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Lesson 5: Pythagorean theorem and distance between points

# 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 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:
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.

## Want to join the conversation?

• who came up with this formula?
• The distance formula is a consequence of the Pythagorean Theorem.
• For those having trouble, it helped me to imagine a scenario that this could be used in. WAR!

Imagine this, you are the admiral of a ship in WW2. You have sighting reports and coordinates of an enemy fleet. You have been tasked to intercept this force and destroy it. One thing that must be done before any action is made is finding the distance between you and the enemy. This can be represented by the "Distance" (Hypotenuse). In order to calculate that distance you must use a coordinate graph to find that distance. You must use the change in X and the change in Y on the coordinate plane.

First, find out how many spaces over he is to the Left/Right of you (X Value) then you must see how many spaces Up/Down he is from you (Y value)

You can then use the distance formula which is just a variant of the Pythagorean theorem to calculate the distance. It starts out as D^2 = (x2 - x1)^2 this is basically taking the distance between the X value of where you are and the X value of where your enemy is. You square it because that is required for the theorem to work. Then you do the same for the Y value. (y2 - y1)^2. Now that you have found Delta X and Delta Y (Change in X and change in Y) You can put it into the formula and launch your attack. The formula is now

D^2 = DeltaX^2 + DeltaY ^2

Then make sure you find out the Square root to get it to a normal number again and there you go!

You now know the distance and the location of your enemy. You can now launch your attack!

(Hope this wasn't too confusing, I tried to put it in a more practical light)
• i figured it out, i knew srry
• 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
• 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"
• bro why do we have to do this
• 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
• 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
• 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
• I don't get it and I have a test tomorrow it's hard for sixth grade