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# 中點公式 (英): Midpoint Formula

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- Let's say I have the point 3 comma negative 4.
- So that would be 1, 2, 3, and then down 4.
- 1, 2, 3, 4.
- So that's 3 comma negative 4.
- And I also had the point 6 comma 1.
- So 1, 2, 3, 4, 5, 6 comma 1.
- So just like that.
- 6 comma 1.
- In the last video, we figured out that we could just use the
- Pythagorean theorem if we wanted to figure out the
- distance between these two points.
- We just drew a triangle there and realized that this was the
- hypotenuse.
- In this video, we're going to try to figure out what is the
- coordinate of the point that is exactly halfway between
- this point and that point?
- So this right here is kind of the distance, the line that
- connects them.
- Now what is the coordinate of the point that is exactly
- halfway in between the two?
- What is this coordinate right here?
- It's something comma something.
- And to do that-- let me draw it really big here.
- Because I think you're going to find out that it's actually
- pretty straightforward.
- At first it seems like a really tough problem.
- Gee, let me use the distance formula with some variables.
- But you're going to see, it's actually one of the simplest
- things you'll learn in algebra and geometry.
- So let's say that this is my triangle right there.
- This right here is the point 6 comma 1.
- This down here is the point 3 comma negative 4.
- And we're looking for the point that is smack dab in
- between those two points.
- What are its coordinates?
- It seems very hard at first. But it's easy when you think
- about it in terms of just the x and the y coordinates.
- What's this guy's x-coordinate going to be?
- This line out here represents x is equal to 6.
- This over here-- let me do it in a little darker color--
- this over here represents x is equal to 6.
- This over here represents x is equal to 3.
- What will this guy's x-coordinate be?
- Well, his x-coordinate is going to be smack dab in
- between the two x-coordinates.
- This is x is equal to 3, this is x is equal to 6.
- He's going to be right in between.
- This distance is going to be equal to that distance.
- His x-coordinate is going to be right in between
- the 3 and the 6.
- So what do we call the number that's right in between
- the 3 and the 6?
- Well we could call that the midpoint, or we could call it
- the mean, or the average, or however you want
- to talk about it.
- We just want to know, what's the average of 3 and 6?
- So to figure out this point, the point halfway between 3
- and 6, you literally just figure out, 3 plus 6 over 2.
- Which is equal to 4.5.
- So this x-coordinate is going to be 4.5.
- Let me draw that on this graph.
- 1, 2, 3, 4.5.
- And you see, it's smack dab in between.
- That's its x-coordinate.
- Now, by the exact same logic, this guy's y-coordinate is
- going to be smack dab between y is equal to negative 4 and y
- is equal to 1.
- So it's going to be right in between those.
- So this is the x right there.
- The y-coordinate is going to be right in between y is equal
- to negative 4 and y is equal to 1.
- So you just take the average.
- 1 plus negative 4 over 2.
- That's equal to negative 3 over 2 or you could say
- negative 1.5.
- So you go down 1.5.
- It is literally right there.
- So just like that.
- You literally take the average of the x's, take the average
- of the y's, or maybe I should say the mean to be a little
- bit more specific.
- A mean of only two points.
- And you will get the midpoint of those two points.
- The point that's equidistant from both of them.
- It's the midpoint of the line that connects them.
- So the coordinates are 4.5 comma negative 1.5.
- Let's do a couple more of these.
- These, actually, you're going to find are very, very
- straightforward.
- But just to visualize it, let me graph it.
- Let's say I have the point 4, negative 5.
- So 1, 2, 3, 4.
- And then go down 5.
- 1, 2, 3, 4, 5.
- So that's 4, negative 5.
- And I have the point 8 comma 2.
- So 1, 2, 3, 4, 5, 6, 7, 8 comma 2.
- 8 comma 2.
- So what is the coordinate of the midpoint
- of these two points?
- The point that is smack dab in between them?
- Well, we just average the x's, average the y's.
- So the midpoint is going to be-- the x values are 8 and 4.
- It's going to be 8 plus 4 over 2.
- And the y value is going to be-- well, we have a 2 and a
- negative 5.
- So you get 2 plus negative 5 over 2.
- And what is this equal to?
- This is 12 over 2, which is 6 comma 2 minus 5 is negative 3.
- Negative 3 over 2 is negative 1.5.
- So that right there is the midpoint.
- You literally just average the x's and average the y's, or
- find their means.
- So let's graph it, just to make sure
- it looks like midpoint.
- 6, negative 5.
- 1, 2, 3, 4, 5, 6.
- Negative 1.5.
- Negative 1, negative 1.5.
- Yep, looks pretty good.
- It looks like it's equidistant from this point and
- that point up there.
- Now that's all you have to remember.
- Average the x, or take the mean of the x, or find the x
- that's right in between the two.
- Average the y's.
- You've got the midpoint.
- What I'm going to show you now is what's in many textbooks.
- They'll write, oh, if I have the point x1 y1, and then I
- have the point-- actually, I'll just stick it in yellow.
- It's kind of painful to switch colors all the time-- and then
- I have the point x2 y2, many books will give you something
- called the midpoint formula.
- Which once again, I think is kind of silly to memorize.
- Just remember, you just average.
- Find the x in between, find the y in between.
- So midpoint formula.
- What they'll really say is the midpoint-- so maybe we'll say
- the midpoint x-- or maybe I'll call it this way.
- I'm just making up notation.
- The x midpoint and the y midpoint is going to be equal
- to-- and they'll give you this formula. x1 plus x2 over 2,
- and then y1 plus y2 over 2.
- And it looks like something you have to memorize.
- But all you have to say is, look.
- That's just the average, or the mean,
- of these two numbers.
- I'm adding the two together, dividing by two, adding these
- two together, dividing by two.
- And then I get the midpoint.
- That's all the midpoint formula is.

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