Hypotenuse leg congruence

WEBVTT
Kind: captions
Language: en

00:00:00.108 --> 00:00:02.448
- [Voiceover] In previous
videos on triangle congruence,

00:00:02.448 --> 00:00:06.824
we talk about how side,
side, angle cannot be used

00:00:06.824 --> 00:00:08.904
to establish triangle congruence.

00:00:08.904 --> 00:00:11.024
What I'll do in this video
is I'll show you an exception

00:00:11.024 --> 00:00:12.804
and the exception is when we are dealing

00:00:12.804 --> 00:00:15.024
with right triangles, in particular,

00:00:15.024 --> 00:00:18.424
when this angle is a right angle.

00:00:18.424 --> 00:00:22.984
So, this is a 90 degree
angle in which case,

00:00:22.984 --> 00:00:26.784
this first side is going
to be the hypotenuse

00:00:26.784 --> 00:00:31.524
and the second is going to be
a leg, is going to be a leg.

00:00:31.524 --> 00:00:33.364
And that's why you'll sometimes see this,

00:00:33.364 --> 00:00:35.766
instead of calling it, you
know, instead of calling it SSA

00:00:35.766 --> 00:00:38.166
for the special case of a right triangle,

00:00:38.166 --> 00:00:40.114
you'll sometimes call this, see this

00:00:40.114 --> 00:00:43.054
as hypotenuse leg congruence.

00:00:43.054 --> 00:00:46.272
Let me do this in the same
colors so I'm consistent.

00:00:46.272 --> 00:00:48.282
So, hypotenuse,

00:00:49.412 --> 00:00:52.652
hypotenuse leg congruence.

00:00:52.652 --> 00:00:53.972
And if you're dealing,

00:00:53.972 --> 00:00:56.326
if someone saying there's a
hypotenuse of that triangle,

00:00:56.326 --> 00:00:58.014
then that triangle by definition is going

00:00:58.014 --> 00:00:59.132
to be a right triangle.

00:00:59.132 --> 00:01:02.790
You can only have a hypotenuse
in a right triangle.

00:01:02.790 --> 00:01:05.950
And to get an intuition for
why two triangles are going

00:01:05.950 --> 00:01:08.050
to be congruent if their hypotenuse,

00:01:08.050 --> 00:01:10.610
if each of their hypotenuses
have the same length.

00:01:10.610 --> 00:01:13.590
Is it hypoteni, hypotenuses?

00:01:13.590 --> 00:01:15.130
If they have the same length,

00:01:15.130 --> 00:01:17.410
and then the leg has the same length,

00:01:17.410 --> 00:01:19.630
we can look at these two right over here.

00:01:19.630 --> 00:01:24.010
So, we're going to assume
that this hypotenuse,

00:01:24.010 --> 00:01:27.280
that this hypotenuse has the same length

00:01:27.280 --> 00:01:29.640
as that hypotenuse.

00:01:29.640 --> 00:01:32.118
We'll say they're both of length c.

00:01:32.118 --> 00:01:34.498
We can assume that this leg here

00:01:34.498 --> 00:01:37.298
has the same length.

00:01:37.298 --> 00:01:40.294
This leg here has the
same length as that leg.

00:01:40.294 --> 00:01:43.354
We could call that length a,

00:01:43.354 --> 00:01:46.594
and you can see that the only
way to draw the next side.

00:01:46.594 --> 00:01:48.424
Remember, there's two
constraints now here.

00:01:48.424 --> 00:01:49.992
We have to connect to
this point right over here

00:01:49.992 --> 00:01:51.470
as we're dealing with the triangle.

00:01:51.470 --> 00:01:53.410
So, we're only going to have three sides,

00:01:53.410 --> 00:01:55.330
and it's a right
triangle, so it has to be,

00:01:55.330 --> 00:01:58.030
the next side I draw
has to be perpendicular.

