who came up with this formula?

The distance formula is a consequence of the Pythagorean Theorem.

okay I understand all you have to do is take your y axis and divide it by your x axis

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"

I still don't understand any of this... :I

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...

how is the formula the same as the Pythagorean theorem

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

I don't get it and I have a test tomorrow it's hard for sixth grade

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.

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.

when you square a negative it becomes a positive

How do you find the distance between two points if it is just a line?

To find the distance between to points if it is just a line, you simply draw a dot where the line ends, then you make a number line and find the coordinates for both points. Finally, you follow the distance formula, plug the values in, and solve. I hope this helps and wasn't a bore!

I prefer the straight up: draw a right triangle, use pythagorean theorem method. Anyone agree?

The distance formula is an application of the Pythagorean Theorem, so what are you doing differently that makes it seem different to you? Are you just finding the x2-x1 and y2-y1 from the graph before putting it into the Pythagorean Theorem?

what is the formula that is used to find distance between two points

bro are u crazy its right above you in the beginning of the lesson

Main content

Basic geometry and measurement

Course: Basic geometry and measurement > Unit 13

Lesson 4: Pythagorean theorem and distance between points

Distance formula

CCSS.Math:

8.G.B.8

Google Classroom

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

The start color #11accd, start text, d, i, s, t, a, n, c, e, end text, end color #11accd between the points left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis is given by the following formula:

square root of, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, start color #e07d10, left parenthesis, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared, end square root

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

Deriving the distance formula

Let's start by plotting the points left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis.

The length of the segment between the two points is the start color #11accd, start text, d, i, s, t, a, n, c, e, end text, end color #11accd between them:

We want to find the start color #11accd, start text, d, i, s, t, a, n, c, e, end text, end color #11accd. If we draw a right triangle, we'll be able to use the Pythagorean theorem!

An expression for the length of the base is start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54:

[Why does this expression work?]

Similarly, an expression for the length of the height is start color #e07d10, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10:

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

start color #11accd, question mark, end color #11accd, squared, equals, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, start color #e07d10, left parenthesis, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared

We can solve for start color #11accd, question mark, end color #11accd by taking the square root of each side:

start color #11accd, question mark, end color #11accd, equals, square root of, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, start color #e07d10, left parenthesis, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared, end square root

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
Posted 6 years ago. Direct link to rose's post “who came up with this for...”
who came up with this formula?
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- Ian Pulizzotto
  Posted 6 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...”
  (60 votes)
Ivan Brown
Posted 5 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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(10 votes)
Answer
- Ethan
  Posted 5 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...”
  (55 votes)
malexander8
Posted 6 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 pageButton navigates to signup page
(11 votes)
Answer
- Marisah visaiz
  Posted 6 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...”
  (18 votes)
jasontheking121
Posted 5 years ago. Direct link to jasontheking121's post “how is the formula the sa...”
how is the formula the same as the Pythagorean theorem
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(9 votes)
Answer
- WeiWei4834
  Posted 5 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 ...”
  (20 votes)
Pink_Ivy_Nikes
Posted 6 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 pageComment on Pink_Ivy_Nikes's post “Sooooo, if I have two poi...”
(4 votes)
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- jameseden
  Posted 5 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...”
  (12 votes)
Eleanor Brown
Posted 6 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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(8 votes)
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- V1rtua1F0X
  Posted 2 years ago. Direct link to V1rtua1F0X'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 V1rtua1F0X's post “I most likely responded w...”
  (6 votes)
Maria Lopes
Posted 5 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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- JalenH
  Posted 3 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
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bro why do we have to do this
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Karen Boyce
Posted 4 years ago. Direct link to Karen Boyce's post “How do you find the dista...”
How do you find the distance between two points if it is just a line?
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- V1rtua1F0X
  Posted 2 years ago. Direct link to V1rtua1F0X's post “To find the distance betw...”
  To find the distance between to points if it is just a line, you simply draw a dot where the line ends, then you make a number line and find the coordinates for both points. Finally, you follow the distance formula, plug the values in, and solve.
  
  I hope this helps and wasn't a bore!
  Comment on V1rtua1F0X's post “To find the distance betw...”
  (2 votes)
Durgen
Posted 2 years ago. Direct link to Durgen's post “I prefer the straight up:...”
I prefer the straight up: draw a right triangle, use pythagorean theorem method. Anyone agree?
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- David Severin
  Posted 2 years ago. Direct link to David Severin's post “The distance formula is a...”
  The distance formula is an application of the Pythagorean Theorem, so what are you doing differently that makes it seem different to you? Are you just finding the x2-x1 and y2-y1 from the graph before putting it into the Pythagorean Theorem?
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  (3 votes)