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Basic geometry and measurement
Unit 13: Lesson 4
Pythagorean theorem and distance between pointsDistance 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:
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?(4 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!(3 votes)
I still don't understand any of this... :I(1 vote)
Are you alive?(7 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(13 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?(3 votes)
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)
