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>> Hi, this Julie Harland
and I'm YourMathGal.
Please visit my website
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
This is part 2
of the Midpoint Formula.
In this video,
we find the midpoint of each
of these pair of points.
All right, in the first video
on the midpoint formula,
we discovered this formula
and so now we're going
to use the formula.
So, we want
to find the midpoint
of the line segment joining
each pair of points
shown below.
So there's 3 different
problems here.
So why don't you try this
on your own first,
put the video on pause
and try it.
Okay, so let's just use
the formula.
So for this first set
of points,
we know the midpoint is simply
the average of the X's
and the average of the Y's.
So you have negative plus 5
over 2, and 3 plus negative 1
over 2.
Remember, you're just taking
the average.
So, what does that give me?
That gives me 0 over 2,
and 2 over 2, right,
which is 0, 1.
So that is the midpoint.
Now, if you draw a picture
that should look correct.
So, if you want to get
out a piece of graph paper
and graph negative 5, 3.
Plot that point
and then plot the 0.5,
negative 1,
and then plot the 0, 1.
It should look like,
like it's directly
between those two points.
And that's a good exercise
to do to sort
of convince yourself your
answer makes sense.
All right,
let's go on to the next one.
All right, so we're going
to begin by taking the average
of the X's.
So, the midpoint will be--
we take the two X values,
negative 4 plus 5 and divided
by 2 and then you take the two
Y values here and add them up
and divided by 2
to take their average.
[ Pause ]
So, that will give you
negative 4 plus 5 is 1,
so that gives you 1/2.
3 plus negative 8 is negative
5 so that gives you
negative 5/2.
And you can leave your answer
like that.
If you want
to write negative 5/2
as negative 2 1/2,
that's fine-- sorry.
[ Pause ]
Same answer.
All right,
let's look at the next one.
So, again we're going
to take the average
of the two X coordinates,
so that will be A plus 0
over 2 and the average
of the B coordinate
which is B plus B over 2.
So that gives us A over 2
which is just really half
of A right?
And we have 2B over 2
[ Pause ]
which is A over 2.
Now, the 2 is canceled, right?
B. Now, one thing you might
notice here is the B values
right here, they're the same
and that makes sense.
That, if the Y value is the
same for both
of these ordered pairs,
you're going
to have the same Y value
for the midpoint.
All right, let's say we wanted
to find the midpoint
of a line segment ending
at those points.
So, let's just look
at the picture.
[ Pause ]
All right,
so here's 4, 2, right?
Now, we've got 4, negative 4.
[ Pause ]
Does it make sense to you
if you're going
to find the midpoint?
So here's--
you know, that line
or actually just like a piece
of line segment
but the X coordinate is not
going to change at all,
it's going to be right
on that vertical line
still anyway.
And when does this midpoint
if you look at it,
it looks like it's 4,
negative 1.
So, notice how the X
coordinates stays the same
because both
of those were the same.
But if we use the formula,
we'll of course get exactly
the same thing.
4 plus 4 over 2,
that's the average
of the X's then we take the
average of the Y coordinates
which is 8 over 2
and negative 2 over 2
which is 4, negative 1.
So, hopefully
that makes sense.
All right, let's do a problem
where we've got
to work a little bit
with fractions.
All right, find the midpoint
of the line segment joining
the point 1 1/2, negative 2/3
and 3, 4, okay?
So, we've got two ordered
pairs here,
try this on your own first.
Okay, now we know you could
just write the average
of the X's, 3 plus 1 1/2
over 2 and 4 plus negative 2/3
over 2.
But you do have to deal
with your complex fractions
here so you got to be careful,
all right?
So, if we simplify this,
we are going
to get the answer.
We'll have 4 1/2 divided by 2
and we have to remember how
to do 4 plus negative 2/3.
So, you could get a common
denominator there,
4 is the same thing
as 12 divided by 3,
12/3 minus 2/3 over 2.
And then how do I do this,
4 1/2 divided by 2?
Well, there's different ways
you could do this.
But, in general you could see
it's going
to start getting messy
with the fractions.
So, I'm going
to give you another possible
way to do this exact
same problem.
Since you see you're going
to be, you know,
having these messy fractions,
really you just want the
average of the X's
and the average of the Y's.
So, instead of writing,
you know, it is all one
ordered pair just figure
out what the X coordinate is,
first.
And that's going
to be the average of the X's
but instead
of writing adding the two
numbers divided by 2,
it's really just half
of 3 plus 1 1/2, okay?
Hope you believe me
that 3 plus 1 1/2 divided
by 2 is the same thing
as 3 plus 1 1/2 times 1/2.
Okay. So then,
we have 1/2 times--
all right, that's 4 1/2,
and then multiplying
with fractions here,
we get a--
just right 4 1/2 is an
improper fraction, 4--
2 times 4 plus 1, that's going
to be 9/2.
So the X coordinate is going
to be 9/4.
I think it's a little bit
easier doing this instead
of trying to do with all
in one ordered pair
because of all the
arithmetic involved.
And the same thing
for the Y value, you're going
to add the Y coordinates
and then take 1/2
of it, right?
Dividing by 2 is the same
thing as taking 1/2 of this,
so we're taking 1/2 of 4 plus,
negative 2/3.
Now in this case,
there's a couple ways you can
do this problem.
You can simplify
in the parenthesis first,
or you could do the
distributive property.
You know, sometimes
when people look
at the problems they decided
to do it one way
over the other.
I'm just kind of do it
by adding in parenthesis first
but I'll show you how you
could also use the
distributive property
in this case.
So I've got
to get a common
denominator here.
So 4 plus negative 2/3,
that's going
to be 12/3 plus negative 2/3,
it's just 1/2 of 10/3.
And I think I might be running
off or my-- you could see it?
And so, what would that be?
Well, if you cancel the 2
and the 10.
So I've got 5/3.
So therefore,
our midpoint should be--
all right,
the X coordinate was 9/4,
the Y coordinate is 5/3
or you could write that,
the each of this is mix
numbers that's 2 1/4, 1 2/3.
So what I was saying
for this second one,
for the Y.
When you have Y equals 1/2
of 4 plus negative 2/3.
I was kind
of just looking ahead
and notice things
that if multiply it
out I get the 2's to cancel.
So 1/2 times 4,
plus 1/2 times negative 2/3
that will give you--
well the 2 cancels
under the 4, 1/2 of 4 is 2,
all right.
And now the 2 is
canceled here.
So you're going
to get 2 plus negative 1/3
and that also is the same
thing as 1 and 2/3.
I'm not showing all the steps
there, but 2 plus negative 1/3
is also 1 and 2/3 or it's 5/3
if you get a
common denominator.
So I'm just saying how you,
you know, do your arithmetic
that's up to you as long
as your arithmetic is going
to make sense.
So if we go back
to our original problem this
is fine.
Starting out like this,
it's just gets a little bit
messy with your arithmetic
right here.
But at this point you might
say, well, 4 1/2 divided
by 2 what's that mean?
That means 4 1/2 is 9/2
and divided
by 2 means times the
reciprocal or 1/2.
Over here I'm going
to get 10/3 in the numerator
and dividing
by 2 means times 1/2.
And you would still get the
same answer,
you would still 9/4 here.
And when you cancel,
you would still get 5/3 here.
So no matter how you look
at it, you're going
to get 9/4, 5/3.
I'm just sort
of working the arithmetic,
check out in different ways
for you.
Oops, that would be 2 and 1/4.
And 1 and 2/3.
I hope I did that correct
on the next page.
Yes. 9/4, 5/3 or 2 1/4,
1 and 2/3.
[ Pause ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos,
which are organized by topic.