Main content

## Get ready for Precalculus

### Unit 5: Lesson 5

Distance formula# Distance formula

CCSS.Math:

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:

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:

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:

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

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?

- How do you find the distance between two points if it is just a line?(5 votes)
- 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!(2 votes)

- what is the formula that is used to find distance between two points(0 votes)
- bro are u crazy its right above you in the beginning of the lesson(10 votes)

- I still don't understand any of this... :I(1 vote)
- Are you alive?(4 votes)

- I prefer the straight up: draw a right triangle, use pythagorean theorem method. Anyone agree?(4 votes)
- 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?(1 vote)

- who came up with this formula?(1 vote)
- .Nevertheless, the theorem came to be credited to Pythagoras. It is also proposition number 47 from Book I of Euclid's Elements. According to the Syrian historian Iamblichus (c. 250–330 ce), Pythagoras was introduced to mathematics by Thales of Miletus and his pupil Anaximander. Brittanica.com states “ Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2. Although the theorem has long been associated with Greek mathematician-philosopher Pythagoras (c. 570–500/490 BCE), it is actually far older. Four Babylonian tablets from circa 1900–1600 BCE indicate some knowledge of the theorem, with a very accurate calculation of the square root of 2 (the length of the hypotenuse of a right triangle with the length of both legs equal to 1) and lists of special integers known as Pythagorean triples that satisfy it (e.g., 3, 4, and 5; 32 + 42 = 52, 9 + 16 = 25). The theorem is mentioned in the Baudhayana Sulba-sutra of India, which was written between 800 and 400 BCE. Nevertheless, the theorem came to be credited to Pythagoras.” I know that’s a lot of words but I had to do it...even tho this isn’t ELA class 😀😂. Hope this helps rose!(7 votes)

- 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.(1 vote)
- Yes it doesn't really matter, because you end up squaring it and ending up with a positive number.(2 votes)

- what is one of the points are a fraction? like how would that work(2 votes)
- You would still use the distance formula or the Pythagorean theorem, except that you would perform the arithmetic with the fractions. Remember that when you square a fraction, you need to square both top and bottom; when you take the square root of a fraction, you need to take the square root of both top and bottom.

Have a blessed, wonderful day!(2 votes)

- Why is it called the Pythagorean theorem? Can't it be something else like the triangle theory?(0 votes)
- It was made by and named after a person called Pythagoros(5 votes)

- this is hard and can you help me.(2 votes)
- okay I understand all you to do is take your Y axis and divide it by your X axis(2 votes)
- That makes it sound like slope more than the distance formula which the video is about. You take the square root of (the change in y)^2 + (the change in x)^2 or d=√((x2-x1)^2+(y2-y1)^2). There is no dividing at all.(1 vote)