<< Chapter < Page Chapter >> Page >
This module covers mathematical distance in preparation for later modules on conic sections.

The key mathematical formula for discussing all the shapes above is the distance between two points.

Many students are taught, at some point, the “distance formula” as a magic (and very strange-looking) rule. In fact, the distance formula comes directly from a bit of intuition...and the Pythagorean Theorem.

The intuition comes in finding the distance between two points that have one coordinate in common.

The distance between two points that have one coordinate in common

The drawing shows the points (2,3) and (6,3). Finding the distance between these points is easy: just count! Take your pen and move it along the paper, starting at (2,3) and moving to the right. Let’s see…one unit gets you over to (3,3); the next unit gets you to (4,3)...a couple more units to (6,3). The distance from (2,3) to (6,3) is 4.

Of course, it would be tedious to count our way from (2,3) to (100,3). But we don’t have to—in fact, you may have already guessed the faster way—we subtract the x coordinates.

  • The distance from (2,3) to (6,3) is 6 - 2 = 4
  • The distance from (2,3) to (100,3) is 100 - 2 = 98

And so on. We can write this generalization in words:

Whenever two points lie on a horizontal line, you can find the distance between them by subtracting their x -coordinates.

This may seem pretty obvious in the examples given above. It’s a little less obvious, but still true, if one of the x coordinates is negative.

The drawing above shows the numbers (-3,1) and (2,1). You can see that the distance between them is 5 (again, by counting). Does our generalization still work? Yes it does, because subtracting a negative number is the same as adding a positive one.

The distance from (-3,1) to (2,1) is 2 - ( -3 ) = 5

How can we express this generalization mathematically? If two points lie on a horizontal line, they have two different x-coordinates: call them x 1 and x 2 . But they have the same y-coordinate, so just call that y. So we can rewrite our generalization like this: “the distance between the points ( x 1 , y ) and ( x 2 , y ) is x 2 x 1 .” In our most recent example, x 1 = –3 , x 2 = 2 , and y = 1 . So the generalization says “the distance between the points (-3,1) and (2,1) is 2 - ( -3 ) ”, or 5.

But there’s one problem left: what if we had chosen x 2 and x 1 the other way? Then the generalization would say “the distance between the points (2,1) and (-3,1) is ( –3 ) -2 ”, or -5. That isn’t quite right: distances can never be negative. We get around this problem by taking the absolute value of the answer. This guarantees that, no matter what order the points are listed in, the distance will come out positive. So now we are ready for the correct mathematical generalization:

Distance between two points on a horizontal line

The distance between the points ( x 1 , y ) and ( x 2 , y ) is | x 2 x 1 |

You may want to check this generalization with a few specific examples—try both negative and positive values of x 1 and x 2 . Then, to really test your understanding, write and test a similar generalization for two points that lie on a vertical line together. Both of these results will be needed for the more general case below.

The distance between two points that have no coordinate in common

So, what if two points have both coordinates different? As an example, consider the distance from (–2,5) to (1,3).

The drawing shows these two points. The (diagonal) line between them has been labeled d : it is this line that we want the length of, since this line represents the distance between our two points.

The drawing also introduces a third point into the picture, the point (–2,3). The three points define the vertices of a right triangle. Based on our earlier discussion, you can see that the vertical line in this triangle is length | 5 3 | = 2 . The horizontal line is length | 1 ( –2 ) | = 3 .

But it is the diagonal line that we want. And we can find that by using the Pythagorean Theorem, which tells us that d 2 = 2 2 + 3 2 . So d = 13

If you repeat this process with the generic points ( x 1 , y 1 ) and ( x 2 , y 2 ) you arrive at the distance formula:

Distance between any two points

If d is the distance between the points ( x 1 , y 1 ) and ( x 2 , y 1 ), then d 2 = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2

x 2 x 1 is the horizontal distance, based on our earlier calculation. y 2 y 1 is the vertical distance, and the entire formula is simply the Pythagorean Theorem restated in terms of coordinates.

And what about those absolute values we had to put in before? They were used to avoid negative distances. Since the distances in the above formulae are being squared , we no longer need the absolute values to insure that all answers will come out positive.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Advanced algebra ii: conceptual explanations' conversation and receive update notifications?