Example calcuating magnitude of vector from graph

WEBVTT
Kind: captions
Language: en

00:00:00.055 --> 00:00:02.341
- [Voiceover] We've already
seen that a vector is defined

00:00:02.341 --> 00:00:04.780
by both its magnitude and its direction.

00:00:04.780 --> 00:00:07.381
What I want to do in this
video is get some practice

00:00:07.381 --> 00:00:10.715
calculating or figuring out
the magnitudes of vectors

00:00:10.715 --> 00:00:12.341
and I have a vector right over here,

00:00:12.341 --> 00:00:14.640
vector u, it is, it is,

00:00:14.640 --> 00:00:17.974
visually depicted here on our corded plane

00:00:17.974 --> 00:00:19.193
and I wanna figure out its magnitude

00:00:19.193 --> 00:00:20.611
and I encourage you to pause the video

00:00:20.611 --> 00:00:24.322
and see if you can figure it out.

00:00:24.322 --> 00:00:25.802
Well the magnitude of a vector,

00:00:25.802 --> 00:00:27.357
just the length of this line

00:00:27.357 --> 00:00:30.400
you could view it as the distance
between the initial point

00:00:30.400 --> 00:00:33.174
and the terminal point right over there

00:00:33.174 --> 00:00:34.881
and so one way to think about it,

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you could view this,

00:00:36.460 --> 00:00:37.737
the magnitude of the vector,

00:00:37.737 --> 00:00:38.700
let me write this.

00:00:38.700 --> 00:00:41.781
The magnitude of the vector.

00:00:41.781 --> 00:00:44.080
Well you could either think
of it as the distance formula,

00:00:44.080 --> 00:00:46.095
which is really, just
comes straight out of the

00:00:46.095 --> 00:00:47.029
Pythagorean theorem.

00:00:47.029 --> 00:00:49.745
It's going to be the square root of our

00:00:49.745 --> 00:00:53.060
change in x squared

00:00:53.205 --> 00:00:56.741
change in x squared

00:00:56.886 --> 00:00:59.092
plus change in y squared

00:00:59.092 --> 00:01:01.994
plus change in y squared.

00:01:01.994 --> 00:01:03.236
Once again this triangle,

00:01:03.236 --> 00:01:05.233
Greek letter delta is just shorthand

00:01:05.233 --> 00:01:06.812
for change in y

00:01:06.812 --> 00:01:08.333
and this once again
comes straight out of the

00:01:08.333 --> 00:01:10.293
distance formula which really comes out of

00:01:10.293 --> 00:01:12.025
the Pythagorean theorem.

00:01:12.025 --> 00:01:13.753
It'll become a little bit
more obvious when I draw

00:01:13.753 --> 00:01:15.773
the change in y and I draw the change in x

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on this diagram.

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So what's our change in y?

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Well we're starting at our initial point.

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We're starting at y is equal to nine

00:01:23.888 --> 00:01:26.663
and we are, to get to the y value

00:01:26.663 --> 00:01:28.149
of our terminal point,

00:01:28.149 --> 00:01:32.358
we're going down to y is equal to two.

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So we have a change in y

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our change in y is equal to,

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going from nine to two,

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our change in y is negative seven.

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Similarly, our change in x

00:01:44.973 --> 00:01:47.877
we're going from x is equal to two,

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to x is equal to five.

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So our change in the horizontal
direction is plus three.

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So our change in x

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where we can either think
of this as the horizontal

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component of the vector.

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This is equal to positive three

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and as we've just seen

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we have drawn a right triangle

00:02:05.605 --> 00:02:07.357
and so we can use a Pythagorean theorem

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to figure out, to figure out the length

00:02:09.487 --> 00:02:10.810
of the hypotenuse and
you might say wait, wait,

00:02:10.810 --> 00:02:13.831
a length of a side of a triangle
can't have a negative value

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and that's why these squareds are valuable

00:02:15.988 --> 00:02:18.356
because it doesn't matter
if you're taking a negative

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seven squared or a positive seven squared,

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you're going to get a positive value here

00:02:21.967 --> 00:02:24.057
and if you really just
view this as a triangle,

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all you care about is
the length of this side

00:02:28.126 --> 00:02:30.348
right over here or it's
the magnitude of this side

00:02:30.348 --> 00:02:32.554
or the absolute value of it
which is just going to be

00:02:32.554 --> 00:02:34.353
positive seven

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and so we can say this
is going to be equal to

00:02:36.885 --> 00:02:41.447
the magnitude of our vector
is going to be equal to

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so three squared is nine,

00:02:45.452 --> 00:02:48.199
nine, and then

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negative seven squared is positive 49.

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So plus 49

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or once again you could view this as our

00:02:56.226 --> 00:02:57.225
change in y squared

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which is negative seven squared

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or you could say,

00:03:00.325 --> 00:03:01.034
well just look,

00:03:01.034 --> 00:03:03.193
the absolute value,
the length of the side,

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we don't wanna think of a side as having

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a negative value, the
negative really just says,

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hey we're going from the top to the bottom

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it gives us our direction,

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but if we just say the length of it

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it's seven, well you're there

00:03:12.388 --> 00:03:13.863
if you just use a Pythagorean theorem.

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Seven squared would also be 49

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and so either way you get

00:03:18.402 --> 00:03:20.631
the magnitude of our vector is equal to

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the square root of nine plus 49

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is going to be

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57, I don't think I can
simplify this radical

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too much, no that's it.

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So the magnitude of this vector

00:03:34.288 --> 00:03:37.083
is the square root of 57.


Equivalent vectors examples

WEBVTT
Kind: captions
Language: en

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- [Voiceover] I have some
example problems here from

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our equivalent vectors
exercise on Khan Academy,

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so let's go through these
and like always, pause

00:00:08.062 --> 00:00:10.578
the video and see if you can
work through them on your own.

00:00:11.578 --> 00:00:16.328
So this first one says, "Are
vectors u and w equivalent?"

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And so we can see vector u here in blue

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and vector w right over here.

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And we have to remember a
vector is defined by both

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having a magnitude and a direction.

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And so for two vectors to be
equivalent they have to have

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the same magnitude and the same direction.

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So when I look at these
two, they are clearly

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pointing in different directions.

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Vector u is pointing to the bottom right,

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that's the direction it's pointing in,

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and vector w is pointing
to the bottom left,

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so they definitely aren't equivalent.

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So they are not equivalent,
scratch that out.

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So they have different
directions, different directions.

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Different directions.

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And also think about the magnitude.

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If I just look at the
length of the arrows,

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just eyeballing it,
they look pretty close.

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Let me verify that.

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So if I'm starting at the
initial point for vector u,

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how much do I move in the x direction?

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Well in the x direction
I go from negative eight

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to negative three, so I
could say my change in x

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is positive five, my x increases by five

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as I go from the initial point,
from the x coordinate of the

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initial point to the x coordinate of the

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terminal point, so that length.

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The magnitude of just
the x component is five.

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And let's see what happens
in the y direction.

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So in the y direction, I start
at y equals negative two,

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right over here, and then I go down

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to y equals negative eight.

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So my change in y, change in y,

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is equal to negative six, negative six.

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Also think about this one over here.

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What's my change in x?

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What's my change in x?

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Well I'm starting at x equals eight,

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and I am going to x equals three.

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So my change in x is negative five.

00:02:27.377 --> 00:02:31.719
I'm gonna write that,
change in x is equal to,

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this is my change in x, change in x.

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And then what's my change in y?

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Well I start at y is equal to eight

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and I go down y is equal to two.

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Do it just like that.

00:02:46.738 --> 00:02:51.683
So my change in y is
equal to negative six.

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Now based on the changes in x's and y's,

00:02:56.350 --> 00:02:59.555
I can figure out the magnitude
of each of these vectors.

00:03:02.785 --> 00:03:07.064
The magnitude of u, so I
could write it like this, the

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magnitude of u, sometimes you'll
see the notation like this,

00:03:10.895 --> 00:03:15.895
the magnitude of u, of
vector u I should say,

00:03:15.956 --> 00:03:18.371
is going to be equal to, we're just gonna

00:03:18.371 --> 00:03:19.695
use the pythagorean theorem here.

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It's going to be the square root.

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The length of this hour is just a

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hypotenuse of the right triangle.

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So it's going to be the
square root of five squared

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plus negative six squared so
it's gonna be the square root

00:03:31.676 --> 00:03:36.676
of 25 plus 36, which is
equal to the square root of,

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what is that?

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It's going to be square root of 61.

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I can make that radical
a little bit smaller,

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square root of 61.

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Now what about vector w?

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So the magnitude of vector w is,

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I could use a double, those double bars.

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The magnitude of vector
w, well that's going to be

00:03:59.860 --> 00:04:03.784
the square root of negative
five squared, which is 25

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plus negative six squared,
which is going to be 36.

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Well that's going to be the
square root of 61 as well.

00:04:10.355 --> 00:04:12.933
So they have the same magnitude,
just different directions.

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So let's see, no they have the

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same magnitude, but different directions.

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Yup, that's the choice we like.

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Let's do one more of these.

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Are vectors u and w equivalent?

00:04:23.586 --> 00:04:27.858
Alright, so these look pretty
equivalent just eyeballing it.

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It looks like they're pointed
in the same direction,

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they're going from the
bottom left to the top right.

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And it looks like they're the same length,

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but we can verify them.

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Once again by looking at
their x and y components.

00:04:39.538 --> 00:04:42.928
So in the x direction, or
thinking about how much we change

00:04:42.928 --> 00:04:44.785
in x and change in y when we go from the

00:04:44.785 --> 00:04:46.852
initial point to the terminal point.

00:04:46.852 --> 00:04:51.852
So in the x direction, for
vector u, we go from negative

00:04:52.122 --> 00:04:57.122
seven to negative four,
so we increased by three.

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And what do we do over here?

00:05:00.086 --> 00:05:03.662
We go from x equals two to x equals five.

00:05:03.662 --> 00:05:08.167
So our x component once again
has a magnitude of three,

00:05:08.167 --> 00:05:10.976
our change in x is a positive three.

00:05:10.976 --> 00:05:14.157
And so our change in y,
what is that going to be?

00:05:14.157 --> 00:05:16.734
Well we're going, on this
vector, from y equals...

00:05:16.734 --> 00:05:18.267
Whoops, I'm using the wrong...

00:05:18.267 --> 00:05:23.267
We're going from y is equal
to one to y is equal to six.

00:05:26.808 --> 00:05:29.571
So the magnitude of the y
component, we could say, is five

00:05:29.571 --> 00:05:31.661
or our change in y is a positive five.

00:05:31.661 --> 00:05:36.026
And then the change in y over
here also is a positive five.

00:05:36.026 --> 00:05:38.696
We go from negative seven to negative two.

00:05:38.696 --> 00:05:40.554
It's also a positive five.

00:05:40.554 --> 00:05:42.225
So notice our change in x is

00:05:42.225 --> 00:05:44.617
the same, it's a positive three.

00:05:44.617 --> 00:05:46.660
Our change in y is the
same, positive five.

00:05:46.660 --> 00:05:48.518
So that let's us know that we have

00:05:48.518 --> 00:05:51.420
the same magnitude and direction.

00:05:51.420 --> 00:05:56.420
And so these are equivalent,
these are equivalent vectors.

00:05:57.689 --> 00:06:00.429
And, in fact, as we'll see in
the future, you can actually

00:06:00.429 --> 00:06:03.703
denote a vector by its components.

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We could say that vector u,
vecotr u, is the vector five,

00:06:10.758 --> 00:06:13.278
oh sorry, the vector three comma five,

00:06:13.278 --> 00:06:18.278
our change in x comma our change in y.

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And that is exactly the same as vector w.

00:06:23.463 --> 00:06:28.463
So that is exactly the same as
vector w, has the same change

00:06:28.780 --> 00:06:31.056
in x, same change of y as
you go from the initial

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point to the tip of the
arrow, the terminal point.


Example finding components of a vector

WEBVTT
Kind: captions
Language: en

00:00:00.331 --> 00:00:04.156
- [Voiceover] Find the
components of vector AB.

00:00:04.156 --> 00:00:05.826
So when they're talking
about the components,

00:00:05.826 --> 00:00:07.135
at least in this context,

00:00:07.135 --> 00:00:09.584
they're just talking about
breaking it down into

00:00:09.584 --> 00:00:12.335
if we start at point A and
we're finishing at point B,

00:00:12.335 --> 00:00:14.880
how much do we have to
move in the X direction?

00:00:14.880 --> 00:00:17.780
So this is going to be
essentially our change in X.

00:00:17.780 --> 00:00:20.112
And then how much do we have
to move in the Y direction

00:00:20.112 --> 00:00:22.214
to go from point A to point B?

00:00:22.214 --> 00:00:25.632
So this one over here is
going to be our change in X.

00:00:25.632 --> 00:00:28.811
I just wrote the Greek
letter Delta for change in X.

00:00:28.811 --> 00:00:33.255
And then, the second
component is going to be

00:00:33.255 --> 00:00:35.332
our change in Y.

00:00:35.332 --> 00:00:36.607
And to think about that,

00:00:36.607 --> 00:00:38.254
let's just think about what our starting

00:00:38.254 --> 00:00:39.692
and final points are,

00:00:39.692 --> 00:00:42.096
our initial and our terminal point are.

00:00:42.096 --> 00:00:44.383
So, this point right over here,

00:00:44.383 --> 00:00:48.572
point A, its coordinates are (4,4).

