Let's start with the perpendicular lines definition. It specifically is talking about the relationship between two different lines who intersect at a 90 degree angle. Two lines that cross at 90 degrees are said to be perpendicular to each other. A good example of this is the familiar x-y axis. The y-axis is perfectly up and down with absolutely no slant to the side, whereas the x-axis is perfectly left to right with absolutely no deviation to the vertical. You can easily see that they form right angles where they meet. However, it is important to realize that the lines themselves can go in any direction, not strictly up/down and left/right. The only critical part is that they intersect at 90 degrees.
On a side note, a corollary to this definition is that the four angles created by the intersection of two perpendicular lines are all 90 degrees. Comparatively, the intersection of two non-perpendicular lines results in the formation of two acute angles (less than 90 degrees) and two obtuse angles (more than 90). This also demonstrates the geometric law which states that the sum of the angles around a point equals 360 degrees. This is a useful rule to remember when solving geometric proofs.
So then, now that you understand what this word means and how to visually recognize it on a graph, you may then wish to prove that your two lines do indeed meet the criteria. How would you even go about that? How can you determine if two lines are perpendicular? Well, to do this, you need to know the mathematical equations of the lines… or, more specifically, you need to know the slopes of the two lines. (Recall that the slope of a line is most simply expressed as "rise over run", which represents the ratio of vertical change to horizontal change. The slope of the line, often abbreviated by "m", is easily solved by comparing two ordered pairs, and then performing the slope calculation m = y2-y1 / x2-x1.) True perpendicular slopes will have the following relationship:
m1 = -1/m2
In words, this means that the slope of the first line is equal to the negative inverse slope of the second line. Looks a bit complicated, but it's not really. Let's take a look at an example.
Consider the lines described by the equations y = 2x - 1, and y = (-1/2)x + 2. Are these perpendicular?
|This is an example of perpendicular lines.|
With this information, you should be able to see that all you need to know about lines are their slopes to be able to say whether they are perpendicular or parallel. Recall, parallel lines have the exact same slope. The trick when working with this type of question is to realize the the intercept values can be 2 and 3847234, or absolutely anything else at all. They equation of the line may look completely and extraordinarily different for each, though the only important part of them is their m values. Keep this in mind, and don't get intimidated by complex and scary-looking equations!
Another type of question might ask you to determine the equation of a line perpendicular to a given line through a specific point. This takes a bit more work, but it is based on the same concepts. Let's try a question like this.
Find the perpendicular line to the line y = 2x - 1 that goes through point (4,0).
Here's the approach I would take to solve this.
- First, recognize that you are given one of the lines' equations, so from that you can easily find its slope.
- Second, from the first slope, you can use the perpendicular relationship to determine the slope of the second line.
- Third, since you now have the slope and a point that lies on the second line, you can substitute numbers into y = mx + b to solve for b, and then rewrite in in terms of x and y to give the final equation.
I will leave the actual work for you to try yourself, but the line in this case is the same as above, y = -1/2x + 2.
A third type of question might ask you to determine the perpendicular bisector for a given line segment. A bisector is a line that perfectly splits another line into two equal pieces, but it can slice through at any angle. On the other hand, a perpendicular bisector is one the does this at precisely 90 degrees. If you can first determine what the exact midpoint of your line segment is, you can then apply the approach that I outline above to solve this question as well.
There is one last important point that I would like to make about this topic, and it is about notation. You are not incorrect to simply state that "line AB is perpendicular to line CD" (or whatever your lines are called), but the shorthand symbol to show this is an upside-down T shape, ⊥. The keyboard character is called the "up tack", though this term is more applicable to lattice theory, type theory, and logic. I believe it is more appropriate to simply call it the "perpendicular sign." So, in this case, you would simply state your answer as AB⊥CD. That's it. It's much simpler!
Finally, I thought I would just throw in a bit of trivia that I came across while researching this topic. Who knows… you might be able to impress your teacher! The word "perpendicular" originally arose in the late 14th century, and its etymology shows that it came from the Latin word "perpendiculum", which means "plumb line", and "perpendicularis", which means "vertical, as a plumb line". A plumb line was a simple device which was composed of a small weight on the end of a string, and when holding it up, gravity pulls the weight straight down and the string represents perfectly vertical. In relation to the ideally perfectly horizontal ground, you can see how they came up with this term. It's not overly useful information, but you never know where extra trivia might come in handy.
So, with that information, you should now know lots about this subject, and now have no problems graphing perpendicular lines or analyzing and identifying them in either graphs or equations. There are several different variations to the questions that you may encounter, but if you understand the basics of what it is that defines two perpendicular lines, then you should have no problems in coming up with the appropriate solutions! Please let me know in the comments below if you would like any further explanation or examples, and don't forget to +1 my post below and follow me on Twitter! I'm @MathConcepts. You can even click here to tweet about my post! Be sure to visit my follow-up post that discusses a bit more of this concept of perpendicularity.