>> Hi, this is Julie Harland
and I'm YourMathGal.
Please visit my website
at yourmathgal.com
where you could search for any
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by topic.
This is Part 1
of the Midpoint Formula.
So in this video,
the formula is given
and we do a problem
and then we actually show
the proof.
You know, how we find the
formula, and so we prove
that that's the
correct formula.
We're going to go
over the midpoint formula.
The midpoint of T points X1,
Y1 and X2, Y2,
so two separate points.
You know, I have arbitrarily
two points here
which I'll call X1,
Y1 and X2, Y2.
It's simply the ordered pair
right between them.
So to get it,
what you could do is just draw
a straight line segment
between those two ordered
pairs X1, Y1 and X2, Y2.
And it's the point
in the middle
so that this length here is
the same as this length here.
So instead of marking
with like three little
slashes, it means
that length is the same
as that length.
So what we're going to try
to figure out is what is
that ordered pair X, Y?
How would you get it in terms
of the X1s and Y1s
and X2s and Y2s?
So I'm going
to give you the formula
and explain it but then,
I'm going to show you why
we're sure
that really is the
midpoint formula.
Okay, so here's the idea.
The midpoint
which I'll call X, Y, okay?
The midpoint,
here's how you get it,
it's a new ordered pair
such that the new X value,
right, for X is really just
the average
of the two X coordinates.
So the average, remember,
you add and divide it by 2
or you can just add
and multiply by one-half
if you have fractions.
And the same thing,
to get the new Y value,
it's the average of the Y
so it's Y2 plus Y1 over 2.
So to get the new X
coordinate,
you would take whatever
numbers these are for X1
and X2 and take their average
by adding and dividing by 2.
And you get the Y coordinate
of the midpoint, we're going
to add the Y coordinates
and divide it by 2.
So the midpoint kind
of makes sense, right?
So let's look
at an example first
and then we'll go
over the proof of it,
where do we get this formula,
how are we sure this
is correct.
And you do have
to know a little bit
of geometry in order
to understand that.
All right, we're going to go
over the midpoint formula.
You would begin
by giving you the formula
and then we'll understand why
that's true by proving it.
And you don't need
to have a little geometry
to understand that.
So the midpoint,
which I'll call M,
of a line segment
within points X1, Y1, and X2,
Y2, so they're two different
ordered pairs.
For instance, here's your X1,
Y1, here's your X2,
Y2 is this M
which we'll call X, Y, right?
So let's just call this
ordered pair here X,
Y. The midpoint is given by--
well, the new--
the X coordinate is X1 plus X2
over 2.
Now, that's just the average
of the X coordinates of each
of these ordered pairs, right?
The average of two things,
you add and divide it by 2
and the Y coordinate is the
average of the Y values,
Y1 plus Y2 over 2.
So in this picture here,
imagine, you have this line
segment between X1,
Y1 and X2, Y2.
If that's the midpoint,
then we know
that that length here is the
same length as this piece
over here.
All right, so you could use
like one hash mark
or two hash marks
or three hash marks
to show that's the
same length.
I just decided to do
or as many hash marks
as you'd like,
some way to show
that that's exactly the
same length.
And we say those are congruent
to each other,
those two pieces here.
All right, before going ahead
and showing why this is
definitely the midpoint,
right, why this formula works,
let's just do a
simple example.
Find the midpoint
of the line segment.
The midpoints--
[ Pause ]
-- 2, negative 3 and 4, 1.
All right,
so let's just use the formula.
So the midpoint will be the
average of the X values,
so the X values are 2 and 4.
So if we add and divide it
by 2, those would be
that average.
And 1 and negative 3 are the Y
values, take that average.
And so for the X coordinate,
we get 2 plus 4
which is 6 over 2.
That will be 3.
And for the Y coordinate,
we'll have Y plus negative 3
which is a negative 2 over 2
which is negative 1.
Now, let's look at a picture
and see if that makes sense.
So we have this ordered pair
2, negative 3
which is right here.
And we have the ordered pair
4, 1 which is right here.
So we have this line segment,
right?