00:00:48.572 --> 00:00:51.334
And then point B, its coordinates are,

00:00:51.334 --> 00:00:56.307
let's see its X coordinate is (-7,-8).

00:00:56.307 --> 00:01:00.386
So let's first think about
what our change in X is,

00:01:00.386 --> 00:01:02.374
and like always, I encourage
you to pause the video

00:01:02.374 --> 00:01:04.536
and try to work through it on your own.

00:01:04.536 --> 00:01:07.901
Well let's see, if we're starting at four

00:01:07.901 --> 00:01:10.619
and then we are going from X equals four.

00:01:10.619 --> 00:01:11.937
That's where we're starting,

00:01:11.937 --> 00:01:16.115
to X equals negative seven.

00:01:16.115 --> 00:01:19.568
So that right over there
is our change in X.

00:01:20.292 --> 00:01:22.935
And there's a couple of
ways you could compute that.

00:01:22.935 --> 00:01:27.011
You could say, "Look, we
finished at negative seven.

00:01:27.011 --> 00:01:28.646
"We started at negative four."

00:01:28.646 --> 00:01:31.871
You take your final point
or where you end up,

00:01:31.871 --> 00:01:34.169
so that's negative seven,

00:01:34.169 --> 00:01:37.554
and you subtract your
initial point, minus four,

00:01:37.554 --> 00:01:40.272
which is going to be equal to negative 11.

00:01:40.272 --> 00:01:44.053
The negative tells us that
we decreased in X by 11.

00:01:44.053 --> 00:01:45.020
And you could see that.

00:01:45.020 --> 00:01:47.022
If you could just visually
count the squares,

00:01:47.022 --> 00:01:48.461
you could say, "Look,
if I'm going from four

00:01:48.461 --> 00:01:50.137
"to negative seven, I have to go down four

00:01:50.137 --> 00:01:51.973
"just to get back to X equals zero,

00:01:51.973 --> 00:01:54.052
"and then I have to go down another seven.

00:01:54.052 --> 00:01:56.357
"So I have to go to the left 11 spaces."

00:01:56.357 --> 00:01:58.005
So that's negative 11.

00:01:58.005 --> 00:02:01.012
So that's my X component, negative 11.

00:02:01.012 --> 00:02:03.520
And what is my change in Y?

00:02:03.520 --> 00:02:05.820
Well I'm going from Y equals four.

00:02:05.820 --> 00:02:07.870
In fact, I'll start at
this point right over here.

00:02:07.870 --> 00:02:10.107
I'm starting at Y equals four.

00:02:10.107 --> 00:02:12.953
And I'm ending up at Y is equal to,

00:02:12.953 --> 00:02:14.911
let me do that in that other color.

00:02:15.617 --> 00:02:18.543
So, I'm starting at Y is equal to four,

00:02:18.543 --> 00:02:22.313
and I'm ending up at Y is
equal to negative eight.

00:02:22.313 --> 00:02:24.062
So our change in Y,

00:02:24.062 --> 00:02:28.226
our change in Y, what's
going to be my final Y value,

00:02:28.226 --> 00:02:29.586
which is negative eight,

00:02:29.586 --> 00:02:32.645
minus my initial Y value, which is four,

00:02:32.645 --> 00:02:36.782
minus four, which is equal to negative 12.

00:02:36.782 --> 00:02:40.220
So negative 12.

00:02:40.220 --> 00:02:41.401
And you could see that here.

00:02:41.401 --> 00:02:42.314
If I'm starting up here,

00:02:42.314 --> 00:02:45.150
I have to go four down just
to get back to the X axis.

00:02:45.150 --> 00:02:46.762
Then I have to go down another eight,

00:02:46.762 --> 00:02:49.127
so I have to go down a total of 12.

00:02:49.127 --> 00:02:50.719
And you can see something interesting

00:02:50.719 --> 00:02:51.992
that I've just set up here.

00:02:51.992 --> 00:02:56.484
You could also view this bigger vector.

00:02:56.484 --> 00:02:59.674
Vector AB is being constructed of this X,

00:02:59.674 --> 00:03:02.277
this vector that goes
purely in the X direction,

00:03:02.277 --> 00:03:04.860
and this vector that goes
purely in the Y direction.

00:03:04.860 --> 00:03:08.594
If you were to add this red
vector to this blue-green,

00:03:08.594 --> 00:03:10.001
dark blue-green vector,

00:03:10.001 --> 00:03:12.468
you would get vector AB,

00:03:12.468 --> 00:03:15.676
but we'll talk more about
that in future videos.


Example comparing x components of vectors

WEBVTT
Kind: captions
Language: en

00:00:00.332 --> 00:00:02.260
- We're asked, "Which of
the following vectors have

00:00:02.260 --> 00:00:05.767
"the same x component as vector a?"

00:00:05.767 --> 00:00:07.966
And they tell us to
"Select all that apply"

00:00:07.966 --> 00:00:09.761
This is vector a right over here

00:00:09.761 --> 00:00:12.865
and we only have to concern
ourselves with the x component.

00:00:12.865 --> 00:00:14.351
That's what they're asking us about.

00:00:14.351 --> 00:00:16.416
Let's think about what
it's x component is.

00:00:16.416 --> 00:00:18.200
We're starting at this
point right over here,

00:00:18.200 --> 00:00:21.120
which has an x value of negative two and

00:00:23.542 --> 00:00:25.000
we're going from x equals negative two

00:00:25.000 --> 00:00:27.316
to x is equal to negative five.

00:00:27.316 --> 00:00:29.986
Our change in x, another
way to think about it,

00:00:29.986 --> 00:00:31.635
our x component.

00:00:31.635 --> 00:00:34.154
We're going from negative
two to negative five.

00:00:34.154 --> 00:00:37.543
Our x value goes down by three.

00:00:39.497 --> 00:00:43.209
Our change in x is
equal to negative three.

00:00:43.209 --> 00:00:45.670
That would also be the
x component of vector a.

00:00:45.670 --> 00:00:50.448
We could say that vector a is equal to,

00:00:50.941 --> 00:00:53.213
it's x component is negative three.

00:00:53.213 --> 00:00:55.047
We're not concerning
ourselves with our y component

00:00:55.047 --> 00:00:58.483
but we see that our y goes up by one.

00:00:58.483 --> 00:01:00.387
So it's negative three comma one

00:01:00.387 --> 00:01:04.190
but we just care about our x component.

00:01:04.428 --> 00:01:06.088
Let's think about what other vectors here

00:01:06.088 --> 00:01:07.759
have an x component of negative three.

00:01:07.759 --> 00:01:09.033
That, if we start at our initial point

00:01:09.033 --> 00:01:11.003
and go to the terminal point,

00:01:11.003 --> 00:01:13.224
our x value has gone down by three.

00:01:13.810 --> 00:01:17.664
We're gonna start with vector b here.

00:01:17.664 --> 00:01:19.803
And let's see, if we start there

00:01:20.815 --> 00:01:23.713
our x value is three at our initial point,

00:01:23.713 --> 00:01:26.008
our starting point for that vector.

00:01:26.008 --> 00:01:29.835
Then, to go to the x value
of the terminal point

00:01:29.835 --> 00:01:33.733
we once again went down by three.

00:01:33.733 --> 00:01:35.335
This has the same x component.

00:01:35.335 --> 00:01:39.068
Vector b is going to be

00:01:39.068 --> 00:01:41.524
negative three comma something.

00:01:42.226 --> 00:01:44.014
I will select that one.

00:01:44.014 --> 00:01:45.893
Let's think about vector c.

00:01:45.893 --> 00:01:49.795
Vector c starts here
at x equals negative 4

00:01:49.795 --> 00:01:52.415
and then it ends at x equals negative one.

00:01:52.930 --> 00:01:56.063
It's x component is going to look,

00:01:56.063 --> 00:01:57.254
one way to think about it, our change

00:01:57.254 --> 00:01:59.477
in x is going to look like this.

00:01:59.477 --> 00:02:01.265
You might be tempted to
say this is the same thing.

00:02:01.265 --> 00:02:03.308
It has a length of three but notice,

00:02:03.308 --> 00:02:04.864
we're not going three to the left,

00:02:04.864 --> 00:02:07.697
the way that we did in
vector a or in vector b.

00:02:07.697 --> 00:02:09.462
We're going three to the right.

00:02:09.462 --> 00:02:13.364
Here our change in x is positive three

00:02:13.364 --> 00:02:16.452
and so vector c is going to be

00:02:16.452 --> 00:02:19.912
positive three comma something.

00:02:19.912 --> 00:02:21.328
It's not gonna be vector c.

00:02:21.328 --> 00:02:23.023
Now let's look at vector d.

00:02:23.023 --> 00:02:26.226
Vector d starts over here, it's x value is

00:02:26.226 --> 00:02:28.363
negative six and we're going from

00:02:28.363 --> 00:02:32.565
x equals negative six to
x equals negative nine.

00:02:32.565 --> 00:02:34.609
Once again we have gone down.

00:02:34.609 --> 00:02:37.442
Our change in x is negative three.

00:02:37.442 --> 00:02:39.254
That has the same x component.

00:02:39.254 --> 00:02:41.206
If I were to say vector d

00:02:41.206 --> 00:02:44.341
I would say it's negative
three comma something.

00:02:44.341 --> 00:02:45.618
I haven't figured out what these are

00:02:45.618 --> 00:02:47.582
but I don't need to for this problem.

00:02:47.582 --> 00:02:49.958
So vector d also has the same x component,

00:02:49.958 --> 00:02:53.272
an x component of negative
three and we're done.


Finite geometric series formula justification

WEBVTT
Kind: captions
Language: en

00:00:00.283 --> 00:00:01.885
- Let's say we are dealing with

00:00:01.885 --> 00:00:05.354
a geometric series.

00:00:05.864 --> 00:00:08.047
There are some things that we know about

00:00:08.047 --> 00:00:10.439
this geometric series.

00:00:10.439 --> 00:00:13.453
For example, we know that the first term

00:00:13.453 --> 00:00:15.867
of our geometric series is a.

00:00:15.867 --> 00:00:19.433
That is a first term.

00:00:19.739 --> 00:00:23.724
We also know the common ratio
of our geometric series.

00:00:23.724 --> 00:00:25.292
We're gonna call that r.

00:00:25.292 --> 00:00:29.368
This is the common ratio.

00:00:30.593 --> 00:00:34.199
We also know that it's a
finite geometric series.

00:00:37.751 --> 00:00:39.946
It has a finite number of terms.

00:00:39.946 --> 00:00:44.946
Let's say that n is equal
to the number of terms.

00:00:47.214 --> 00:00:52.198
We're going to use a notation S sub n

00:00:52.758 --> 00:00:57.264
to denote the sum of first.

00:01:00.360 --> 00:01:02.240
n terms.

00:01:02.240 --> 00:01:05.003
The goal of this whole video
is using this information,

00:01:05.003 --> 00:01:08.185
coming up with a general
formula for the sum

00:01:08.185 --> 00:01:09.885
of the first n terms.

00:01:09.885 --> 00:01:13.999
A formula for evaluating
a geometric series.

00:01:13.999 --> 00:01:15.231
Let's write out S sub n.

00:01:15.231 --> 00:01:17.509
Just get a feeling for
what it would look like.

00:01:17.509 --> 00:01:19.765
S sub n is going to be equal to,

00:01:19.765 --> 00:01:21.417
you'll have your first term here,

00:01:21.417 --> 00:01:23.514
which is an a and then what's

00:01:23.514 --> 00:01:25.233
our second term going to be?

00:01:25.933 --> 00:01:27.273
This is a geometric series so it's

00:01:27.273 --> 00:01:30.786
going to be a times the common ratio.

00:01:30.786 --> 00:01:33.530
It's going to be the first
term times the common ratio.

00:01:33.530 --> 00:01:36.018
The first term times r.

00:01:36.018 --> 00:01:39.298
Now, what's the third term going to be?

00:01:39.298 --> 00:01:41.438
Well, it's going to be
the second term times

00:01:41.438 --> 00:01:42.559
our common ratio again.

00:01:42.559 --> 00:01:46.562
It's going to be ar times r or ar squared.

00:01:48.077 --> 00:01:50.490
We could go all the way to our nth term.

00:01:50.959 --> 00:01:52.794
We're gonna go all the way

00:01:52.794 --> 00:01:55.255
to the nth term and you might be

00:01:55.255 --> 00:01:59.575
tempted to say it's going to
be a times r to the nth power

00:01:59.575 --> 00:02:01.843
but we have to be careful here.

00:02:01.843 --> 00:02:03.945
Because notice, our first term is really

00:02:03.945 --> 00:02:05.577
ar to the zeroth power,

00:02:05.577 --> 00:02:07.527
second term is ar to the firsth power,

00:02:07.527 --> 00:02:09.895
third term is ar to the second power.

00:02:09.895 --> 00:02:12.291
So whatever term we're on the exponent

00:02:12.291 --> 00:02:15.727
is that term number minus one.

00:02:15.727 --> 00:02:17.171
If we're on the nth term it's going to be

00:02:17.171 --> 00:02:20.634
ar to the n minus oneth power.

00:02:21.081 --> 00:02:23.336
We want to come up with
a nice clean formula

00:02:23.336 --> 00:02:25.922
for evaluating this and we're gonna use

00:02:25.922 --> 00:02:27.432
a little trick to do it.

00:02:27.432 --> 00:02:31.958
To do it we're gonna think
about what r times the sum is.

00:02:32.660 --> 00:02:34.081
We're gonna subtract that out.