00:01:58.030 --> 00:02:00.210
So, the only way to draw that,

00:02:00.210 --> 00:02:03.190
the only way to draw
that is just like that.

00:02:03.190 --> 00:02:06.530
And to feel good that, I
guess more analytically,

00:02:06.530 --> 00:02:08.978
that this side, this green
side is congruent to this side,

00:02:08.978 --> 00:02:10.658
we can just use the Pythagorean theorem.

00:02:10.658 --> 00:02:13.038
The Pythagorean theorem would tell us

00:02:13.038 --> 00:02:16.098
that this green side,
right over here, is going

00:02:16.098 --> 00:02:18.486
to be the square root

00:02:18.486 --> 00:02:22.566
of the hypotenuse-squared, c-squared.

00:02:22.566 --> 00:02:26.294
c-squared minus a-squared,
minus the square

00:02:26.294 --> 00:02:28.914
of the other leg, and
this one is also going

00:02:28.914 --> 00:02:31.274
to be the square root of c-squared.

00:02:31.274 --> 00:02:32.674
Actually, I'm just gonna draw an arrow.

00:02:32.674 --> 00:02:33.854
This is gonna be true for that

00:02:33.854 --> 00:02:37.300
and it's going to be true from that.

00:02:37.300 --> 00:02:39.710
That comes straight out of
the Pythagorean theorem.

00:02:39.710 --> 00:02:43.010
So, from that, we know that the third leg

00:02:43.010 --> 00:02:45.950
or the third side, that they,

00:02:45.950 --> 00:02:49.490
those two corresponding
sides have to be congruent.

00:02:49.490 --> 00:02:50.520
And so, that's where we get

00:02:50.520 --> 00:02:52.560
the hypotenuse leg congruence from,

00:02:52.560 --> 00:02:54.830
which is really just
a special case of SSA,

00:02:54.830 --> 00:02:57.204
which we've said in the past,
you cannot apply generally

00:02:57.204 --> 00:02:59.544
to establish triangle congruence.


Proof: Radius is perpendicular to tangent line

WEBVTT
Kind: captions
Language: en

00:00:00.702 --> 00:00:02.883
- [Voiceover] So what
we have here is a circle

00:00:02.883 --> 00:00:05.400
with the center at Point O,

00:00:05.400 --> 00:00:07.457
and then we have a tangent
line to the circle.

00:00:07.457 --> 00:00:09.431
Let me actually label this line.

00:00:09.431 --> 00:00:11.776
Let's call this Line L.

00:00:11.776 --> 00:00:15.066
And we see at Point A is the point

00:00:15.066 --> 00:00:18.469
that the tangent line
intersects with the circle,

00:00:18.469 --> 00:00:20.120
and then we've drawn a radius

00:00:20.120 --> 00:00:23.130
from the center of the circle to Point A.

00:00:23.130 --> 00:00:25.268
Now what we want to do in this video

00:00:25.268 --> 00:00:28.865
is prove to ourselves that this radius

00:00:28.865 --> 00:00:33.370
and that this tangent line
intersect at a right angle.

00:00:33.370 --> 00:00:34.879
We want to prove to ourselves

00:00:34.879 --> 00:00:37.774
that they intersect at a right angle.

00:00:37.774 --> 00:00:40.174
And the first step to doing that

00:00:40.174 --> 00:00:41.477
is we're going to feel good,

00:00:41.477 --> 00:00:44.497
we're going to prove to
ourselves that Point A

00:00:44.497 --> 00:00:48.438
is the closest point on Line L

00:00:48.438 --> 00:00:51.168
to the center of our circle.

00:00:51.168 --> 00:00:53.333
So I want to prove,

00:00:53.333 --> 00:00:57.677
prove that Point A

00:00:57.677 --> 00:01:01.845
is closest, closest point

00:01:02.628 --> 00:01:06.268
on L to Point O.

00:01:06.268 --> 00:01:08.010
To Point O.

