Parallel and Perpendicular Lines

Parallel and perpendicular lines are everywhere in our daily lives, from the stripes on a road to the walls of a building. Parallel lines are two lines that are always the same distance apart and never meet, no matter how far they are extended. For example, railroad tracks are parallel lines. Perpendicular lines, on the other hand, intersect at a 90-degree angle. This can be seen in many places, such as the corners of a room or the cross-section of a bridge. In mathematics, understanding how to identify and work with parallel and perpendicular lines is crucial. It helps in areas such as geometry, trigonometry, and calculus. For example, architects use the concept of parallel and perpendicular lines to design structures that are both aesthetically pleasing and structurally sound.

But how do you know which lines are truly parallel or perpendicular? Understanding their equations can help. Let's review the different types of lines and walk through a few examples.

What are parallel lines?

Parallel lines are straight coplanar lines that never intersect and have the same slope. Here is an example of parallel lines:

What are perpendicular lines?

Perpendicular lines are coplanar and have two distinct straight lines that intersect one another at 90 degrees (right angle). Here is an example of perpendicular lines:

You'll see perpendicular lines in a variety of places, from the tiles in your bathroom to the four-square grid outside on a playground.

Identifying and finding parallel and perpendicular lines

Being familiar with the slope-intercept form $y = m x + b$ of a line will help you determine whether a set of lines is parallel or perpendicular. Let's look at the following lines:

$y = 3 x + 8$

$y = 3 x - 4$

These lines are parallel because they have the same slope of 3 ("m" represents the slope and "b" represents the y-intercept). Here's how the two lines are graphed:

If you want to find the equation of any line parallel to another line, switch out the y-intercept but don't change the slope ex. $y = 5 x + 6$ and $y = 5 x - 2$ .

Now, let's look at these two lines:

$y = - \frac{1}{3} x + 6$

$y = 3 x - 4$

The slopes of perpendicular lines are always the opposite reciprocal of one another, which means you can multiply them and always get -1. The lines above are perpendicular because their slopes are negative reciprocals: $(- \frac{1}{3}), (3)$ . Here's how the two lines are graphed:

If you want to find the equation of a line perpendicular to another line, always switch out the slope ("m") for an opposite reciprocal.

Parallel and perpendicular lines that pass through a point

Let's go over how to find the equations of parallel and perpendicular lines that pass through specific points or coordinates $(x_{1}, y_{1})$ . Suppose you have the line $y = 2 x + 3$ and want to find the equation for a parallel line that passes through the point (6,5).

You can find the equation using the point-slope form: $y - y_{1} = m (x - x_{1})$ .

Start by replacing m with 2 for the slope:

$y - y_{1} = 2 (x - x_{1})$

Next, add the point (6,5) in the proper places ( $x_{1}$ is 6 and $y_{1}$ is 5):

$y - 5 = 2 (x - 6)$

Use the distributive property:

$y - 5 = 2 x - 12$

Add 5 to each side:

$y - 5 + 5 = 2 x - 12 + 5$

Answer: $y = 2 x - 7$

Now, suppose you have the same line $y = 2 x + 3$ and want to find the equation for a perpendicular line that passes through the point $(6, 5)$ . Remember, for perpendicular lines, the slope (m) of one line is the opposite reciprocal of the slope of the other line.

Start with point-slope form: $y - y_{1} = n (x - x_{1})$ .

Replace m with the negative reciprocal $- \frac{1}{2}$ for the slope:

$y - y_{1} = - \frac{1}{2} (x - x_{1})$

Add the point (6,5) in the proper places ( $x_{1}$ is 6 and $y_{1}$ is 5):

$y - 5 = - \frac{1}{2} (x - 6)$

Use the distributive property:

$y - 5 = - \frac{1}{2} x + \frac{6}{2}$ or $y - 5 = - \frac{1}{2} x + 3$

Add 5 to each side:

$y - 5 + 5 = - \frac{1}{2} x + 3 + 5$

Answer: $y = - \frac{1}{2} x + 8$

Practice questions on identifying and finding parallel and perpendicular lines

Are the following lines parallel, perpendicular, or neither?

a. $y = 4 x + 2$ and $y = - \frac{1}{4} x - 3$

Answer: Perpendicular

b. $y = 5 x + 2$ and $y = 5 x - 7$

Answer: Parallel

c. $y = -3 x + 6$ and $y = \frac{1}{3} x - 4$

Answer: Perpendicular

d. $y = 2 x + 7$ and $y = - 2 x + 7$

Answer: Neither

Find the equation for the following:

e. A line parallel to $y = - 5 x + 3$ that passes through the point $(1, 4)$

$y - 4 = - 5 (x - 1)$

$y - 4 = - 5 x + 5$

$y - 4 + 4 = - 5 x + 5 + 4$

Answer: $y = -5 x + 9$

f. Find a line perpendicular to $y = - \frac{1}{3} x + 6$ that passes through the point $(3, 5)$ .

$y - 5 = 3 (x - 3)$

$y - 5 = 3 x - 9$

$y - 5 + 5 = 3 x - 9 + 5$

Answer: $y = 3 x - 4$

Topics related to the Parallel and Perpendicular Lines

Parallel Lines

Midpoint Formula

Distance between Parallel Lines

Flashcards covering the Parallel and Perpendicular Lines

Advanced Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Parallel and Perpendicular Lines

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

Get help learning about parallel and perpendicular lines

Learning about parallel and perpendicular lines can be a bit intimidating at first, especially when tasked with identifying lines or finding equations for lines. Fortunately, working with a skilled tutor can help your student improve their understanding of different types of lines and even assist with assignments and test preparations. Find out more about the perks of tutoring and get your student signed up to start by reaching out to the Educational Directors at Varsity Tutors today.