00:02:34.081 --> 00:02:37.215
We're gonna take the r times that sum,

00:02:37.215 --> 00:02:39.486
r times

00:02:41.106 --> 00:02:42.838
the sum of the first nth terms.

00:02:42.838 --> 00:02:45.534
Actually, let's just multiply negative r.

00:02:46.636 --> 00:02:48.102
Something that we can
just add these two things

00:02:48.102 --> 00:02:50.633
and you'll see that it
cleans this thing up nicely.

00:02:50.633 --> 00:02:52.816
So what is this going to be equal to?

00:02:52.816 --> 00:02:54.902
This is going to be equal to,

00:02:54.902 --> 00:02:57.222
well if you multiply

00:02:58.912 --> 00:03:01.268
a times negative r,

00:03:01.268 --> 00:03:05.533
we will get negative ar.

00:03:05.993 --> 00:03:07.925
I'm just gonna write it
right underneath this one.

00:03:07.925 --> 00:03:09.795
So if you multiply this times negative r.

00:03:09.795 --> 00:03:11.550
I'm just gonna multiply everyone of these

00:03:11.550 --> 00:03:12.676
terms by negative r.

00:03:12.676 --> 00:03:13.804
That's the equivalent of multiplying

00:03:13.804 --> 00:03:15.011
negative times the sum.

00:03:15.011 --> 00:03:17.015
I'm distributing the negative r.

00:03:17.708 --> 00:03:19.962
If I multiply it times this
term, a times negative r,

00:03:22.082 --> 00:03:26.508
that's going to be negative ar.

00:03:27.126 --> 00:03:30.366
Then, if I multiply ar times negative r

00:03:30.366 --> 00:03:33.136
that's going to be negative ar squared.

00:03:35.742 --> 00:03:37.493
You might see where this is going.

00:03:38.126 --> 00:03:40.484
And just to be clear what's going on,

00:03:40.484 --> 00:03:42.965
that's that term times negative r.

00:03:45.826 --> 00:03:47.998
This is that term times negative r.

00:03:47.998 --> 00:03:50.831
And we would keep going all the way

00:03:50.831 --> 00:03:53.555
to the term

00:03:56.091 --> 00:03:59.053
before this times negative r.

00:03:59.053 --> 00:04:03.028
So the term before this times
negative r is going to be,

00:04:05.578 --> 00:04:07.183
let me put subtraction signs,

00:04:07.183 --> 00:04:11.125
it's going to be negative a

00:04:11.125 --> 00:04:15.646
times r to the n minus one power.

00:04:18.252 --> 00:04:19.542
That was the term right before this.

00:04:19.542 --> 00:04:22.861
That was a times r to the n
minus two times negative r

00:04:22.861 --> 00:04:24.045
is gonna give us this.

00:04:24.045 --> 00:04:25.523
It's gonna get us right over there

00:04:25.523 --> 00:04:27.659
and then finally we take this last term

00:04:27.659 --> 00:04:29.076
and you multiply it by negative r,

00:04:29.076 --> 00:04:30.087
what do you get?

00:04:30.087 --> 00:04:31.341
You get,

00:04:31.341 --> 00:04:32.758
negative a

00:04:35.464 --> 00:04:37.721
times r to the n.

00:04:40.664 --> 00:04:41.882
You multiply this times the negative,

00:04:41.882 --> 00:04:43.344
you get the negative a and then

00:04:43.344 --> 00:04:45.715
r to the n minus one times r,

00:04:45.715 --> 00:04:47.139
or times r to the first,

00:04:47.139 --> 00:04:49.089
well this is going to be r to the n.

00:04:49.089 --> 00:04:50.946
Now what's interesting here is we can

00:04:50.946 --> 00:04:54.475
add up the left side and we can
add up the right hand sides.

00:04:54.475 --> 00:04:55.584
Let's do that.

00:04:57.373 --> 00:04:59.980
On the left hand side we get,

00:05:00.530 --> 00:05:04.087
S sub n minus r

00:05:05.764 --> 00:05:07.535
times S sub n

00:05:09.135 --> 00:05:10.181
and on the right hand side

00:05:10.181 --> 00:05:12.261
we have something very cool happening.

00:05:12.261 --> 00:05:15.252
Notice, this a, we still have that.

00:05:15.252 --> 00:05:17.805
The a sits there but everything else,

00:05:17.805 --> 00:05:18.944
except for this last thing,

00:05:18.944 --> 00:05:21.344
is going to cancel out.

00:05:22.093 --> 00:05:25.503
These two are gonna cancel out.

00:05:26.713 --> 00:05:29.906
These two are gonna cancel out.

00:05:29.927 --> 00:05:31.524
All we're gonna have left with is

00:05:31.524 --> 00:05:33.742
negative ar to the n.

00:05:33.742 --> 00:05:37.106
It's going to be a minus a

00:05:37.106 --> 00:05:40.751
times r to the nth power.

00:05:40.751 --> 00:05:43.040
Now we can just solve for S sub n

00:05:43.040 --> 00:05:45.914
and we have our formula,
what we were looking for.

00:05:45.914 --> 00:05:47.879
Let's see, we can factor out an S sub n

00:05:47.879 --> 00:05:49.432
on the left hand side.

00:05:49.432 --> 00:05:53.741
You get an S sub n, the
sub of our first n terms.

00:05:53.741 --> 00:05:55.016
You factor that out,

00:05:55.016 --> 00:05:56.677
it's gonna be times

00:05:57.787 --> 00:06:01.338
one minus r is going to be equal to

00:06:01.878 --> 00:06:02.686
and on the right hand side we can

00:06:02.686 --> 00:06:04.541
actually factor out an a.

00:06:04.541 --> 00:06:08.015
It's going to be a times

00:06:09.042 --> 00:06:11.732
one minus r to the n.

00:06:11.732 --> 00:06:14.399
To solve for S sub n, the
sum of our first n terms,

00:06:14.399 --> 00:06:16.372
we deserve a little bit
of a drum roll here,

00:06:16.372 --> 00:06:18.184
S sub n is going to be equal to

00:06:18.184 --> 00:06:21.085
this divided by one minus r.

00:06:21.085 --> 00:06:25.324
It's going to be a times

00:06:26.329 --> 00:06:28.930
one minus r to the n

00:06:28.930 --> 00:06:32.690
over one minus r.

00:06:32.690 --> 00:06:33.501
And we're done.

00:06:33.501 --> 00:06:37.141
We have figured out
our formula for the sum

00:06:37.141 --> 00:06:40.155
or for the sum of a finite

00:06:40.951 --> 00:06:43.666
geometric series.

00:06:43.666 --> 00:06:45.229
In the next few videos or in future videos

00:06:45.229 --> 00:06:46.756
we will apply this and I encourage you,

00:06:46.756 --> 00:06:49.776
whenever you use this
formula it's very important,

00:06:49.776 --> 00:06:51.298
now that you know where it came from,

00:06:51.298 --> 00:06:53.054
that you really keep
close track of how many

00:06:53.054 --> 00:06:54.751
terms you are actually summing up.

00:06:54.751 --> 00:06:57.078
Sometimes you might have a sigma notation

00:06:57.078 --> 00:06:59.583
and it might start it's index at zero

00:06:59.583 --> 00:07:00.698
and then goes up to a number,

00:07:00.698 --> 00:07:01.568
in which case you're gonna have that

00:07:01.568 --> 00:07:02.952
number plus one term.

00:07:02.952 --> 00:07:03.742
So you're going to have
to be very careful.

00:07:03.742 --> 00:07:05.345
This is the number of terms.

00:07:05.345 --> 00:07:06.784
This is the first term here,

00:07:06.784 --> 00:07:07.876
we define it up here.

00:07:07.876 --> 00:07:11.057
N is the number of
terms, the first n terms,

00:07:11.057 --> 00:07:13.123
r is our common ratio.


Worked example: arithmetic series (recursive formula)

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.203
- [Voiceover] So I'm going
to recursively define

00:00:02.203 --> 00:00:04.084
an arithmetic sequence.

00:00:04.084 --> 00:00:08.078
So we're going to say that
they ith term of the sequence

00:00:08.078 --> 00:00:12.095
is equal to the i minus oneth term

00:00:12.095 --> 00:00:16.487
of the sequence plus 11.

00:00:17.598 --> 00:00:21.218
So each term is going to be 11 more

00:00:21.218 --> 00:00:23.654
than the term before it.

00:00:23.654 --> 00:00:25.792
Now we have to establish a base case here,

00:00:25.792 --> 00:00:28.079
and so we're going to
say that the first term

00:00:28.079 --> 00:00:30.443
of our arithmetic sequence is going

00:00:30.443 --> 00:00:32.747
to be equal to four.

00:00:32.747 --> 00:00:34.627
So given this recursive definition

00:00:34.627 --> 00:00:37.413
of our arithmetic
sequence right over here,

00:00:37.413 --> 00:00:39.525
what I challenge you to do

00:00:39.525 --> 00:00:44.525
is to find the sum of the first 650 terms

00:00:45.992 --> 00:00:47.374
of the sequence.

00:00:47.374 --> 00:00:48.047
Let me write that down.

00:00:48.047 --> 00:00:53.047
Find the sum of first

00:00:54.269 --> 00:00:59.269
650 terms of the sequence,

00:00:59.618 --> 00:01:01.557
of this arithmetic sequence

00:01:01.557 --> 00:01:04.157
that we have just defined.

00:01:04.157 --> 00:01:05.516
And like always, pause the video

00:01:05.516 --> 00:01:07.896
and see if you can work that out.

00:01:07.896 --> 00:01:10.415
All right, so how can we think about this?

00:01:10.415 --> 00:01:13.269
Well, in many videos we give our intuition

00:01:13.269 --> 00:01:16.417
for the sum of an arithmetic sequence,

00:01:16.417 --> 00:01:19.622
and we came up with a formula

00:01:19.622 --> 00:01:22.988
for evaluating a sum of
an arithmetic sequence,

00:01:22.988 --> 00:01:24.880
which we call an arithmetic series,

00:01:24.880 --> 00:01:28.955
and that sum of the first n terms

00:01:28.955 --> 00:01:31.080
is going to be the first term

00:01:31.080 --> 00:01:33.320
plus the last term over two,

00:01:33.320 --> 00:01:36.154
so really the average of
the first and last terms,

00:01:36.154 --> 00:01:38.729
times the number of terms we have.

00:01:38.729 --> 00:01:42.700
And this is only the case
when it's an arithmetic series

00:01:42.700 --> 00:01:44.476
where each term that we're adding

00:01:44.476 --> 00:01:46.926
is a fixed amount larger or less than

00:01:46.926 --> 00:01:49.934
the term before it, or that
we have a fixed difference.

00:01:49.934 --> 00:01:51.896
So what about this one right over here?

00:01:51.896 --> 00:01:54.415
What is the first and the last term

00:01:54.415 --> 00:01:55.611
going to be, and what is our n?

00:01:55.611 --> 00:01:58.609
Well we know that n is 650,

00:01:58.609 --> 00:02:01.104
we know that n is 650,

00:02:01.104 --> 00:02:02.160
and we know what the first term

00:02:02.160 --> 00:02:02.706
is going to be.

00:02:02.706 --> 00:02:04.760
The first terms is going to be four.

00:02:04.760 --> 00:02:07.690
We need to figure out
what the nth term is,

00:02:08.315 --> 00:02:11.119
or we need to figure out
what the 650th term is

00:02:11.119 --> 00:02:12.408
going to be.

00:02:12.408 --> 00:02:14.324
Let's think about this a little bit.

00:02:14.324 --> 00:02:17.006
So this is going to be,
if we're taking the sum,

00:02:17.006 --> 00:02:20.825
it's going to be four plus the next term,

00:02:20.825 --> 00:02:22.926
the second term,

00:02:22.926 --> 00:02:27.001
so if a sub two is going
to be a sub one plus 11.

00:02:27.001 --> 00:02:31.343
So it's going to be four
plus 11, which is 15.

00:02:31.343 --> 00:02:32.922
We're going to add 11,
which is going to get us

00:02:32.922 --> 00:02:37.286
to 26, and we're going to keep adding 11.

00:02:37.286 --> 00:02:40.363
Now how many times are we going to add 11?

00:02:40.363 --> 00:02:42.255
Well to get to the second term,

00:02:42.255 --> 00:02:44.554
we add 11 once.

00:02:44.554 --> 00:02:45.715
To get to the third term,

00:02:45.715 --> 00:02:48.431
we add 11 twice.

00:02:48.431 --> 00:02:53.272
So to get to the 650th term,

00:02:53.272 --> 00:02:57.869
so this is a sub 650,

00:02:57.869 --> 00:03:00.052
a sub 650,

00:03:00.052 --> 00:03:02.049
we're going to have to add 11,

00:03:02.049 --> 00:03:03.352
look, to get to the second term,

00:03:03.352 --> 00:03:04.859
we added 11 once,

00:03:04.859 --> 00:03:06.773
third term, add 11 twice.

00:03:06.773 --> 00:03:09.375
So to get to the 650th term,

00:03:09.375 --> 00:03:11.766
we are going to add 11,

00:03:11.766 --> 00:03:16.073
we are going to add 11
650 minus one times,

00:03:16.073 --> 00:03:18.314
or 649 times.

00:03:18.314 --> 00:03:19.753
Notice, to get to each term,

00:03:19.753 --> 00:03:20.694
to get to the first term,

00:03:20.694 --> 00:03:23.515
you added one minus one,

00:03:23.515 --> 00:03:25.070
you had an 11 one minus one times,

00:03:25.070 --> 00:03:26.974
you added 11 zero times.