00:01:08.010 --> 00:01:09.580
And I encourage you to pause the video

00:01:09.580 --> 00:01:12.893
and see if you can prove that to yourself.

00:01:12.893 --> 00:01:14.607
Well, to think about
that, just think about

00:01:14.607 --> 00:01:16.592
any other point on Line L.

00:01:16.592 --> 00:01:19.109
Pick any other arbitrary point on Line L.

00:01:19.109 --> 00:01:20.615
It could be this point right over here.

00:01:20.615 --> 00:01:22.172
It could be this point right over here.

00:01:22.172 --> 00:01:23.458
It could be this point right over here.

00:01:23.458 --> 00:01:27.162
And you immediately see that
it sits outside of the circle.

00:01:27.162 --> 00:01:28.792
And if it's sitting outside of the circle,

00:01:28.792 --> 00:01:30.071
I'll pick this point here,

00:01:30.071 --> 00:01:32.810
just so it'll become a little
bit clearer on our diagram,

00:01:32.810 --> 00:01:35.227
if it's sitting outside of the circle,

00:01:35.227 --> 00:01:37.918
in order to get from
Point O to this point,

00:01:37.918 --> 00:01:41.244
I'm going to call this
Point B right over here,

00:01:41.244 --> 00:01:44.792
you have to go the length of the radius,

00:01:44.792 --> 00:01:46.939
you have to go the length of the radius,

00:01:46.939 --> 00:01:49.123
and then you have to go some more.

00:01:49.123 --> 00:01:52.049
So this length, the length of Segment O B,

00:01:52.049 --> 00:01:55.077
is clearly going to be longer
than the length of the radius,

00:01:55.077 --> 00:01:56.511
because you have to go to the radius

00:01:56.511 --> 00:01:58.630
to get to the circle itself

00:01:58.630 --> 00:02:00.348
and then you have to
go a little bit further

00:02:00.348 --> 00:02:02.645
for any point that sits
outside of the circle.

00:02:02.645 --> 00:02:04.652
So Point A is the only point,

00:02:04.652 --> 00:02:06.292
by definition this is a tangent line,

00:02:06.292 --> 00:02:08.660
it's the only point
that sits on the circle.

00:02:08.660 --> 00:02:12.860
Every other point on Line L
sits outside of the circle,

00:02:12.860 --> 00:02:14.581
so it's going to be further.

00:02:14.581 --> 00:02:17.507
So Point A, hopefully
this makes you feel good,

00:02:17.507 --> 00:02:18.645
because you pick any other point,

00:02:18.645 --> 00:02:19.805
it's going to sit outside of the circle,

00:02:19.805 --> 00:02:21.663
so you have to go to the
radius and then some.

00:02:21.663 --> 00:02:24.420
So hopefully this makes
you feel good that Point A

00:02:24.420 --> 00:02:29.247
is the closest point on L
to the center of the circle.

00:02:29.897 --> 00:02:33.039
Now, we're not done yet, we
now have to convince ourselves

00:02:33.039 --> 00:02:35.884
that if we have a point and a line,

00:02:35.884 --> 00:02:37.862
that the segment connecting the point

00:02:37.862 --> 00:02:42.440
to the closest point on the
line, to that original point,

00:02:42.440 --> 00:02:45.362
that, that's going to be
perpendicular to the line.

00:02:45.362 --> 00:02:47.919
So let me give ourselves some space here.

00:02:47.919 --> 00:02:51.862
We want to prove, we want to prove

00:02:51.862 --> 00:02:55.095
that if we, that the segment,

00:02:55.741 --> 00:02:59.022
segment connecting,

00:02:59.022 --> 00:03:03.940
connecting a point off the line,

00:03:04.470 --> 00:03:06.305
off the line,

00:03:06.305 --> 00:03:08.916
and closest point on the line,

00:03:08.916 --> 00:03:12.580
closest point on the line,

00:03:13.381 --> 00:03:15.485
is perpendicular,

00:03:15.485 --> 00:03:19.772
is perpendicular to the line.

