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- In this video, we're going to learn how to take the distance
- between any two points in our x, y coordinate plane, and
- we're going to see, it's really just an application of
- the Pythagorean theorem.
- So let's start with an example.
- Let's say I have the point, I'll do it in a darker color
- so we can see it on the graph paper.
- Let's say I have the point 3 comma negative 4.
- So if I were to graph it, I'd go 1, 2, 3, and
- then I'd go down 4.
- 1, 2, 3, 4, right there, is 3 comma negative 4.
- And let's say I also have the point 6 comma 0.
- So 1, 2, 3, 4, 5, 6, and then there's no movement in the
- y-direction.
- We're just sitting on the x-axis.
- The y-coordinate is 0, so that's 6 comma 0.
- And what I want to figure out is the distance between these
- two points.
- How far is this blue point away from this orange point?
- And at first, you're like, gee, Sal, I don't think I've
- ever seen anything about how to solve for a
- distance like this.
- And what are you even talking about the Pythagorean theorem?
- I don't see a triangle there!
- And if you don't see a triangle, let
- me draw it for you.
- Let me draw this triangle right there, just like that.
- Let me actually do several colors here, just to really
- hit the point home.
- So there is our triangle.
- And you might immediately recognize
- this is a right triangle.
- This is a right angle right there.
- The base goes straight left to right, the right side goes
- straight up and down, so we're dealing with a right triangle.
- So if we could just figure out what the base length is and
- what this height is, we could use the Pythagorean theorem to
- figure out this long side, the side that is opposite the
- right angle, the hypotenuse.
- This right here, the distance is the hypotenuse of this
- right triangle.
- Let me write that down.
- The distance is equal to the hypotenuse
- of this right triangle.
- So let me draw it a little bit bigger.
- So this is the hypotenuse right there.
- And then we have the side on the right, the side that goes
- straight up and down.
- And then we have our base.
- Now, how do we figure out-- let's
- call this d for distance.
- That's the length of our hypotenuse.
- How do we figure out the lengths of this up and down
- side and the base side right here?
- So let's look at the base first. What is this distance?
- You could even count it on this graph paper, but here,
- where x is equal to-- let me do it in the green.
- Here, we're at x is equal to 3 and here we're at x is equal
- to 6, right?
- We're just moving straight right.
- This is the same distance as that distance right there.
- So to figure out that distance, it's literally the
- end x point.
- And you could actually go either way, because you're
- going to square everything, so it doesn't matter if you get
- negative numbers, so the distance here is going to be 6
- minus 3, right?
- 6 minus 3.
- That's this distance right here, which is equal to 3.
- So we figured out the base.
- And to just remind ourselves, that is equal to
- the change in x.
- That was equal to your finishing x minus your
- starting x.
- 6 minus 3.
- This is our delta x.
- Now, by the same exact line of reasoning, this height right
- here is going to be your change in y.
- Up here, you're at y is equal to 0.
- That's kind of where you finish.
- That's your higher y point.
- And over here, you're at y is equal to negative 4.
- So change in y is equal to 0 minus negative 4.
- I'm just taking the larger y-value minus the smaller
- y-value, the larger x-value minus the smaller x-value.
- But you're going to see we're going to square it in a
- second, so even if you did it the other way around, you'd
- get a negative number, but you'd still get the same
- answer, so this is equal to 4.
- So this side is equal to 4.
- You can even count it on the graph paper if you like.
- And this side is equal to 3.
- And now we can do the Pythagorean theorem.
- This distance is the distance squared.
- Be careful.
- The distance squared is going to be equal to this delta x
- squared, the change in x squared plus
- the change in y squared.
- This is nothing fancy.
- Sometimes people will call this the distance formula.
- It's just the Pythagorean theorem.
- This side squared plus that side squared is equal to
- hypotenuse squared, because this is a right triangle.
- So let's apply it with these numbers, the numbers that we
- have at hand.
