Graphing Perpendicular Lines

Do you know all about graphing perpendicular lines? How can you tell if two lines are exactly perpendicular to each other? Sometimes, you can look at a graph of lines and be reasonably sure, but maybe it’s just “almost” perpendicular but it isn’t “exactly.” After all, two lines at 89 degrees are still not two lines at 90 degrees! In this post, I’m going to go over all that you need to know so that you can solve one of the favourite exam questions of teachers: determine if two lines are perpendicular to each other?

First of all, let me explain the perpendicular lines definition. If you have two lines whose intersection forms a 90 degree angle, your lines are perpendicular. In addition, when two lines cross at a right angle like this, the four angles around the intersection are all 90 degrees. The most obvious example of perpendicularity is the familiar x-y axis in graphing. You can easily see that the y-axis is perfectly vertical without leaning to either side, and similarly, the x-axis is perfectly horizontal without any variation to it’s height. Where the two axes cross, at the origin, is a right angle. However, perpendicular lines do not necessarily need to be perfectly left-right and up-down. They can very easily be rotated into any orientation. They key concept about this, however, is that the angle of intersection of your two lines must be 90 degrees. Your lines can form an X or a +, or anywhere in between, as long as the angle is a right angle. I’ve written extensively on the definition of perpendicular lines elsewhere previously, so if you would like more information about this particular math concept, I highly recommend that you take a look at that article before continuing here.

So then, now armed with a full understanding of perpendicularity, how can we determine if two lines are perpendicular? First, you need to know a few things first about each of the lines. Actually, you only need to know one key piece of information about each line to be able to solve this problem with confidence: their slopes. Thankfully, there is a simple relationship check that you can do to decide if you have a perfect 90 degree intersection or not. For the sake of this example, let’s consider line 1 and line 2, and their respective slopes are m1 and m2. If the lines are indeed perpendicular, their slopes will have the following relationship:

perpendicular equation

 You can simply say about this that one slope is the negative inverse of the other.

Try it out with a few examples until you are comfortable, then check out their graphs to see that it works.

Graphing Perpendicular Lines

Example: Are the lines described by the following equations perpendicular:

perpendicular lines example

Looks interesting, right? One line seems to be fairly reasonable, but the only has a y-intercept of -12309 so it’s waaaay down low where we’re not used to looking! It doesn’t matter though! On a two-dimensional (x-y) plane, all straight lines will cross exactly once UNLESS they are parallel. Parallel lines will never cross because they run forever at the same angle as each other – they never get any closer or further away from each other. However, every other set of lines will intersect eventually, at some point. So then, based on this information, can we say for certain whether or not the lines described by these two equations are perpendicular or not? As I stated above, all we need to do is analyze their slopes!

In this case, line 1 has a slope of 4. Line 2 has a slope of -1/4. Does that agree with the relationship I have pictured above? Of course it does. The slope of line two is the negative inverse of the slope of line 1! So, that’s all that needs to be done! We can tell at a glance of these equations that the lines run perfectly perpendicular to each other.

Here’s a graph to prove to you that these two lines indeed form 90 degrees. Graphing perpendicular lines is the same as graphing any old line. In this case, the lines are presented to you in slope-intercept form. This makes it easy, at a glance, to be able to plot the y-intercept and then the slope of your line. To fill in more points for confidence, you can always construct a table of values, but figuring out what y is when you substitute in a variety of x values.

perpendicular graphs 1

In this case, we can now confidently state that line 1 is perpendicular to line 2! Of course, since this is mathematics, there is also a symbol to represent this. It looks like an upside-down capital T: ⟘. So, in this case, we could write something like “line 1 ⟘ line 2”. And that’s as easy as it gets!

Let’s try one more:

Example: Are the lines described by the following equations perpendicular:

perpendicular example

 

For this one, I’ll just get right to the point. 🙂

The slope of line 1 can readily be seen to be 21. The slope of line 2 is equally obviously -1/20. Does this satisfy the relationship shown above? Unfortunately, no it doesn’t. It ALMOST does! But it doesn’t. Here is the graph of these two lines. You should take away from this that even graphs that LOOK like they are perpendicular may not be so exactly. It is always best to assess their equations, analyze their slopes, and then confidently say yes or no.

perpendicular example

I think that these examples should hopefully be sufficient to demonstrate what you need to know about graphing perpendicular lines! I’ve explained to you that all you really need is to know the slopes of your two lines in order to properly assess your situation. If you know the equations of the lines, you can simply read the slope from the equation, where slope is equal to the m value. Alternately, you may have work to do to uncover the slopes, in cases where you are only provided the ordered pairs of a set of points that you can plot on the graph. Although, I’ve already covered how you can calculate the slope in such cases, so that is still a straight forward exercise. If you can see, or calculate, the slopes of your lines, then that is a much faster method of evaluating their perpendicularity than creating a table of values and manually graphing the lines. Besides, as my second example demonstrated above, even that method isn’t foolproof, because you could very easily mistake that graph as being perpendicular! It is always best to figure out the slopes, and perform the analysis that I did in this post, and you will be able to confidently say whether you are graphing perpendicular lines or not!

If you’ve read all the way here to the bottom, thank you for reading my post! Please leave me a comment to let me know that this is helpful for you, and I always appreciate people subscribing or following me, which you can do on the right side of this post! Thanks again, and I encourage you to explore my site to see if there is anything else I can help explain for you!

In my next post, I will demonstrate to you what you need to know about graphing parallel lines.

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