3x + 6

h (x) = 3x + 1

p (x) = 3x + 1/4

If we decide to solve the equation of the line that runs parallel to f and runs across the line (1, 7), we can use the slope, which is 3 and will only have to determine what value for b will yield the right line. You can start by using the point-slope formula of an equation for a line. Then, we can rewrite it in slope-intercept format.

y – y₁ = m (x – x₁)

y – 7 = 3 (x – 1)

y – 7 = 3x – 3

y = 3x + 4

So, g (x) = 3x + 4 passes through the point (1, 7) and is parallel to f (x) = 3x + 1.

Writing Equations for Perpendicular Lines

It is possible to use a similar procedure to create the equation for an angle perpendicular to the given line. Instead of applying the same slope, we instead use an opposite reciprocal to the slope.

Let’s say we have the following equation:

f (x) = 2x + 4

This line’s slope equals 2 while its negative reciprocal is -1/2. Any function that has a slope of -1/2 can be assumed to be perpendicular with f(x). The lines generated by any of the functions listed below are perpendicular to f(x).

g (x) = – 12x + 4

h (x) = – 12x + 2

p (x) = – 12x – 12

As mentioned before, we can reduce our options of a particular perpendicular line when we know it runs through a specific point. Let’s say we want to formulate the equation for an x-axis that is perpendicular to f(x) and that passes across the line (4, 0). We know the slope is 1/2. Then we can make use of the point to determine the y-intercept by converting these values in the slope-intercept format of a line, and then solving for the value b.

g (x) = mx + b

0 = -12 (4) + b

0 = -2 + b

2 = b

b = 2

The formula for the function that has an inclination of -1/2 and an y-intercept of 2 is

g (x) = -12x + 2

Thus, we can say that g (x) = -12x + 2 passes through (4, 0) points and meets at a right angle to f (x) = 2x + 4.

Conclusion

We can call two lines parallel if they do not intersect. Also, we can say that the lines have the same slope.

f (x) =  m₁ x + b₁ and g (x) =  m2 x + b2 are parallel if m₁ = m₂

However, if b1 = b2  and m1 = m2 , we can prove that the lines will intersect. So, they are the same lines. Two lines that intersect at right angles are called perpendicular lines.

f (x) =  m₁ x + b₁ and g (x) =  m₂ x + b₂ are perpendicular if m₁ x m₂= -1 and m₂ = -1/m₁