00:03:26.974 --> 00:03:27.845
You started with the four,

00:03:27.845 --> 00:03:28.994
didn't add 11 at all.

00:03:28.994 --> 00:03:29.772
Then the second term,

00:03:29.772 --> 00:03:31.188
you added 11 once.

00:03:31.188 --> 00:03:33.033
Third term, you added 11 twice.

00:03:33.033 --> 00:03:34.880
Fourth term, you added 11 three times.

00:03:34.880 --> 00:03:39.802
650th term, you added 11 649 times.

00:03:39.802 --> 00:03:42.102
And so if you add 11 649 times,

00:03:42.102 --> 00:03:43.180
what do you get?

00:03:43.180 --> 00:03:48.180
So four plus 649 times 11

00:03:49.286 --> 00:03:50.227
is going to be equal to,

00:03:50.227 --> 00:03:52.282
I'll get my calculator out for this,

00:03:52.282 --> 00:03:55.521
so this is going to be equal to

00:03:55.521 --> 00:03:59.967
649 times 11 is equal to,

00:03:59.967 --> 00:04:04.967
now plus four is equal to 7,143,

00:04:05.319 --> 00:04:07.478
7,143.

00:04:07.478 --> 00:04:10.043
So that's the 650th term.

00:04:10.043 --> 00:04:15.043
7, 143.

00:04:15.476 --> 00:04:17.832
And so now we can just evaluate this.

00:04:17.832 --> 00:04:19.888
So I'll get the calculator out for that.

00:04:19.888 --> 00:04:23.324
So we have 7,143 plus four

00:04:23.324 --> 00:04:25.391
plus the first term

00:04:25.391 --> 00:04:28.432
plus four is equal to that.

00:04:28.432 --> 00:04:30.232
We're going to divide by two.

00:04:30.232 --> 00:04:35.232
So divide by two gets us 3,573.5.

00:04:35.641 --> 00:04:37.743
We're going to multiply that times 650.

00:04:37.743 --> 00:04:39.182
That's how many terms we have.

00:04:39.182 --> 00:04:42.560
Times 650 is equal to,

00:04:42.560 --> 00:04:44.025
that's a pretty large number,

00:04:44.025 --> 00:04:49.025
is going to be equal to 2,322,775.

00:04:51.373 --> 00:04:55.576
2,322,775.

00:04:55.576 --> 00:04:57.179
I already forgot it.

00:04:57.179 --> 00:04:59.533
I have trouble remembering things

00:04:59.533 --> 00:05:00.810
when I take it off my screen.

00:05:00.810 --> 00:05:01.867
All right,

00:05:04.357 --> 00:05:09.357
2,322,775.

00:05:13.272 --> 00:05:14.468
I'm glad I had a calculator in hand

00:05:14.468 --> 00:05:15.873
for that one, but you could do it by hand.

00:05:15.873 --> 00:05:16.849
I always encourage you to do it.

00:05:16.849 --> 00:05:19.456
It never hurts to practice the arithmetic.


Worked example: arithmetic series (sum expression)

WEBVTT
Kind: captions
Language: en

00:00:00.214 --> 00:00:01.630
- [Voiceover] So we
have the sum negative 50

00:00:01.630 --> 00:00:04.788
plus negative 44, plus
negative 38, all the way,

00:00:04.788 --> 00:00:09.223
we keep adding all the way
up to 2,038, and then 2,044.

00:00:09.223 --> 00:00:14.223
So see if you can pause this
video and evaluate this sum.

00:00:14.377 --> 00:00:16.606
So let's work through this together,

00:00:16.606 --> 00:00:18.325
and let's just think
about what's going on.

00:00:18.325 --> 00:00:20.507
So the first term here is negative 50,

00:00:20.507 --> 00:00:22.179
and then we go to negative 44,

00:00:22.179 --> 00:00:24.478
so the second term is negative 50

00:00:24.478 --> 00:00:29.478
plus 6, and then the third
term, we add 6 again.

00:00:29.493 --> 00:00:33.208
Negative 44 plus 6 is negative 38, and we

00:00:33.208 --> 00:00:35.367
go all the way to here, we keep adding 6,

00:00:35.367 --> 00:00:39.175
and to go from 2,038 to 2,044,
to get to that last term,

00:00:39.175 --> 00:00:42.263
we add 6 once again.

00:00:42.263 --> 00:00:44.655
And so each successive term is just 6

00:00:44.655 --> 00:00:46.513
more than the term before it.

00:00:46.513 --> 00:00:48.416
So we are dealing right over here,

00:00:48.416 --> 00:00:52.015
this sum is an arithmetic series.

00:00:52.015 --> 00:00:55.104
It's a sum of an arithmetic sequence.

00:00:55.104 --> 00:00:58.145
Each term is 6 more, is a constant amount

00:00:58.145 --> 00:01:01.767
more than the term before that.

00:01:01.767 --> 00:01:03.578
So we know how to take the sum of

00:01:03.578 --> 00:01:05.181
an arithmetic sequence.

00:01:05.181 --> 00:01:06.922
We know that if we have,

00:01:06.922 --> 00:01:08.849
if we are taking the sum of,

00:01:08.849 --> 00:01:10.242
let me do this in a new color,

00:01:10.242 --> 00:01:12.100
just to have a little bit
of variety on the screen,

00:01:12.100 --> 00:01:14.886
if we're taking the sum of,

00:01:14.886 --> 00:01:19.832
of the first n terms of
an arithmetic sequence

00:01:19.832 --> 00:01:21.295
or, if we're taking

00:01:21.295 --> 00:01:23.965
or if we're evaluating the
first n terms of an arithmetic

00:01:23.965 --> 00:01:25.915
series, I could say,

00:01:25.915 --> 00:01:28.910
it's going to be the first term

00:01:28.910 --> 00:01:30.675
plus the last term

00:01:30.675 --> 00:01:31.743
divided by 2.

00:01:31.743 --> 00:01:33.833
You could view this of
the average of the first

00:01:33.833 --> 00:01:34.854
and the last terms,

00:01:34.854 --> 00:01:37.966
times the number of terms

00:01:37.966 --> 00:01:39.359
that we're dealing with.

00:01:39.359 --> 00:01:41.983
So over here, we know what
our first and last terms are,

00:01:41.983 --> 00:01:43.028
we know this right over here.

00:01:43.028 --> 00:01:45.024
That is a1,

00:01:45.024 --> 00:01:47.996
and this is our last term, 2,044,

00:01:47.996 --> 00:01:50.016
so that is our a-sub-n,

00:01:50.016 --> 00:01:52.617
so the other question is, well, what is n?

00:01:52.617 --> 00:01:55.032
How many terms do we actually deal with?

00:01:55.032 --> 00:01:56.378
And to think about that, we just say,

00:01:56.378 --> 00:01:58.840
well how many times do we have to add 6

00:01:58.840 --> 00:02:03.158
to go from negative 50 to 2,044?

00:02:03.158 --> 00:02:07.686
Well, 2,044 minus negative 50,

00:02:07.686 --> 00:02:10.589
minus negative 50,

00:02:10.589 --> 00:02:14.814
well that's the same
thing as 2,044 plus 50

00:02:14.814 --> 00:02:18.971
or 2,094, and the whole
reason I calculated this

00:02:18.971 --> 00:02:20.712
is I want to figure out
how far do I have to go

00:02:20.712 --> 00:02:23.382
from negative 50 to 2,044?

00:02:23.382 --> 00:02:26.099
I have to go up 50 just
to get back to zero

00:02:26.099 --> 00:02:28.189
and then go up another 2,044.

00:02:28.189 --> 00:02:29.837
So I have to go 50 just
to get back to zero,

00:02:29.837 --> 00:02:31.695
then go up another 2,044

00:02:31.695 --> 00:02:34.574
so I have to go up by 2,094,

00:02:34.574 --> 00:02:35.433
so if I'm going,

00:02:35.433 --> 00:02:37.848
if I'm adding 6 on every term,

00:02:37.848 --> 00:02:39.380
how many times do I have to add 6

00:02:39.380 --> 00:02:42.399
to increase by 2,094?

00:02:42.399 --> 00:02:46.230
Well, let's just take
2,094 and divide it by 6

00:02:46.230 --> 00:02:47.692
to figure that out.

00:02:47.692 --> 00:02:49.829
So 6 goes into 20 three times,

00:02:49.829 --> 00:02:51.477
3 times 6 is 18,

00:02:51.477 --> 00:02:54.078
subtract, 20 minus 18 is 2,

00:02:54.078 --> 00:02:55.610
bring down the 9,

00:02:55.610 --> 00:02:58.605
6 goes into 29 four times,

00:02:58.605 --> 00:03:00.974
4 times 6 is 24,

00:03:00.974 --> 00:03:04.015
subtract, 29 minus 24 is 5,

00:03:04.015 --> 00:03:05.478
bring down the 4,

00:03:05.478 --> 00:03:06.755
we have 54,

00:03:06.755 --> 00:03:09.449
6 goes into 54 nine times,

00:03:09.449 --> 00:03:11.863
9 times 6 is 54

00:03:11.863 --> 00:03:13.907
and we are done.

00:03:13.907 --> 00:03:16.484
So to go from negative 50 to 2,044,

00:03:16.484 --> 00:03:20.780
I have to add 6 to 349 times,

00:03:20.780 --> 00:03:23.009
so I add it once, I add it twice,

00:03:23.009 --> 00:03:24.007
and then this right over here,

00:03:24.007 --> 00:03:27.397
this is the 349th time that I'm adding 6,

00:03:27.397 --> 00:03:29.556
so how many terms do I have?

00:03:29.556 --> 00:03:32.668
Now, you might be
tempted to say 349 terms,

00:03:32.668 --> 00:03:35.199
but really, you have 349 plus 1 terms.

00:03:35.199 --> 00:03:38.310
You have the 349 for
every time you added 6,

00:03:38.310 --> 00:03:39.866
so this is the first time you added 6,

00:03:39.866 --> 00:03:40.864
second time you added 6,

00:03:40.864 --> 00:03:43.511
all the way to the 349th time you added 6,

00:03:43.511 --> 00:03:44.347
so let me make it clear,

00:03:44.347 --> 00:03:45.810
this, this is,

00:03:45.810 --> 00:03:48.433
oh, actually, this is the
349th time I added 6 to get

00:03:48.433 --> 00:03:51.173
to this, but we haven't counted
the first term just yet,

00:03:51.173 --> 00:03:52.288
so we're going to have,

00:03:52.288 --> 00:03:53.658
so we have 300,

00:03:53.658 --> 00:03:57.419
we have the first term and
then we add 6 349 times,

00:03:57.419 --> 00:04:01.854
so we have 350 terms in this sum,

00:04:01.854 --> 00:04:06.382
so in this case, n is
going to be equal to 350.

00:04:06.382 --> 00:04:10.073
N is equal to 350.

00:04:10.073 --> 00:04:13.765
And so we can say the
sum of the first 300...

00:04:13.765 --> 00:04:16.343
I'll do this in green,

00:04:16.343 --> 00:04:20.034
the sum of the first 350 terms

00:04:20.034 --> 00:04:22.867
is going to be equal to the
average of the first and last

00:04:22.867 --> 00:04:25.561
term, so negative 50

00:04:25.561 --> 00:04:30.228
plus 2,044 over 2,

00:04:30.228 --> 00:04:31.737
over 2,

00:04:31.737 --> 00:04:34.314
times 350,

00:04:34.314 --> 00:04:35.313
so let's see,

00:04:35.313 --> 00:04:38.261
negative 50 plus 2,044,

00:04:38.261 --> 00:04:39.260
that's going to be what?

00:04:39.260 --> 00:04:41.698
2,094,

00:04:41.698 --> 00:04:46.698
2,094 divided by 2

00:04:46.713 --> 00:04:48.153
times,

00:04:48.153 --> 00:04:49.430
times 350,

00:04:49.430 --> 00:04:50.567
so let's see, if I just take,

00:04:50.567 --> 00:04:54.236
so this is going to be, 'cause
right, this is 294 times,

00:04:54.236 --> 00:04:54.724
let's see,

00:04:54.724 --> 00:04:57.347
350 divided,

00:04:57.347 --> 00:04:58.694
oh, sorry, not 294,

00:04:58.694 --> 00:04:59.600
what am I?

00:04:59.600 --> 00:05:00.807
My brain is not working.

00:05:00.807 --> 00:05:02.223
2,000,

00:05:02.223 --> 00:05:03.779
this is 2--

00:05:03.779 --> 00:05:08.376
actually, this is going to be 1994.

00:05:08.376 --> 00:05:11.743
My brain really wasn't
working a little while ago,

00:05:11.743 --> 00:05:16.743
so this is going to be
1994 divided by 2 times 350

00:05:16.851 --> 00:05:18.500
and so let's see,

00:05:18.500 --> 00:05:21.217
350 divided by 2 is 175,

00:05:21.217 --> 00:05:24.746
so this is going to be 1,900--

00:05:24.746 --> 00:05:29.413
1,994 times

00:05:29.413 --> 00:05:32.338
175.

00:05:32.338 --> 00:05:33.453
Which is equal to,

00:05:33.453 --> 00:05:35.241
and I'll use a calculator for this one,

00:05:35.241 --> 00:05:37.447
so, let me get the calculator out,

00:05:37.447 --> 00:05:41.974
so, I have 1,994

00:05:41.974 --> 00:05:45.062
times 175

00:05:45.062 --> 00:05:48.963
gives us 348,950.

