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中點公式 (英)

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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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中點公式 (英)

中點公式 (英) : Midpoint Formula