00:03:21.102 --> 00:03:23.239
So what we want to do,
is we want to say, hey,

00:03:23.239 --> 00:03:25.204
if we have some line here,

00:03:25.204 --> 00:03:27.326
if we have some line here, L,

00:03:27.326 --> 00:03:30.888
and if you were to take
a point off the line,

00:03:30.888 --> 00:03:33.829
so let's say, so you wanted
a point off the line,

00:03:33.829 --> 00:03:36.246
so let's say that, that's
this point right over here,

00:03:36.246 --> 00:03:39.423
Point O, and you want the
segment connecting the point

00:03:39.423 --> 00:03:42.984
off the line to the
closest point on the line.

00:03:42.984 --> 00:03:45.861
So the closest point on the line,

00:03:45.861 --> 00:03:49.407
so let's say that this is the
closest point on the line,

00:03:49.407 --> 00:03:52.594
we want to feel good that
this segment connecting them,

00:03:52.594 --> 00:03:54.710
so let me do this in a new color,

00:03:54.710 --> 00:03:56.211
that the segment connecting them

00:03:56.211 --> 00:03:58.758
is going to be perpendicular to the line,

00:03:58.758 --> 00:04:00.947
that it goes just straight down like that,

00:04:00.947 --> 00:04:03.761
that it's going to be perpendicular.

00:04:03.761 --> 00:04:06.184
And I'm gonna prove this by contradiction.

00:04:06.184 --> 00:04:07.984
I'm going to assume that
it's not perpendicular.

00:04:07.984 --> 00:04:10.573
So assume,

00:04:10.573 --> 00:04:13.811
assume that the segment,

00:04:14.531 --> 00:04:16.322
segment connecting,

00:04:17.363 --> 00:04:19.151
connecting, this is kind of wordy,

00:04:19.151 --> 00:04:23.536
a point off line

00:04:24.183 --> 00:04:27.283
and closest point to the line,

00:04:27.283 --> 00:04:30.159
closest point to line,

00:04:31.899 --> 00:04:35.877
is not, is not perpendicular

00:04:36.510 --> 00:04:39.480
to the line.

00:04:39.480 --> 00:04:41.536
So how can I visualize that?

00:04:41.536 --> 00:04:44.592
Well, I could draw my
line right over here.

00:04:44.592 --> 00:04:47.136
So that's Line L,

00:04:47.136 --> 00:04:51.118
and let's say I have my
Point O right over here.

00:04:51.118 --> 00:04:54.630
Point O, and let's say the
closest point on Line L

00:04:54.630 --> 00:04:57.570
to Point O, let's say that it's not,

00:04:57.570 --> 00:04:58.902
so let's say it's over here,

00:04:58.902 --> 00:05:00.783
that if I were to
connect these two points,

00:05:00.783 --> 00:05:03.035
that it's not perpendicular to Line L.

00:05:03.035 --> 00:05:04.405
So this is the closest point,

00:05:04.405 --> 00:05:06.364
let's call this Point A,

00:05:06.364 --> 00:05:08.514
and let's say that the
segment connecting these two

00:05:08.514 --> 00:05:11.580
is not perpendicular to the line.

00:05:11.580 --> 00:05:16.580
So let me get, so let's assume
this is not perpendicular.

00:05:16.618 --> 00:05:19.358
So this angle, this angle,

00:05:19.358 --> 00:05:24.327
this angle right here, is
not, is not 90 degrees.

00:05:25.217 --> 00:05:26.672
So if we assume that,

00:05:26.672 --> 00:05:28.940
the reason that this is going
to be a proof by contradiction

00:05:28.940 --> 00:05:31.474
is I can show that if
this is not 90 degrees,

00:05:31.474 --> 00:05:34.477
that I can always find a point
that is going to be closer,

00:05:34.477 --> 00:05:37.288
another point on Line L
that is going to be closer

00:05:37.288 --> 00:05:40.120
to Point O, which contradicts the fact

00:05:40.120 --> 00:05:42.158
that this was supposed
to be the closest point,

00:05:42.158 --> 00:05:44.937
A was supposed to be the
closest point on Line L to O.