00:05:48.963 --> 00:05:53.444
348,950.

00:05:53.444 --> 00:05:55.534
And we could express this
in sigma notation now,

00:05:55.534 --> 00:05:57.043
now that we know what the n is.

00:05:57.043 --> 00:05:59.365
We found our answer, this
is what we were looking for,

00:05:59.365 --> 00:06:01.084
but just in case you're curious,

00:06:01.084 --> 00:06:02.987
we could write this as the sum

00:06:02.987 --> 00:06:05.797
from, let's say, k is equal to 1

00:06:05.797 --> 00:06:10.797
to k is equal to 350 of,

00:06:10.975 --> 00:06:13.854
let's see, we could write this as,

00:06:13.854 --> 00:06:17.523
as negative 50,

00:06:17.523 --> 00:06:22.523
negative 50 plus 6 times k minus 1,

00:06:23.490 --> 00:06:26.114
'cause the first term,
we don't want to actually

00:06:26.114 --> 00:06:27.414
add the 6,

00:06:27.414 --> 00:06:28.528
and then the last term,

00:06:28.528 --> 00:06:31.268
we want to add the 6 349 times.

00:06:31.268 --> 00:06:33.474
Which we saw, so we're
going to add it 349 times,

00:06:33.474 --> 00:06:34.426
and there you have it.

00:06:34.426 --> 00:06:35.123
That,

00:06:35.123 --> 00:06:39.325
this arithmetic series
written in sigma notation,

00:06:39.325 --> 00:06:41.113
so hopefully you enjoyed that.

00:06:41.113 --> 00:06:43.342
Pardon my little mental error earlier.

00:06:43.342 --> 00:06:45.589
I don't know what was
going on in my brain.


Worked example: arithmetic series (sigma notation)

WEBVTT
Kind: captions
Language: en

00:00:00.167 --> 00:00:01.978
- [Voiceover] So I have
a finite series here

00:00:01.978 --> 00:00:04.022
expressed in sigma notation
and I encourage you

00:00:04.022 --> 00:00:06.622
to pause the video and
see if you can figure out

00:00:06.622 --> 00:00:08.062
what this evaluates to.

00:00:08.062 --> 00:00:11.916
This is going to evaluate to a number.

00:00:11.916 --> 00:00:14.168
So assuming you've had a go at it,

00:00:14.168 --> 00:00:15.283
let's work through this together.

00:00:15.283 --> 00:00:19.068
So this is a sum from k
equals 1 to k equals 550,

00:00:19.068 --> 00:00:21.808
so we're going to have 550 terms here

00:00:21.808 --> 00:00:24.896
and it's the sum from k
equals 1 to k equals 550

00:00:24.896 --> 00:00:27.450
of 2k plus 50,

00:00:27.450 --> 00:00:29.238
so whenever I try to evaluate a series,

00:00:29.238 --> 00:00:31.397
I like to just expand
out the sum a little bit

00:00:31.397 --> 00:00:34.137
just so that I can get a
feel for what it looks like,

00:00:34.137 --> 00:00:34.740
so let's see.

00:00:34.740 --> 00:00:35.948
This is going to look like,

00:00:35.948 --> 00:00:38.038
when k is equal 1, this is going to be

00:00:38.038 --> 00:00:40.894
2 times 1 plus 50.

00:00:40.894 --> 00:00:42.589
When k is equal to 2, it's going to be

00:00:42.589 --> 00:00:45.212
2 times 2 plus 50.

00:00:45.212 --> 00:00:46.652
When k is equal to 3, it's going to be

00:00:46.652 --> 00:00:49.485
2 times 3 plus 50.

00:00:49.485 --> 00:00:51.505
And we're going to keep
going all the way until

00:00:51.505 --> 00:00:52.921
we get to the last term,

00:00:52.921 --> 00:00:55.266
when k is equal to 550, it's going to be

00:00:55.266 --> 00:00:59.654
2 times 550 plus 50.

00:00:59.654 --> 00:01:00.328
So let's see.

00:01:00.328 --> 00:01:01.744
This first term is going to be,

00:01:01.744 --> 00:01:05.761
it evalutes to 52

00:01:05.761 --> 00:01:08.362
plus, this next term is 2 times 2 plus 50

00:01:08.362 --> 00:01:10.312
is going to be 54,

00:01:10.312 --> 00:01:12.146
plus the next term, 2 times 3 is 6

00:01:12.146 --> 00:01:14.352
plus 50 is 56,

00:01:14.352 --> 00:01:16.767
and we're going to go all the way,

00:01:16.767 --> 00:01:19.832
all the way to our last term,

00:01:19.832 --> 00:01:23.384
2 times 550 is 1100 plus 50

00:01:23.384 --> 00:01:25.776
is going to be 1150.

00:01:25.776 --> 00:01:27.749
So that gives us a good feel for this sum,

00:01:27.749 --> 00:01:28.492
for this series.

00:01:28.492 --> 00:01:31.256
We're going to start 52, and
we're just going to keep adding

00:01:31.256 --> 00:01:33.606
2 for each successive term, all the way

00:01:33.606 --> 00:01:35.185
until we get to 1150,

00:01:35.185 --> 00:01:38.320
and we're going to take
the sum of all of these,

00:01:38.320 --> 00:01:40.549
and since each successive
term, we're increasing

00:01:40.549 --> 00:01:41.663
by the same amount,

00:01:41.663 --> 00:01:43.776
we're increasing by 2,

00:01:43.776 --> 00:01:46.237
we're increasing by 2,
we can recognize this

00:01:46.237 --> 00:01:48.931
as an arithmetic series.

00:01:48.931 --> 00:01:51.578
We are increasing by the same amount

00:01:51.578 --> 00:01:52.832
each time.

00:01:52.832 --> 00:01:54.434
And there is a formula for the sum

00:01:54.434 --> 00:01:56.384
of an arithmetic series, and
first we're just going to

00:01:56.384 --> 00:01:57.406
apply the formula,

00:01:57.406 --> 00:01:58.985
but then we're going to get
a little bit of an intuitive

00:01:58.985 --> 00:02:01.098
sense for why that formula works,

00:02:01.098 --> 00:02:02.932
and actually, in other
videos, we have proved

00:02:02.932 --> 00:02:05.440
this formula, but it's always
good to get a sense that,

00:02:05.440 --> 00:02:06.818
you know, that this formula
just doesn't come out of

00:02:06.818 --> 00:02:10.232
thin air, so the formula for the sum

00:02:10.232 --> 00:02:12.855
of an arithmetic series,

00:02:12.855 --> 00:02:16.222
so the sum of the first
n terms is going to be

00:02:16.222 --> 00:02:20.634
the first term plus the nth term

00:02:20.634 --> 00:02:22.932
over 2, so it's really the,

00:02:22.932 --> 00:02:25.394
it's really the arithmetic
mean of the first and

00:02:25.394 --> 00:02:27.414
the last terms,

00:02:27.414 --> 00:02:28.226
you could say the average,

00:02:28.226 --> 00:02:29.991
in, I guess, everyday language,

00:02:29.991 --> 00:02:31.988
average of the first and last terms

00:02:31.988 --> 00:02:36.771
and then times the number
of terms you actually have,

00:02:36.771 --> 00:02:38.977
so if we were to try to
apply it to this case,

00:02:38.977 --> 00:02:40.974
we're trying to take the sum,

00:02:40.974 --> 00:02:43.458
sum of the first 550 terms,

00:02:43.458 --> 00:02:45.966
I'll do this in a new
color just for kicks.

00:02:45.966 --> 00:02:47.870
All right, so we're going to take the sum

00:02:47.870 --> 00:02:52.165
of the first 550 terms,

00:02:52.165 --> 00:02:54.464
and it's going to be
equal to the first term,

00:02:54.464 --> 00:02:57.552
so that's 52 plus the last term,

00:02:57.552 --> 00:03:00.803
the nth term, 1150,

00:03:00.803 --> 00:03:02.451
it's really just the average of those two,

00:03:02.451 --> 00:03:04.750
the average of the first and the last term

00:03:04.750 --> 00:03:06.468
and then times the
number of terms we have,

00:03:06.468 --> 00:03:09.579
times 550,

00:03:09.579 --> 00:03:10.787
so what is this going to be?

00:03:10.787 --> 00:03:12.180
Well, we could simplify this a little bit.

00:03:12.180 --> 00:03:15.709
If we're going to take 550 divided by 2,

00:03:15.709 --> 00:03:17.265
this is going to be,

00:03:17.265 --> 00:03:18.495
I could write this as times,

00:03:18.495 --> 00:03:19.262
actually, let me just,

00:03:19.262 --> 00:03:21.584
let me just simply this
in a different way.

00:03:21.584 --> 00:03:24.532
So this is going to be the same thing as,

00:03:24.532 --> 00:03:25.508
I could write this,

00:03:25.508 --> 00:03:26.738
52 divided,

00:03:26.738 --> 00:03:27.899
well, let me just add first.

00:03:27.899 --> 00:03:32.899
This is going to be 1,202

00:03:33.170 --> 00:03:35.120
over 2.

00:03:35.120 --> 00:03:36.026
All right, did I do that right?

00:03:36.026 --> 00:03:39.091
Yeah, 1,202 over 2

00:03:39.091 --> 00:03:42.179
times 550.

00:03:42.179 --> 00:03:47.179
Now, 1202 divided by 2 is going to be 601,

00:03:47.217 --> 00:03:49.795
so this is equal to 601

00:03:49.795 --> 00:03:52.929
times 550.

00:03:52.929 --> 00:03:55.112
And let's see, I can multiply that out.

00:03:55.112 --> 00:03:58.711
So, let me just do 550

00:03:58.711 --> 00:04:01.822
times 601,

00:04:01.822 --> 00:04:05.537
so 1 times 550 is 550,

00:04:05.537 --> 00:04:06.977
and then I have a zero here,

00:04:06.977 --> 00:04:08.277
but I just have a zero there,

00:04:08.277 --> 00:04:09.763
so zero times 550,

00:04:09.763 --> 00:04:11.481
I'm just going to get a bunch of zeros,

00:04:11.481 --> 00:04:14.059
and then I go to the hundreds place.

00:04:14.059 --> 00:04:17.541
6 times zero is zero.

00:04:17.541 --> 00:04:21.303
6 times 5 is 30.

00:04:21.303 --> 00:04:24.879
6 times 5 is 30 plus 3 is 33.

00:04:24.879 --> 00:04:25.993
You add it all together,

00:04:25.993 --> 00:04:27.711
we get a zero, we get a 5,

00:04:27.711 --> 00:04:29.430
we get a 5, we get a zero,

00:04:29.430 --> 00:04:31.357
we get a 3, we get a 3.

00:04:31.357 --> 00:04:35.258
We get 330,550.

00:04:35.258 --> 00:04:36.976
That's what this whole thing,

00:04:36.976 --> 00:04:39.716
that's what this whole thing sums up to.

00:04:39.716 --> 00:04:42.780
Now, I just said that we'll
get a little bit of intuition

00:04:42.780 --> 00:04:45.451
for why we were able to
just apply this formula,

00:04:45.451 --> 00:04:47.146
and let's just think about what the sum

00:04:47.146 --> 00:04:48.887
of the first 550 terms is,

00:04:48.887 --> 00:04:50.745
and I just wrote it down up here.

00:04:50.745 --> 00:04:52.602
So, let me write it,

00:04:52.602 --> 00:04:54.529
I'm just going to switch colors again.

00:04:54.529 --> 00:04:58.593
So, we're going to have the
sum of the first 550 terms,

00:04:58.593 --> 00:05:00.288
which is what we just wrote over here,

00:05:00.288 --> 00:05:01.240
we already said this is going to be

00:05:01.240 --> 00:05:05.512
52 plus 54 plus 56

00:05:05.512 --> 00:05:08.020
plus, and we're going to
keep going all the way

00:05:08.020 --> 00:05:09.854
to 1150.

00:05:09.854 --> 00:05:11.108
I'm going to write it again,

00:05:11.108 --> 00:05:13.592
the sum of the first 550 terms,

00:05:13.592 --> 00:05:15.148
but I'm going to just write it in reverse.

00:05:15.148 --> 00:05:17.284
We can obviously swap the
order in which we add.

00:05:17.284 --> 00:05:19.513
It's going to be 1150

00:05:19.513 --> 00:05:23.298
plus 1150 minus 2, which is

00:05:23.298 --> 00:05:26.920
1148, plus that minus 2,

00:05:26.920 --> 00:05:30.519
which is 1146,

00:05:30.519 --> 00:05:34.373
and we go all the way to the first term,

00:05:34.373 --> 00:05:37.345
all the way to 52.

00:05:37.345 --> 00:05:38.576
Now, what I want to do is just

00:05:38.576 --> 00:05:39.899
I want to add these two sums,

00:05:39.899 --> 00:05:42.546
so I'm just going to get 2
times the sum of the first 550,

00:05:42.546 --> 00:05:43.963
so if I had the two left sides,

00:05:43.963 --> 00:05:47.445
I would have 2 times the
sum of the first 550 terms,

00:05:47.445 --> 00:05:50.278
and we do this generally
when we prove this formula

00:05:50.278 --> 00:05:50.998
in previous videos,

00:05:50.998 --> 00:05:53.157
but I always like to get a
sense of where it comes from

00:05:53.157 --> 00:05:55.549
and so this is going to be equal to,

00:05:55.549 --> 00:05:58.892
well if I add these two
terms right over here,

00:05:58.892 --> 00:05:59.844
I get what?

