<< 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

the art of managing the production, distribution and consumption.
Satangthem Reply
what is economics
Khawar Reply
marginal utility is the additional satisfaction one derives from consuming additional unit of a good or service.
It's the allocation of scarce resources.
marginal utility please?
marginal utility is the additional satisfaction one derives from consuming additional unit of a good or service.
I know the definition, but I don't understand its meaning.
what is the must definition of economic please?
demand lfs
Economics is derived from the word Oikonomia which means management of household things. Thus, Economics is a study of household things with the constrains of allocating scare resources.
what is Open Market Operation
Adu Reply
dominating middlemen men activities circumstances
Christy Reply
what Equilibrium price
Adji Reply
what is gap
who is good with the indifference curve
What is diseconomic
Alixe Reply
what are the types of goods
how can price determination be the central problem of micro economics
simon Reply
marginal cost formula
Nandu Reply
you should differentiate the total cost function in order to get marginal cost function then you can get marginal cost from it
What about total cost
how can price determination be the central problem if micro economics
formula of cross elasticity of demand
Theresia Reply
what is ceteris paribus
Priyanka Reply
what is ceteris parabus
Ceteris paribus - Literally, "other things being equal"; usually used in economics to indicate that all variables except the ones specified are assumed not to change.
What is broker
land is natural resources that is made by nature
What is broker
what is land
What is broker
land is natural resources that is made by nature
whats poppina nigga turn it up for a minute get it
amarsyaheed Reply
what is this?
am from nigeria@ pilo
am from nigeria@ pilo
what is production possibility frontier
it's a summary of opportunity cost depicted on a curve.
please help me solve this question with the aid of appropriate diagrams explain how each of the following changes will affect the market price and quantity of bread 1. A
Manuela Reply
please l need past question about economics
Prosper Reply
ok let me know some of the questions please.
ok am not wit some if den nw buh by tommorow I shall get Dem
Hi guys can I get Adam Smith's WEALTH OF NATIONS fo sale?
hello I'm Babaisa alhaji Mustapha. I'm studying Economics in the university of Maiduguri
my name is faisal Yahaya. i studied economics at Kaduna state university before proceeding to West African union university benin republic for masters
Hi guys..I am from Bangladesh..
Wat d meaning of management
igwe Reply
disaster management cycle
Gogul Reply
cooperate social responsibility
Fedric Wilson Taylor also define management as the act of knowing what to do and seeing that it is done in the best and cheapest way
Difference between extinct and extici spicies
Amanpreet 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?