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## High school geometry

### Course: High school geometry>Unit 6

Lesson 1: Distance and midpoints

# Distance formula

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

## 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 first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two.
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:
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points.
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!
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle.
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:
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle. The hypotenuse of the right triangle is unknown and the side made from the point at x one, y one and x two, y one is labeled x two minus x one.
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:
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle. The hypotenuse of the right triangle is unknown and the side made from the point at x one, y one and x two, y one is labeled x two minus x one. The third side is labeled y two minus y one.
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.

## Want to join the conversation?

• who came up with this formula?
• The distance formula is a consequence of the Pythagorean Theorem.
• 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...
• 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"
• 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
• 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
• 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.
• 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
• 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!