00:05:59.844 --> 00:06:02.422
I get 1202.

00:06:02.422 --> 00:06:04.209
That number should look familiar.

00:06:04.209 --> 00:06:07.646
And then if I add these
two, right over here,

00:06:07.646 --> 00:06:08.691
what do I get?

00:06:08.691 --> 00:06:11.082
I get 1202.

00:06:11.082 --> 00:06:14.403
And then if I add, I think
you see where this is going,

00:06:14.403 --> 00:06:17.862
and then if I add these two characters,

00:06:17.862 --> 00:06:18.628
what do I get?

00:06:18.628 --> 00:06:22.645
I get 1202, all the way

00:06:22.645 --> 00:06:24.735
to these last two characters.

00:06:24.735 --> 00:06:25.571
You add them together, what do you get?

00:06:25.571 --> 00:06:27.939
You get 1202,

00:06:27.939 --> 00:06:31.004
so how many 1202s do I have?

00:06:31.004 --> 00:06:33.256
Well I have 550 of them.

00:06:33.256 --> 00:06:35.485
There are 550 of these terms.

00:06:35.485 --> 00:06:36.995
So this is going to be equal to

00:06:36.995 --> 00:06:41.476
550 times 1202.

00:06:41.476 --> 00:06:44.007
And so if you just wanted
to solve for this sum,

00:06:44.007 --> 00:06:46.584
you just divide all
these sides divided by 2,

00:06:46.584 --> 00:06:48.720
so you divide by 2.

00:06:48.720 --> 00:06:51.042
You divide by 2.

00:06:51.042 --> 00:06:54.780
You divide by 2, and that's
exactly what we did over here.

00:06:54.780 --> 00:06:58.426
550 times 1202, divided by 2.

00:06:58.426 --> 00:07:00.928
So hopefully that gives you
an intuition for things.


Area of a circle intuition

WEBVTT
Kind: captions
Language: en

00:00:00.253 --> 00:00:01.765
- [Voiceover] What I'd
like to do in this video

00:00:01.765 --> 00:00:03.747
is make an informal argument

00:00:03.747 --> 00:00:06.446
for why the formula for
the area of a circle

00:00:06.446 --> 00:00:09.525
is Pi r squared.

00:00:09.525 --> 00:00:10.523
And we're gonna start

00:00:10.523 --> 00:00:14.901
just with the most traditional
definition of the number Pi

00:00:14.901 --> 00:00:16.563
and that's that and I'll just do it here

00:00:16.563 --> 00:00:18.391
in a corner some place

00:00:18.391 --> 00:00:21.311
that Pi is equal to the ratio

00:00:21.311 --> 00:00:24.110
of the circumference and
the diameter of the circle

00:00:24.110 --> 00:00:25.312
or the ratio of the circumference

00:00:25.312 --> 00:00:27.143
to the diameter of a circle

00:00:27.143 --> 00:00:28.165
or, of course, we could write this

00:00:28.165 --> 00:00:29.426
as ratio of the circumference

00:00:29.426 --> 00:00:30.937
to, instead of the diameter,

00:00:30.937 --> 00:00:32.919
I could write two times the radius

00:00:32.919 --> 00:00:35.425
or I can multiply both sides
times two times the radius

00:00:35.425 --> 00:00:38.605
and we get our traditional
formula for the circumference,

00:00:38.605 --> 00:00:41.708
for the circumference of a circle,

00:00:41.708 --> 00:00:44.367
but once again, this literally
just comes straight out

00:00:44.367 --> 00:00:46.936
of the definition for the number Pi.

00:00:46.936 --> 00:00:49.449
The number Pi is defined as the ratio

00:00:49.449 --> 00:00:51.551
between the circumference
and the diameter,

00:00:51.551 --> 00:00:53.173
so you just multiply both sides out of it

00:00:53.173 --> 00:00:55.845
times the diameter, you
get the circumference

00:00:55.845 --> 00:00:59.317
is equal to Pi times the diameter.

00:00:59.317 --> 00:01:01.899
Now, we have the circumference
formula right here,

00:01:01.899 --> 00:01:04.721
once again, this comes out
of the definition of Pi,

00:01:04.721 --> 00:01:07.388
but from this, I'd like to at
least get an intuitive feel

00:01:07.388 --> 00:01:11.997
for why the area formula
is given by Pi r squared

00:01:11.997 --> 00:01:14.027
and to think about that,
we're going to approximate

00:01:14.027 --> 00:01:17.844
the area of polygons,
the areas of polygons,

00:01:17.844 --> 00:01:20.213
that are inscribed in a circle.

00:01:20.213 --> 00:01:21.835
So over here, I have this, what's this?

00:01:21.835 --> 00:01:24.921
There's a five-sided
polygon right over here

00:01:24.921 --> 00:01:28.491
and its area is going to be equal to,

00:01:28.491 --> 00:01:30.660
so the area of this polygon,

00:01:30.660 --> 00:01:32.692
it's going to be five times the area

00:01:32.692 --> 00:01:34.017
of each of those triangles

00:01:34.017 --> 00:01:36.486
and the area of each of those triangles,

00:01:36.486 --> 00:01:39.402
the height is a, the base is b,

00:01:39.402 --> 00:01:41.634
so it's going to be base times height

00:01:41.634 --> 00:01:46.138
or height times base times 1/2.

00:01:46.138 --> 00:01:48.655
So this, once again,
five times ab over two,

00:01:48.655 --> 00:01:50.553
it's not that good of an approximation.

00:01:50.553 --> 00:01:54.801
This would be the area of
just the inscribed polygon,

00:01:54.801 --> 00:01:56.506
so we're definitely underestimating

00:01:56.506 --> 00:01:58.251
the area of the entire circle

00:01:58.251 --> 00:01:59.853
or leaving out all of these little,

00:01:59.853 --> 00:02:01.855
these little chunks
outside of the polygon,

00:02:01.855 --> 00:02:03.603
but still inside of the circle.

00:02:03.603 --> 00:02:05.835
But as we add more size to the polygon,

00:02:05.835 --> 00:02:08.368
we see that we're leaving out less.

00:02:08.368 --> 00:02:09.666
We see, when we have now,

00:02:09.666 --> 00:02:11.991
this is a one, two, three, four,

00:02:11.991 --> 00:02:14.951
five, six, seven-sided polygon,

00:02:14.951 --> 00:02:17.971
we're leaving a little bit less.

00:02:17.971 --> 00:02:19.696
We're underestimating still,

00:02:19.696 --> 00:02:21.474
but we're underestimating by less.

00:02:21.474 --> 00:02:22.976
This area that we're giving up

00:02:22.976 --> 00:02:26.018
isn't as large as this
area right over here.

00:02:26.018 --> 00:02:29.315
So in this approximation,
we have, what did I say?

00:02:29.315 --> 00:02:31.257
Seven triangles? One, two, three, four,

00:02:31.257 --> 00:02:33.339
five, six, seven triangles.

00:02:33.339 --> 00:02:34.901
And the area of each of those triangles

00:02:34.901 --> 00:02:38.094
is once again ab over two.

00:02:38.094 --> 00:02:41.606
Now, a and b here are different
than a and b over here,

00:02:41.606 --> 00:02:43.624
and notice what's happening.

00:02:43.624 --> 00:02:45.482
As we increase,

00:02:45.482 --> 00:02:48.792
as we increase the number of triangles,

00:02:48.792 --> 00:02:52.612
not only is it approximating
the area of the circle better,

00:02:52.612 --> 00:02:55.008
but a is getting longer.

00:02:55.008 --> 00:02:56.870
And you can see, you could imagine,

00:02:56.870 --> 00:02:59.028
as we increase many,
many, many more triangles,

00:02:59.028 --> 00:03:01.694
a is going to approach r.

00:03:01.694 --> 00:03:05.048
Now another thing to think
about is what is seven,

00:03:05.048 --> 00:03:07.377
what is seven times b approaching?

00:03:07.377 --> 00:03:09.426
So we're saying that a is approaching r

00:03:09.426 --> 00:03:12.268
as we add more of these
sides of the polygon,

00:03:12.268 --> 00:03:13.663
as we add more triangles,

00:03:13.663 --> 00:03:16.486
now what is the number of triangles

00:03:16.486 --> 00:03:19.577
times the base of the triangle,
what is that approaching?

00:03:19.577 --> 00:03:22.849
Well this is going to
approach the perimeter of the,

00:03:22.849 --> 00:03:24.904
or this is going to be the
perimeter of the polygon,

00:03:24.904 --> 00:03:28.585
so seven times b is that plus,

00:03:28.585 --> 00:03:30.087
let me actually, let me draw this,

00:03:30.087 --> 00:03:33.900
it's that plus that

00:03:33.900 --> 00:03:35.014
plus that,

00:03:35.014 --> 00:03:35.935
I think you get the point,

00:03:35.935 --> 00:03:37.730
plus that,

00:03:37.730 --> 00:03:39.519
plus that,

00:03:39.519 --> 00:03:41.354
plus that,

00:03:41.354 --> 00:03:43.122
plus that.

00:03:43.122 --> 00:03:46.893
So once again, seven, let
me write this at seven,

00:03:46.893 --> 00:03:50.830
times b, that is the
perimeter of the polygon,

00:03:50.830 --> 00:03:52.932
perimeter,

00:03:52.932 --> 00:03:57.824
perimeter of the polygon.

00:03:57.824 --> 00:03:59.306
So think about what's happening.

00:03:59.306 --> 00:04:01.868
As we have more and more
sides of the polygon,

00:04:01.868 --> 00:04:06.082
our a, our height of
each of our triangles,

00:04:06.082 --> 00:04:09.058
is going to approach our radius,

00:04:09.058 --> 00:04:11.955
is going to approach the radius.

00:04:11.955 --> 00:04:14.164
It's going to get the height
of each of the triangles.

00:04:14.164 --> 00:04:15.165
It's gonna get longer and longer,

00:04:15.165 --> 00:04:16.867
it's gonna approach the radius as we have,

00:04:16.867 --> 00:04:19.512
as we approach an infinite
number of triangles,

00:04:19.512 --> 00:04:22.315
and then the number of polygons we have,

00:04:22.315 --> 00:04:23.162
not the number of polygons,

00:04:23.162 --> 00:04:26.449
the number of sides we
have times the bases,

00:04:26.449 --> 00:04:28.665
that's going to be the
perimeter of the polygon

00:04:28.665 --> 00:04:30.657
and as we add more and more sides,

00:04:30.657 --> 00:04:31.698
as we add more and more sides,

00:04:31.698 --> 00:04:35.348
the perimeter of the polygon
is going to approach,

00:04:35.348 --> 00:04:38.781
is going to approach,

00:04:38.781 --> 00:04:41.764
is going to approach the
circumference of the circle.

00:04:41.764 --> 00:04:44.784
I'll write it out, circumference,

00:04:44.784 --> 00:04:46.326
circumference,

00:04:46.326 --> 00:04:49.889
and you'll see that even
more clearly right over here.

00:04:49.889 --> 00:04:53.845
So, once again, how many
sides do I have here?

00:04:53.845 --> 00:04:57.665
I have one, two, three, four, five, six,

00:04:57.665 --> 00:05:00.671
seven, eight, nine, 10 sides,

00:05:00.671 --> 00:05:03.404
so this, I can write the
perimeter of the polygon

00:05:03.404 --> 00:05:06.973
as 10 times b

00:05:06.973 --> 00:05:11.344
and then if I multiply
that times a over two,

00:05:11.344 --> 00:05:14.047
if I multiply that times,

00:05:14.047 --> 00:05:15.391
we'll use another color.

00:05:15.391 --> 00:05:17.448
a over,

00:05:17.448 --> 00:05:18.938
let me just write it like this,

00:05:18.938 --> 00:05:22.421
times a over two,

00:05:22.421 --> 00:05:25.792
I'm once again approximating
the area of the circle

00:05:25.792 --> 00:05:28.174
'cause a times b over two,

00:05:28.174 --> 00:05:31.394
that's the area of each of these triangles

00:05:31.394 --> 00:05:33.933
and then I have 10 of these triangles,

00:05:33.933 --> 00:05:36.309
but now let's think about
this more generally.

00:05:36.309 --> 00:05:39.038
Let's think about it if I were to have n,

00:05:39.038 --> 00:05:41.534
if I were to have an n-sided polygon,

00:05:41.534 --> 00:05:43.679
so I have n-sided polygon,

00:05:43.679 --> 00:05:46.366
then I'd be approximating the area

00:05:46.366 --> 00:05:50.480
as n times b,

00:05:50.480 --> 00:05:52.441
n times b,

00:05:52.441 --> 00:05:53.319
we see this right over here.

00:05:53.319 --> 00:05:55.665
When n is equal to 10,
you have 10 times b.

00:05:55.665 --> 00:05:58.191
So it's n times b

00:05:58.191 --> 00:06:01.861
times a over two.

00:06:01.861 --> 00:06:04.373
Times a over two.

00:06:04.373 --> 00:06:06.005
I just wrote,

00:06:06.005 --> 00:06:07.697
this isn't something mysterious.

00:06:07.697 --> 00:06:11.253
The base times the height divided by two,

00:06:11.253 --> 00:06:12.197
this right over here,

00:06:12.197 --> 00:06:14.517
that's the area of each triangle

00:06:14.517 --> 00:06:16.949
and then I'm gonna have
n of these triangles,

00:06:16.949 --> 00:06:19.856
so this is our approximation for the area,

00:06:19.856 --> 00:06:20.887
so let me write this,

00:06:20.887 --> 00:06:24.367
the area is going to be
approximately that right over there.