And it looks
like the midpoint is
about right here
and it does happen to be 3,
negative 1.
So that seems
like the reasonable answer.
Now, you could prove it
to yourself if you want
to simply use the distance
formula and make sure those
two are congruent
to each other.
That's one thing you could do.
All right, so now,
let's see why
that formula really
always works.
So let's look at this diagram.
We have let's say some ordered
pair X1, Y1,
and some other ordered pair
X2, Y2, right?
And we're trying
to find the midpoint.
So let's say this is the
midpoint right in the middle,
and I'm just going to call
that X, Y. So that means,
the distance from here
to here is the same
as the distance from here
to here, so those are two
congruent segments.
All right, next,
you have to know a little bit
about geometry that's called
the transversal
of parallel lines.
So imagine I drew
up here a parallel line,
two parallel lines right here.
So these are parallel, okay?
And this line segment is doing
what's called cutting the
transversal,
it's going
through two parallel lines.
And when that happens,
what we say is
that this angle,
these are corresponding
angles, have to be the same.
Now, if you look
at the picture, they do look
like they're the same, right?
So we've got
that those two angles are
the same.
Now, if we take two other
parallel lines, all right,
so I'm going
to take these parallel
lines now.
Here's a parallel line
to this point right here,
okay?
So we'll say these two green
lines are parallel.
By the same token,
we say these transversal is
cutting through these two
parallel lines,
so this angle is the same
as this angle, okay?
We also made two parallel
lines in the horizontal
direction and then we cut two
parallel lines
in the vertical directions
so we actually made a right
triangle right here--
right here.
Actually, I don't need
to show it here,
it's kind of unnecessary.
Okay, so what we have is a two
congruent triangles.
If you know
about the side angle side,
if you have two triangles
where one angle--
I'm sorry,
this is angle side angle.
You have an angle,
then a congruent side
and another angle that's the
same in some other triangle,
that means all
of the corresponding sides
and angles are the same.
So I can say for instance
that this length is congruent
to this length as well.
And I could say this length
over here is congruent
to this length.
All right, so it looks kind
of confusing
but what we have basically is
we have two
congruent triangles.
All right, so now,
the key is how long is this
little piece, all right?
And how long is this
little piece?
Well, this is the difference
of the X and the X1.
Now, so it looks like X is
to the right of X1.
So this distance right here is
X minus X1.
And the same thing here,
this distance here is the
difference
of the X coordinates
so that's X2 minus X, okay?
So we know those are congruent
to each other so I'm going
to set those equal
to each other.
X minus X1 has
to be the same length
as X2 minus X. And if we solve
that by adding X to both sides
and adding X1 to both sides,
we'll get 2X except that X2,
I'm meant to say add X1,
we will have 2X equals--
all right, this cancel
so you have X2 plus X1.
I'm just going to write
that as X1 plus X2.
It doesn't matter.
And if I divide it by 2,
I get my value of X
which is X1 plus X2 over X.
And that was the X coordinate
of the midpoint up here,
all right?
Now, we do the same thing
for Y, okay?
So this little piece here is
the difference
in the Y values.
So that's F
at Y2 minus Y, right?
And this little piece right
here is the difference
of these Y values, right?
Y minus Y1.
So by the same token, I say--
let's say those are equal
to each other.
So I could write Y minus Y1
equals Y2 minus Y.
And if we do the same
reasoning putting the Ys
on one side and the Y1
on the other side, all right,
add Y to both sides and add Y1
to both sides.
I get 2Y equals Y1 plus Y2
and divide it by 2.
And that tells you the two
coordinates for the midpoint.
So that tells you the midpoint
X, Y is actually X1 plus X2
over 2, Y1 plus Y2 over 2,
and the way I remember this is
it's just the average
of the X coordinates
and the average
of the Y coordinates.
All right,
we'll do some more problems
on the next page.
Actually, understanding why
this is the formula is the
hard part.
But if you know the formula
and can use it,
it's really not a hard formula
and it's easy to use.
But if you want
to understand it,
it's this picture right here
which I know is a little bit
messy and complicated
but it's cool
to really understand
where formulas come from.
[ Pause ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.