Thursday, April 5, 2007

Graphing - Parallel and Perpendicular Lines

How can you determine if two lines are perpendicular?  How can you determine if two lines are parallel?  If you have two lines on a graph, and you have determined their equations or slopes, you may be asked if the two lines are parallel or perpendicular to each other.  These are two favorite questions of teachers and you will undoubtedly have to answer them!  Keep reading to find out how to easily answer them, and please remember to click the +1 button at the end.

Parallel lines are at the same angle and will never cross... like two railroad tracks. It doesn't matter what direction the lines travel. As long as they are going the same way, they are parallel. In mathematical terms, two lines are said to be parallel if they have the exact same slope.

Remember our equation for a line, y = mx + b.  Two parallel lines, each defined by their own equation, will have the same value for m, the slope.  So, y = 3x + 5 and y = 3x + 200 are parallel lines (they differ in their y-intercepts, but they have the same slope m).  You can plot this out for yourself quickly to see that this is indeed the case.

The opposite of parallel lines are perpendicular lines.  But how can you determine if two lines are perpendicular?  Perpendicular lines have a bit of a twist to them. Two lines are perpendicular if they cross (remember, any two straight lines that are NOT parallel will cross at only a single point.  They cannot ever intersect again unless they curve back on themselves, in which case they are not straight!) and they form a 90 degree angle, or rather, a T-shape. For example: The x-axis and y-axis are perpendicular to each other. Mathematically, if line 1 has a slope of m1, then a perpendicular line 2 will have a slope m2=(-1/m1)... that is, it's slope will be the negative inverse of the first.

Try it out... y = 2x + 1 and y = (-1/2)x + 5... m1 = 2 and m2 = (-1/2). Check it out on the graph to see that they indeed form a 90 degree angle where they intersect.  If you don't believe that this is correct, go get a protractor and measure it for yourself!  You can then convince yourself that the relationship holds true.

example of perpendicular lines
Perpendicular graph

Also of interest to you: the symbol for "perpendicular" is ┴, an upside-down capital T, whereas the symbol for "parallel" is two vertical lines next to each other, like ||.  You would write AB┴CD to say that AB is perpendicular to CD, and similarly AB||CD to say that they are parallel.

So, I've shown you that to in order to determine if lines are parallel or perpendicular, all you need to know is their slopes!  If you know the equations of the lines, then this is only a matter of simply reading the m value, or at the worst, performing some basic mathematical rearrangements to find this info.  And once you have the numbers, you can easily see if either relationship applies.  This is much faster than manually creating a table of values and plotting out the graphs by hand!  Now, you should have no problem answering questions that ask you to quickly identify true parallel or perpendicular line relationships!  If this post helped solve your questions, please remember to do me a favor and click the +1 button below to help me get the word out!  Thanks!


  1. Mathematically, if line 1 has a slope of m1, then a perpendicular line 2 will have a slope m2=(-1/m1).


  2. Think about it like this... perpendicular lines are 90 degrees to each other. So, take a line (line 1), and from that determine it's rise and it's run (ie. its slope). Now, if you were to rotate the things 90 degrees and then overlap this perpendicular line (line 2) on to the original line 1, you would see that rise2=run1, and run2=rise1. You're essentially dealing with the same numbers, just in different places. Also realize that if you have a line rising to the right, its corresponding perpendicular line will fall to the right, and so its slope will be negative. (Technically, the negative comes from the (y2-y1)/(x2-x1), but the numerical value is the same.) So, you can see that:

    m1 = rise1/run1

    m2 = rise2/run2
    = -(run1/rise1)...

    which is the same as saying m2=(-1/m1).

    I hope that clears things up a bit for you.

  3. Another way to look at it is by noticing that both lines have a point in common, if we equal both line's equations we get the final relation between the two different slopes...

    Great Blog!!!!

    PD: I'm from Argentina...sorry for my english...

  4. Hey i just want to say tht was a amazing expanation. And a rlly awesome blog. you really helped me out :).. thanks so much!! :) :)

  5. great explanation! thanks a lot!!

  6. Your blog is very informative and I am here to discuss the exact definition of parallel lines-Two lines are said to be parallel when they are on a plane and never meet. They always remain at the same distance apart.


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