Distance formula | Analytic geometry (article)

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Walk through deriving a general formula for the distance between two points.

The $distance$ ‍ between the points $(x_{1}, y_{1})$ ‍ and $(x_{2}, y_{2})$ ‍ is given by the following formula:

$\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$ ‍

In this article, we're going to derive this formula!

Deriving the distance formula

Let's start by plotting the points $(x_{1}, y_{1})$ ‍ and $(x_{2}, y_{2})$ ‍.

The length of the segment between the two points is the $distance$ ‍ between them:

We want to find the $distance$ ‍. If we draw a right triangle, we'll be able to use the Pythagorean theorem!

An expression for the length of the base is $x_{2} - x_{1}$ ‍:

Most people have to stop and think about why this expression works. For example, think about if $x_{1} = 3$ ‍ and $x_{2} = 7$ ‍. Here is how we would find the length of the base:

$\begin{aligned} x_{2} - x_{1} \\ = 7 - 3 \\ = 4 \end{aligned}$ ‍

This makes sense because the distance from $3$ ‍ to $7$ ‍ is $4$ ‍.

Similarly, an expression for the length of the height is $y_{2} - y_{1}$ ‍:

Now we can use the Pythagorean theorem to write an equation:

$?^{2} = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ ‍

We can solve for $?$ ‍ by taking the square root of each side:

$? = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$ ‍

That's it! We derived the distance formula!

Interestingly, a lot of people don't actually memorize this formula. Instead, they set up a right triangle, and use the Pythagorean theorem whenever they want to find the distance between two points.