00:06:24.367 --> 00:06:27.656
It's going to be n, the
number of triangles I have,

00:06:27.656 --> 00:06:29.685
times the area of each triangle.

00:06:29.685 --> 00:06:30.881
Now what's going to happen,

00:06:30.881 --> 00:06:33.046
as n approaches infinity,

00:06:33.046 --> 00:06:36.546
as I approach having an
infinite sided polygon,

00:06:36.546 --> 00:06:38.304
as I have an infinite number of triangles,

00:06:38.304 --> 00:06:40.085
so let's just think this
through a little bit.

00:06:40.085 --> 00:06:41.216
'cause this is where it gets interesting.

00:06:41.216 --> 00:06:43.106
Now this is the informal argument.

00:06:43.106 --> 00:06:46.378
To do this better, I'd have to dig out

00:06:46.378 --> 00:06:49.032
a little bit of calculus, but
this gives you the essence.

00:06:49.032 --> 00:06:50.844
So let's just think about what happens

00:06:50.844 --> 00:06:54.069
as n approaches infinity.

00:06:54.069 --> 00:06:56.275
So, as n approaches infinity,

00:06:56.275 --> 00:06:58.738
we've already said, as we
have more and more sides

00:06:58.738 --> 00:07:01.250
and we have more and more triangles,

00:07:01.250 --> 00:07:03.221
a approaches r,

00:07:03.221 --> 00:07:05.122
so let's write that down.

00:07:05.122 --> 00:07:07.773
So, a is going to approach r,

00:07:07.773 --> 00:07:11.785
the height of the triangles is
going to approach the radius

00:07:11.785 --> 00:07:14.257
and what else is going to happen?

00:07:14.257 --> 00:07:18.309
Well, n times b, the
perimeter of the polygon,

00:07:18.309 --> 00:07:19.850
the perimeter of the polygon

00:07:19.850 --> 00:07:22.102
is going to approach the circumference.

00:07:22.102 --> 00:07:23.903
So, a is going to approach r

00:07:23.903 --> 00:07:28.385
and n times b is going to
approach the circumference,

00:07:28.385 --> 00:07:30.274
is going to approach the circumference,

00:07:30.274 --> 00:07:31.816
or another way of thinking about it,

00:07:31.816 --> 00:07:33.748
if it's approaching the circumference,

00:07:33.748 --> 00:07:37.479
we could say that n times b

00:07:37.479 --> 00:07:40.031
is going to approach
two Pi times the radius

00:07:40.031 --> 00:07:41.573
'cause that's what the circumference

00:07:41.573 --> 00:07:44.163
is going to be equal to.

00:07:44.163 --> 00:07:47.245
So, if a is approaching the radius

00:07:47.245 --> 00:07:50.736
and nb is approaching two Pi r,

00:07:50.736 --> 00:07:53.667
well then, what is the entire,

00:07:53.667 --> 00:07:56.668
what is the area of,

00:07:56.668 --> 00:07:59.217
what is the area of our polygon

00:07:59.217 --> 00:08:03.859
or what is the area of
our polygon going to,

00:08:03.859 --> 00:08:06.221
or, the area of our circle going to be?

00:08:06.221 --> 00:08:10.452
Well, it is going to approach,

00:08:10.452 --> 00:08:13.104
it is going, or I should
say, the area of our polygon

00:08:13.104 --> 00:08:14.515
is going to approach,

00:08:14.515 --> 00:08:17.417
nb is going to approach two Pi r,

00:08:17.417 --> 00:08:19.419
nb is going to approach two Pi r.

00:08:19.419 --> 00:08:21.901
Instead of nb, I'm writing two Pi r there.

00:08:21.901 --> 00:08:24.142
a is going to approach r.

00:08:24.142 --> 00:08:26.433
a is going to approach r,

00:08:26.433 --> 00:08:28.725
and then I'm dividing it,

00:08:28.725 --> 00:08:31.725
and then I am dividing it by two.

00:08:31.725 --> 00:08:33.792
So as n approaches infinity,

00:08:33.792 --> 00:08:35.954
as we have infinite number
of sides of our polygon,

00:08:35.954 --> 00:08:37.726
an infinite number of triangles,

00:08:37.726 --> 00:08:40.428
the area of our polygon
will approach this,

00:08:40.428 --> 00:08:42.129
which is equal to what?

00:08:42.129 --> 00:08:43.720
Well you have two divided two

00:08:43.720 --> 00:08:45.811
and then Pi r times r

00:08:45.811 --> 00:08:49.662
is equal to Pi r squared.

00:08:49.662 --> 00:08:53.474
So as we approach having
infinite number of triangles

00:08:53.474 --> 00:08:55.326
and infinite, I wanna keep that there,

00:08:55.326 --> 00:08:57.107
infinite number of sides,

00:08:57.107 --> 00:09:00.219
we see that we approach
the area of the circle

00:09:00.219 --> 00:09:02.341
and as we approach the area of the circle,

00:09:02.341 --> 00:09:04.161
we are approaching Pi r squared.

00:09:04.161 --> 00:09:06.023
So hopefully this gives
you an intuitive sense

00:09:06.023 --> 00:09:07.414
why this right over here

00:09:07.414 --> 00:09:11.759
is the formula for the area of a circle.

00:09:11.759 --> 00:09:14.136
You could think about it as the area

00:09:14.136 --> 00:09:17.908
of an infinite sided polygon

00:09:17.908 --> 00:09:19.470
that is inscribed in the circle,

00:09:19.470 --> 00:09:24.042
which will be equal to
the area of the circle.


Radius & diameter from circumference

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:02.676
- [Voiceover] Let's say that
we know that the circumference

00:00:04.921 --> 00:00:09.653
of a circle is 49 pi.

00:00:10.282 --> 00:00:12.499
Based on that, let's
see if we can figure out

00:00:12.499 --> 00:00:16.313
what the radius of that
same circle is going to be.

00:00:16.313 --> 00:00:17.811
And I encourage you,

00:00:17.811 --> 00:00:18.984
and I'll write equals here.

00:00:18.984 --> 00:00:20.586
And I encourage you to pause the video,

00:00:20.586 --> 00:00:22.481
and see if you can figure
it out on your own.

00:00:23.229 --> 00:00:25.893
Let's just draw the circle
to help visualize it.

00:00:27.361 --> 00:00:28.987
I'll just do a hand-drawn circle,

00:00:28.987 --> 00:00:32.191
clearly not a perfect
circle right over here.

00:00:32.191 --> 00:00:36.257
We know that if its radius is of length r,

00:00:37.201 --> 00:00:38.624
that the circumference

00:00:40.069 --> 00:00:42.450
is going to be two pi times r.

00:00:42.955 --> 00:00:44.690
So, I could write the circumference

00:00:44.690 --> 00:00:48.255
is equal to two pi times r.

00:00:48.597 --> 00:00:50.582
In fact, the number pi,

00:00:52.322 --> 00:00:54.104
the standard definition for it,

00:00:54.104 --> 00:00:58.991
is just the ratio between the
circumference and the diameter

00:00:58.991 --> 00:00:59.925
of a circle.

00:00:59.925 --> 00:01:00.843
Now, why is that?

00:01:00.843 --> 00:01:03.222
Well, if the diameter here

00:01:04.819 --> 00:01:06.641
is two r, right?

00:01:06.641 --> 00:01:08.522
We have r and then have another r.

00:01:08.522 --> 00:01:12.594
We see that the circumference
is pi times two r,

00:01:13.122 --> 00:01:16.187
or we can say that the ratio
between the circumference

00:01:16.187 --> 00:01:17.185
and the diameter,

00:01:19.555 --> 00:01:23.100
which is the ratio between c and two r,

00:01:23.686 --> 00:01:26.513
that's just going to be pi.

00:01:26.785 --> 00:01:29.702
Anyway, I've gone on longer than I need to

00:01:29.702 --> 00:01:31.248
just to solve this problem.

00:01:32.054 --> 00:01:34.237
We can go to this original formula here,

00:01:34.237 --> 00:01:36.768
saying the circumference
is two pi times r,

00:01:36.768 --> 00:01:39.974
and we can just substitute in
49 pi for the circumference.

00:01:40.583 --> 00:01:42.755
So, we could say 49 pi

00:01:43.446 --> 00:01:45.948
is going to be equal to
two pi times the radius.

00:01:48.542 --> 00:01:51.084
Now, let's see, we can
divide both sides by two pi

00:01:51.084 --> 00:01:52.669
to solve for r.

00:01:53.418 --> 00:01:55.503
So, dividing both sides by two pi.

00:01:55.503 --> 00:01:57.720
On the right-hand side,
the two pis cancel out.

00:01:57.720 --> 00:02:01.075
On the left-hand side, pi
divided by pi cancels out.

00:02:01.075 --> 00:02:05.168
49 divided by two is 24.5.

00:02:05.881 --> 00:02:09.585
So, if the circumference
is 49 pi whatever units,

00:02:09.585 --> 00:02:13.068
then the radius is going to be 24.5

00:02:13.433 --> 00:02:14.967
of those units.

00:02:15.460 --> 00:02:16.609
Let's do one more of these.

00:02:16.609 --> 00:02:18.212
Let's say that we have a circle

00:02:18.212 --> 00:02:19.930
whose circumference,

00:02:19.930 --> 00:02:23.577
I'll just say C, is equal to 1600 pi.

00:02:24.139 --> 00:02:26.410
My question is what is the diameter?

00:02:26.410 --> 00:02:29.280
The diameter of the
circle is equal to what?

00:02:29.889 --> 00:02:31.920
Just as we said that the circumference

00:02:31.920 --> 00:02:34.625
could be written as two pi r

00:02:34.625 --> 00:02:37.249
or as pi times two r,

00:02:37.249 --> 00:02:38.781
two r is just the diameter.

00:02:38.781 --> 00:02:41.742
So, we could say that the
circumference is equal to

00:02:41.742 --> 00:02:44.263
pi times the diameter.

00:02:44.651 --> 00:02:47.194
Once again, that comes out of
that traditional definition

00:02:47.194 --> 00:02:49.272
of pi as the ratio
between the circumference

00:02:49.272 --> 00:02:50.038
and the diameter.

00:02:50.038 --> 00:02:52.093
You could say that the ratio
between the circumference

00:02:52.093 --> 00:02:55.134
and the diameter is equal to pi.

00:02:55.134 --> 00:02:58.338
Circles are this very fundamental
thing in the universe,

00:02:58.338 --> 00:02:59.673
and you take the ratio

00:02:59.673 --> 00:03:01.972
of the circumference and the diameter,

00:03:01.972 --> 00:03:04.479
you get this magical and
mystical number that we see

00:03:04.479 --> 00:03:06.615
that keeps popping up in mathematics.

00:03:06.615 --> 00:03:08.081
Anyway, back to the problem.

00:03:08.771 --> 00:03:11.267
If we say the circumference is 1600 pi,

00:03:13.077 --> 00:03:15.461
and this is equal to
pi times the diameter,

00:03:15.461 --> 00:03:17.727
we can just divide both sides by pi

00:03:20.260 --> 00:03:21.453
to get the diameter,

00:03:21.667 --> 00:03:23.312
which is going to be 1600.

00:03:23.312 --> 00:03:25.831
The circumference is 1600 pi units,

00:03:25.831 --> 00:03:27.630
whatever units those are, maybe meters.

00:03:27.630 --> 00:03:30.033
Then, the diameter is
just going to be 1600

00:03:30.033 --> 00:03:32.799
of those units, or in
this case, maybe meters.

00:03:33.628 --> 00:03:34.679
And we're all done.


Tangents of circles problem (example 1)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.187
- [Voiceover] So we're
told that angle A is

00:00:02.187 --> 00:00:05.316
circumscribed about circle O.

00:00:05.316 --> 00:00:07.937
So this is angle A right over here,

00:00:07.937 --> 00:00:10.639
we're talking about this
angle right over there.

00:00:10.639 --> 00:00:14.479
And when they say it's
circumscribed about circle O

00:00:14.479 --> 00:00:16.368
that means that the two sides of the angle

00:00:16.368 --> 00:00:19.578
they're segments that would
be part of tangent lines,

00:00:19.578 --> 00:00:23.195
so if we were to continue, so for example

00:00:23.195 --> 00:00:27.312
that right over there, that
line is tangent to the circle

00:00:27.312 --> 00:00:30.762
and (mumbles)

00:00:30.762 --> 00:00:34.769
and this line is also
tangent to the circle.

00:00:34.769 --> 00:00:37.146
So you see that the sides of angle A

00:00:37.146 --> 00:00:41.291
are parts of those tangents,
and points B and point C

00:00:41.291 --> 00:00:45.435
are where those tangents
actually sit on the circle.

00:00:45.435 --> 00:00:47.345
So given all of that they're asking us

00:00:47.345 --> 00:00:49.174
what is the measure of angle A,

00:00:49.174 --> 00:00:50.961
so we're trying to figure
this out right here

00:00:50.961 --> 00:00:52.363
and I encourage you to pause the video

00:00:52.363 --> 00:00:54.395
and figure it out on your own.

00:00:54.395 --> 00:00:57.402
So the key insight here
and there's multiple ways

00:00:57.402 --> 00:00:59.556
that you could approach
this is to realize,

00:00:59.556 --> 00:01:04.228
that a radius is going to be
perpendicular to a tangent

00:01:04.228 --> 00:01:06.453
or a radius that intersects a tangent

00:01:06.453 --> 00:01:08.351
is going to be perpendicular to it.