- So the distance squared is going to be equal to delta x
- squared is 3 squared plus delta y squared plus 4
- squared, which is equal to 9 plus 16, which is equal to 25.
- So the distance is equal to-- let me write that-- d squared
- is equal to 25.
- d, our distance, is equal to-- you don't want to take the
- negative square root, because you can't have a negative
- distance, So it's only the principal root, the positive
- square root of 25, which is equal to 5.
- So this distance right here is 5.
- Or if we look at this distance right here, that was the
- original problem.
- How far is this point from that point?
- It is 5 units away.
- So what you'll see here, they call it the distance formula,
- but it's just the Pythagorean theorem.
- And just so you're exposed to all of the ways that you'll
- see the distance formula, sometimes people will say, oh,
- if I have two points, if I have one point, let's call it
- x1 and y1, so that's just a particular point.
- And let's say I have another point that is x2 comma y2.
- Sometimes, you'll see this formula, that the distance--
- you'll see it in different ways.
- But you'll see that the distance is equal to-- and it
- looks as though there's this really complicated formula,
- but I want you to see that this is really just the
- Pythagorean theorem.
- You see that the distance is equal to x2 minus x1 minus x1
- squared plus y2 minus y1 squared.
- You'll see this written in a lot of textbooks as the
- distance formula.
- And it's a complete waste of your time to memorize it
- because it's really just the Pythagorean theorem.
- This is your change in x.
- And it really doesn't matter which x you pick to be first
- or second, because even if you get the negative of this
- value, when you square it, the negative disappears.
- This right here is your change in y.
- So it's just saying that the distance squared-- remember,
- if you square both sides of this equation, the radical
- will disappear and this will be the distance squared is
- equal to this expression squared, delta x squared,
- change in x-- delta means change in-- delta x squared
- plus delta y squared.
- I don't want to confuse you.
- Delta y just means change in y.
- I should have probably said that earlier in the video.
- But let's apply it to a couple more, and I'll just pick some
- points at random.
- Let's say I have the point, let's see, 1, 2, 3, 4, 5, 6.
- Negative 6 comma negative 4.
- And let's say I want to find the distance between that and
- 1 comma 1, 2, 3, 4, 5, 6, 7, and the point 1 comma 7, so I
- want to find this distance right here.
- So it's the exact same idea.
- We just use the Pythagorean theorem.
- You figure out this distance, which is our change in x, this
- distance, which is our change in y.
- This distance squared plus this distance squared is going
- to equal that distance squared.
- So let's do it.
- So our change in x, you just take-- you
- know, it doesn't matter.
- In general, you want to take the larger x-value minus the
- smaller x-value, but you could do it either way.
- So we could write the distance squared is equal to-- what's
- our change in x?
- So let's take the larger x minus the smaller x, 1 minus
- negative 6.
- 1 minus negative 6 squared plus the change in y.
- The larger y is here.
- It's 7.
- 7 minus negative 4.
- 7 minus negative 4 squared.
- And I just picked these numbers at random, so they're
- probably not going to come out too cleanly.
- So we get that the distance squared is equal to 1 minus
- negative 6.
- That is 7, 7 squared, and you'll even see it over here,
- if you count it.
- You go, 1, 2, 3, 4, 5, 6, 7.
- That's that number right here.
- That's what your change in x is.
- Plus 7 minus negative 4.
- That's 11.
- That's this distance right here, and you can count it on
- the blocks.
- We're going up 11.
- We're just taking 7 minus negative 4 to get
- a distance of 11.
- So plus 11 squared is equal to d squared.
- So let me just take the calculator out.
- So the distance if we take 7 squared plus 11 squared is
- equal to 170, that distance is going to be the square root of
- that, right? d squared is equal to 170.
- So let's take the square root of 170 and we get 13.0,
- roughly 13.04.
- So this distance right here that we tried to
- figure out is 13.04.
- Hopefully, you found that helpful.