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rose
7 years agoPosted 7 years ago. Direct link to rose's post “who came up with this for...”
who came up with this formula?
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(47 votes)
- Ian Pulizzotto
  See Also
  11.1 Distance and Midpoint Formulas; Circles - Intermediate Algebra | OpenStax 11.2: Distance and Midpoint Formulas and Circles Midpoint formula | Analytic geometry (article) | Khan Academy Section 1.1 The Distance and Midpoint Formulas
  7 years agoPosted 7 years ago. Direct link to Ian Pulizzotto's post “The distance formula is a...”
  The distance formula is a consequence of the Pythagorean Theorem.
  Comment on Ian Pulizzotto's post “The distance formula is a...”
  (104 votes)
Ivan Brown
7 years agoPosted 7 years ago. Direct link to Ivan Brown's post “I still don't understand ...”
I still don't understand any of this... :I
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(21 votes)
- Ethan
  7 years agoPosted 7 years ago. Direct link to Ethan's post “I haven't read any of the...”
  I haven't read any of the article on this so I really hope I don't say the exact same thing he says.... here goes:
  Here is the graph I am referring to in my explanation: https://www.desmos.com/calculator/juthaysfbl
  -- only look at the graph, ignore everything on the sides and bottom --
  (intuitive solution, and how I learned this)
  Think of the Pythagorean theorem. The formula is a^2 + b^2 = c^2 . Now, imagine two points, let's say they are (0,0) and (3,4) to keep it simple. Look at the blue line going from (0,0) to (3,0). This is the base, with a distance of 3 units. How did we find this? We took one of the x values (3) and subtracted it by the other (0). 3 - 0 = 0. Next, we must find the height. The red line represents it, and it is a length of 4 units. We found this, again, by subtracting the y values (4 - 0 = 0). We can now find the hypotenuse, if we replace a and b with the base height length, so we get 3^2 + 4^2 = c^2 (where c is the orange line, or hypotenuse). The hypotenuse is the distance of the two points.
  Of course, we can square root both sides so we get c = sqrt( 3^2 + 4^2). We can expand this even further if we replace the 3 and 4 with how we got there, so c = sqrt( (3 - 0)^2 + (4 - 0)^2). But what do 3 and 0 and 4 and 0 mean? The two x values and y values, respectively. Therefore, we replace the numbers so we get c (hypotenuse) = ( ( x1 - x2) ^2 + (y1 - y2) ^2) .
  I really hope this helped you, I spent a long time explaining this lmao...
  Comment on Ethan's post “I haven't read any of the...”
  (77 votes)
bpjblr
a year agoPosted a year ago. Direct link to bpjblr's post “POV: your teacher makes y...”
POV: your teacher makes you show your work
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(38 votes)
- Hailey Bodhi
  7 months agoPosted 7 months ago. Direct link to Hailey Bodhi's post “The worst feeling in the ...”
  The worst feeling in the world is when you lose a point because you "Didn't show your work" like. Ow
  Comment on Hailey Bodhi's post “The worst feeling in the ...”
  (21 votes)
ur mom
a year agoPosted a year ago. Direct link to ur mom's post “bro why do we have to do ...”
bro why do we have to do this
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(19 votes)
- Jackandmoo
  a year agoPosted a year ago. Direct link to Jackandmoo's post “I cannot say why you have...”
  I cannot say why you have to, but I can say why this might be useful in the future.
  Lets say for whatever reason you needed to make a video game, and let’s also say you needed to calculate the distance between a player and an object in order to make an action occur. I imagine this would be useful for those purposes
  Comment on Jackandmoo's post “I cannot say why you have...”
  (30 votes)
  See Also
  8.2: Distance and Midpoint Formulas and Circles
malexander8
7 years agoPosted 7 years ago. Direct link to malexander8's post “okay I understand all you...”
okay I understand all you have to do is take your y axis and divide it by your x axis
Button navigates to signup page•Button navigates to signup page
(13 votes)
- Marisah visaiz
  7 years agoPosted 7 years ago. Direct link to Marisah visaiz's post “If you were to get two pe...”
  If you were to get two perfect squares under the giant square root after subtracting the two points within each parentheses, would you be able to separate them in order to pull them out of the square root and make them rational?
  For example, if I got "the square root of (6)^squared + (6)^squared" would I first square them and get "the square root of 36+36?" or could I separate them into "the square root of 36 + the square root of 36"
  Comment on Marisah visaiz's post “If you were to get two pe...”
  (27 votes)
jasontheking121
6 years agoPosted 6 years ago. Direct link to jasontheking121's post “how is the formula the sa...”
how is the formula the same as the Pythagorean theorem
Button navigates to signup page•Comment on jasontheking121's post “how is the formula the sa...”
(12 votes)
- WeiWei4834
  6 years agoPosted 6 years ago. Direct link to WeiWei4834's post “The x and the y axis are ...”
  The x and the y axis are perpendicular, so if you imagine a right triangle when you find a distance, and the hypotenuse is the distance
  Comment on WeiWei4834's post “The x and the y axis are ...”
  (26 votes)
Pink_Ivy_Nikes
7 years agoPosted 7 years ago. Direct link to Pink_Ivy_Nikes's post “Sooooo, if I have two poi...”
Sooooo, if I have two points, (1, 2) and (-1, 4), it does not matter in which order I subtract as long as I do the x with the x, and so on? Because it doesn't look that way.
Button navigates to signup page•Comment on Pink_Ivy_Nikes's post “Sooooo, if I have two poi...”
(7 votes)
- jameseden
  7 years agoPosted 7 years ago. Direct link to jameseden's post “when you square a negativ...”
  when you square a negative it becomes a positive
  Comment on jameseden's post “when you square a negativ...”
  (22 votes)
Eleanor Brown
7 years agoPosted 7 years ago. Direct link to Eleanor Brown's post “I don't get it and I have...”
I don't get it and I have a test tomorrow it's hard for sixth grade
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(13 votes)
- spec.
  3 years agoPosted 3 years ago. Direct link to spec.'s post “I most likely responded w...”
  I most likely responded wayyyy to late for this, but I thoroughly recommend you go through both the article and video ( maybe the practices!) and study hard.
  Comment on spec.'s post “I most likely responded w...”
  (11 votes)
Maria Lopes
6 years agoPosted 6 years ago. Direct link to Maria Lopes's post “what is the formula that ...”
what is the formula that is used to find distance between two points
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(0 votes)
- JalenH
  4 years agoPosted 4 years ago. Direct link to JalenH's post “bro are u crazy its right...”
  bro are u crazy its right above you in the beginning of the lesson
  Comment on JalenH's post “bro are u crazy its right...”
  (34 votes)
Jermaine jones
a year agoPosted a year ago. Direct link to Jermaine jones's post “pov:i still dont understa...”
pov:i still dont understand but agree with everything ;-;
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(13 votes)

Distance formula | Analytic geometry (article) | Khan Academy (2024)

Deriving the distance formula

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