00:01:08.351 --> 00:01:11.675
So let me label this,

00:01:11.675 --> 00:01:14.175
so this is going to be a right angle

00:01:14.175 --> 00:01:16.944
and this is going to be a right angle.

00:01:16.944 --> 00:01:19.946
In any quadrilateral,
the sum of the angles

00:01:19.946 --> 00:01:22.283
are going to add up to 360 degrees,

00:01:22.283 --> 00:01:24.477
and if you wonder where that comes from

00:01:24.477 --> 00:01:27.545
well you can divide a
quadrilateral into two triangles,

00:01:27.545 --> 00:01:29.353
where the sum of all the interior angles

00:01:29.353 --> 00:01:31.385
of a triangle are 180 and
since you have two of them,

00:01:31.385 --> 00:01:34.493
it's 360 degrees.

00:01:34.493 --> 00:01:39.146
So this is 92 + 90 + 90 + question mark

00:01:39.146 --> 00:01:41.503
is going to be equal to 360 degrees.

00:01:41.503 --> 00:01:43.333
So let me write that down.

00:01:43.333 --> 00:01:47.203
92 plus 90 plus 90 so plus 180.

00:01:48.103 --> 00:01:51.212
Plus, let's just call this X. X degrees.

00:01:51.212 --> 00:01:55.093
Plus X, they all have to
add up to be 360 degrees

00:01:55.093 --> 00:01:59.110
so let's see, we could
subtract 180 from both sides

00:01:59.110 --> 00:02:00.918
and so, if we do that we would have

00:02:00.918 --> 00:02:04.751
92 plus X is equals to 180

00:02:04.751 --> 00:02:07.520
and if we subtract 92 from both sides

00:02:07.520 --> 00:02:11.807
we get X is equals to let's
see 180 minus 90 would be 90

00:02:11.807 --> 00:02:13.087
and then we're just going
to subtract two more

00:02:13.087 --> 00:02:16.004
so it's going to be X is equals to 88.

00:02:16.004 --> 00:02:20.259
So the measure of angle A is 88 degrees.

00:02:20.259 --> 00:02:22.046
Now let's do one more of these.

00:02:22.046 --> 00:02:25.399
These are surprisingly fun. All right.

00:02:25.399 --> 00:02:30.399
So it says, angle A is
circumscribed about circle O,

00:02:30.417 --> 00:02:32.368
we have seen that before
in the last question

00:02:32.368 --> 00:02:34.498
and they said what is
the measure of angle D.

00:02:34.498 --> 00:02:38.808
So we want to find, let me make
sure I'm on the right layer.

00:02:38.808 --> 00:02:42.546
We want to find the measure of that angle

00:02:42.546 --> 00:02:45.459
and let's call that,
let's call that x again.

00:02:45.459 --> 00:02:47.719
So what can we figure out?

00:02:47.719 --> 00:02:49.149
Well just like in the last question

00:02:49.149 --> 00:02:52.644
we have a quadrilateral here,
quadrilateral A, B, O, C

00:02:52.644 --> 00:02:53.883
and we know two of the angles,

00:02:53.883 --> 00:02:55.163
we know this is going to be a right angle

00:02:55.163 --> 00:02:57.134
we have a radius
intersecting with a tangent,

00:02:57.134 --> 00:02:59.775
or part of a tangent
I guess you could say.

00:02:59.775 --> 00:03:01.604
And then this would be a right angle

00:03:01.604 --> 00:03:04.692
and so by the same logic as
we saw in the last question,

00:03:04.692 --> 00:03:07.658
this angle, plus this
angle, plus this angle,

00:03:07.658 --> 00:03:11.661
plus the central angle are
going to add up to 360 degrees.

00:03:11.661 --> 00:03:14.017
So let's call the measure
of the central angle

00:03:14.017 --> 00:03:16.433
let's call that Y over there.

00:03:16.433 --> 00:03:20.844
So we have Y plus 80 degrees or

00:03:20.844 --> 00:03:22.490
we'll just assume
everything is in degrees,

00:03:22.490 --> 00:03:27.222
so Y plus 80 plus 90
plus 90, so I could say

00:03:27.222 --> 00:03:32.120
plus another 180 is going
to be equal to 360 degrees

00:03:32.120 --> 00:03:35.490
sum of the interior
angles of a quadrilateral.

00:03:35.490 --> 00:03:38.617
And so let's see you have, we could have

00:03:38.617 --> 00:03:42.047
Y plus 80 is equal to 180,

00:03:42.047 --> 00:03:43.836
if I just subtract 180 from both sides

00:03:43.836 --> 00:03:47.971
if I subtract 80 from both
sides, we get Y is equal to 100

00:03:47.971 --> 00:03:50.547
or the measure of this is 100,

00:03:50.547 --> 00:03:52.471
the measure of this interior angle

00:03:52.471 --> 00:03:54.564
right over here is 100 degrees,

00:03:54.564 --> 00:03:58.251
which also tells us that
the measure of this arc

00:03:58.251 --> 00:04:01.201
cause that interior angle intercepts...

00:04:01.201 --> 00:04:04.491
intercepts arc C, B, right over here,

00:04:04.491 --> 00:04:05.911
that tells us that the measure

00:04:05.911 --> 00:04:08.674
of arc C, B, is also 100 degrees.

00:04:08.674 --> 00:04:10.726
And so if we're trying to find this angle,

00:04:10.726 --> 00:04:14.200
the measure of angle D,
that's an inscribed angle

00:04:14.200 --> 00:04:16.683
that intercepts the same arc.

00:04:16.683 --> 00:04:19.331
And we've seen in previous
videos that an inscribed angle

00:04:19.331 --> 00:04:20.722
that intercepts that arc is going to have

00:04:20.722 --> 00:04:22.266
half the arc's measure.

00:04:22.266 --> 00:04:24.765
So if this is a 100 degree measured arc,

00:04:24.765 --> 00:04:26.126
then the measure of this angle

00:04:26.126 --> 00:04:29.519
right over here is going to be 50 degrees.

00:04:29.519 --> 00:04:32.608
So the measure of angle D is 50 degrees.

00:04:32.608 --> 00:04:35.554
And we are done.


Proof: Segments tangent to circle from outside point are congruent

WEBVTT
Kind: captions
Language: en

00:00:00.086 --> 00:00:01.490
- [Voiceover] So I have a circle here,

00:00:01.490 --> 00:00:05.345
with a center at point O and
let's pick an arbitrary point

00:00:05.345 --> 00:00:07.109
that sits outside of the circle.

00:00:07.109 --> 00:00:09.524
So let me just pick this
point right over here.

00:00:09.524 --> 00:00:11.010
Point A.

00:00:11.010 --> 00:00:13.726
And if I have an arbitrary
point outside of the circle,

00:00:13.726 --> 00:00:16.083
I can actually draw two
different tangent lines

00:00:16.083 --> 00:00:19.949
that contain A, that are
tangent to this circle.

00:00:19.949 --> 00:00:21.272
Let me draw them.

00:00:21.272 --> 00:00:24.835
So one of them would look like this.

00:00:24.835 --> 00:00:27.464
Let me just start right over here.

00:00:27.464 --> 00:00:29.208
So I make a tangent to the circle.

00:00:29.208 --> 00:00:31.604
So it could look like that.

00:00:31.604 --> 00:00:35.841
And then the other one
would look like this,

00:00:35.841 --> 00:00:38.105
would look like,

00:00:38.105 --> 00:00:39.997
would look like that.

00:00:39.997 --> 00:00:41.182
Now let's say that the point

00:00:41.182 --> 00:00:44.362
that the tangent lines
intersect the circle,

00:00:44.362 --> 00:00:49.238
let's say this point right
over here, is point B,

00:00:49.238 --> 00:00:53.522
and this point right over here is point C.

00:00:53.522 --> 00:00:55.716
This is point C. Right over here.

00:00:55.716 --> 00:00:57.283
What I want to prove

00:00:57.283 --> 00:01:01.962
What I want to prove
is that the segment AB

00:01:01.962 --> 00:01:04.414
is congruent to the segment AC,

00:01:04.414 --> 00:01:07.825
or another way of thinking
about it, I want to prove,

00:01:07.825 --> 00:01:10.588
let me do this in a new color
that I haven't used yet,

00:01:10.588 --> 00:01:15.588
I want to prove that this
segment, right over here,

00:01:16.590 --> 00:01:20.084
is congruent to this segment...

00:01:20.084 --> 00:01:23.683
is congruent to that segment.

00:01:23.683 --> 00:01:25.888
I encourage you to pause the video

00:01:25.888 --> 00:01:29.661
and try to work it out on your own

00:01:29.661 --> 00:01:32.006
before I work through it.

00:01:32.006 --> 00:01:34.467
Alright, now let's try to
work through this together.

00:01:34.467 --> 00:01:35.490
And to work through this together,

00:01:35.490 --> 00:01:38.394
I'm going to actually
set up two triangles.

00:01:38.394 --> 00:01:39.532
Two triangles.

00:01:39.532 --> 00:01:40.735
And they're going to be right triangles

00:01:40.735 --> 00:01:41.916
as we'll see in a second.

00:01:41.916 --> 00:01:46.246
So let me draw some lines
here to set up our triangles.

00:01:46.246 --> 00:01:47.907
So I'm going to draw one line,

00:01:47.907 --> 00:01:52.907
just like that, and then we draw that.

00:01:53.241 --> 00:01:56.583
And then, let me draw that.

00:01:56.583 --> 00:01:59.071
Now what do we know

00:01:59.071 --> 00:02:00.472
about these triangles?

00:02:00.472 --> 00:02:01.435
Well, as I mentioned,

00:02:01.435 --> 00:02:02.979
we're going to be dealing
with right triangles.

00:02:02.979 --> 00:02:04.012
How do I know that?

00:02:04.012 --> 00:02:05.115
Well, in a previous video, we saw that

00:02:05.115 --> 00:02:09.039
if we have a radius
intersecting a tangent line,

00:02:09.039 --> 00:02:11.999
that they intersect at right angles.

00:02:11.999 --> 00:02:13.636
We've proven this.

00:02:13.636 --> 00:02:15.253
So that's a radius. That's a tangent line.

00:02:15.253 --> 00:02:16.949
They're going to intersect
at a right angle.

00:02:16.949 --> 00:02:21.906
So radius, tangent line, they
intersect at a right angle.

00:02:21.906 --> 00:02:26.341
We also know, since OB
and OC are both radii,

00:02:26.341 --> 00:02:28.500
that they're both the length
of the radius of the circle.

00:02:28.500 --> 00:02:30.705
So this side, right over here,

00:02:30.705 --> 00:02:32.598
let me do some new colors here,

00:02:32.598 --> 00:02:36.278
so this side is going to
be congruent to that side.

00:02:36.278 --> 00:02:39.586
And you can see that the
hypotenuse of both circles

00:02:39.586 --> 00:02:42.582
is the same side, side OA,

00:02:42.582 --> 00:02:44.848
so of course, it is equal to itself.

00:02:44.848 --> 00:02:46.367
This is equal to itself.

00:02:46.367 --> 00:02:51.185
So we see triangle ABO and triangle ACO,

00:02:51.185 --> 00:02:55.108
they're both right triangles
that have two sides in common,

00:02:55.108 --> 00:02:56.943
in particular, they both have a hypotenuse

00:02:56.943 --> 00:02:57.999
that are equal to each other

00:02:57.999 --> 00:03:00.379
and they both have a base, or a leg.

00:03:00.379 --> 00:03:03.200
And we know from Hypotenuse-Leg Congruence

00:03:03.200 --> 00:03:04.767
that if you have two right triangles,

00:03:04.767 --> 00:03:07.333
where the hypotenuses are equal,

00:03:07.333 --> 00:03:08.847
and you have a leg that are equal,

00:03:08.847 --> 00:03:12.307
then the both triangles
are going to be congruent.

00:03:12.307 --> 00:03:17.008
So triangle ABO is
congruent to triangle ACO.

00:03:17.008 --> 00:03:18.924
And in that proof where
we prove it, as well,

00:03:18.924 --> 00:03:20.897
the Pythagorean theorem tells us

00:03:20.897 --> 00:03:22.047
if you know two sides of a right triangle

00:03:22.047 --> 00:03:24.380
that determines what the third side is.

00:03:24.380 --> 00:03:26.981
So the third side, so length of AB

00:03:26.981 --> 00:03:29.418
is going to be the same
thing as the length of AC.

00:03:29.418 --> 00:03:30.986
Once again, it just comes out of,

00:03:30.986 --> 00:03:33.284
these are both right
triangles, if two sides,

00:03:33.284 --> 00:03:35.641
if two corresponding sides
of these two right triangles

00:03:35.641 --> 00:03:37.220
are congruent, then the third side has to,

00:03:37.220 --> 00:03:39.182
that comes straight from
the Pythagorean theorem.

00:03:39.182 --> 00:03:40.203
And there you go.

00:03:40.203 --> 00:03:42.722
We hopefully feel good about the fact that

00:03:42.722 --> 00:03:45.137
AB is going to be congruent to AC.

00:03:45.137 --> 00:03:45.984
Or another way to think about it,

00:03:45.984 --> 00:03:47.761
if I take a point outside of a circle,

00:03:47.761 --> 00:03:50.698
and I construct segments that
are tangent to the circle,

00:03:50.698 --> 00:03:52.323
that those two segments are going to be

00:03:52.323 --> 00:03:54.767
congruent to each other.