00:05:44.937 --> 00:05:47.646
And how do I always find a closer point?

00:05:47.646 --> 00:05:49.823
Well, I construct a right triangle.

00:05:49.823 --> 00:05:52.519
I can construct a right triangle,

00:05:53.456 --> 00:05:55.323
just like this.

00:05:56.456 --> 00:05:59.991
I can construct a right
triangle like that,

00:05:59.991 --> 00:06:01.826
and we see that this distance,

00:06:01.826 --> 00:06:04.194
let's call this distance
right over here a,

00:06:04.194 --> 00:06:07.143
and we could call the
base of this triangle b,

00:06:07.143 --> 00:06:08.814
let me do this in a different color.

00:06:08.814 --> 00:06:13.738
So a, b, that's the base
of the right triangle,

00:06:13.738 --> 00:06:17.127
and the hypotenuse is
the distance from O to A.

00:06:17.127 --> 00:06:19.038
We could call that c.

00:06:19.038 --> 00:06:21.655
We know from the Pythagorean Theorem

00:06:21.655 --> 00:06:24.511
that a squared plus b squared,

00:06:25.111 --> 00:06:26.807
plus b squared,

00:06:26.807 --> 00:06:30.013
let me do this, plus b squared,

00:06:30.013 --> 00:06:32.440
is going to be equal to c squared.

00:06:32.440 --> 00:06:35.566
Is going to be equal to c squared.

00:06:35.566 --> 00:06:37.150
And so b squared,

00:06:37.150 --> 00:06:39.347
if we have a non-degenerate
triangle right over here,

00:06:39.347 --> 00:06:43.065
this is going to be some
positive value over here,

00:06:43.065 --> 00:06:45.939
and so a is going to be less than c.

00:06:45.939 --> 00:06:48.827
So this gets us to the conclusion,

00:06:48.827 --> 00:06:51.491
because if this is some
positive value here,

00:06:51.491 --> 00:06:53.070
and a and c are positive,

00:06:53.070 --> 00:06:54.626
everything are positive distances,

00:06:54.626 --> 00:06:57.245
then this tells us that
a has to be less than c.

00:06:57.245 --> 00:06:59.522
That a non-hypotenuse
side of a right triangle,

00:06:59.522 --> 00:07:01.080
of a non-degenerate right triangle,

00:07:01.080 --> 00:07:02.539
assuming it has some area,

00:07:02.539 --> 00:07:04.413
is going to be shorter
than the hypotenuse.

00:07:04.413 --> 00:07:06.573
The hypotenuse is the longest side.

00:07:06.573 --> 00:07:10.497
So a is going to be less than c,

00:07:10.497 --> 00:07:12.816
which would tell us, if a is less than c,

00:07:12.816 --> 00:07:14.693
that we've found another point,

00:07:14.693 --> 00:07:16.242
let's call this point, I don't know,

00:07:16.242 --> 00:07:17.574
I've used a lot of letters here,

00:07:17.574 --> 00:07:18.889
let's call that Point D.

00:07:18.889 --> 00:07:21.072
D is going to be closer.

00:07:21.072 --> 00:07:22.821
D is closer.

00:07:23.421 --> 00:07:27.560
D is closer.

00:07:27.796 --> 00:07:30.137
So we've just set up a contradiction.

00:07:30.137 --> 00:07:34.406
We assumed A was the closest
point on Line L to Point O,

00:07:34.406 --> 00:07:36.568
but we assumed the segment connecting them

00:07:36.568 --> 00:07:38.277
is not at a 90-degree angle.

00:07:38.277 --> 00:07:40.100
If it's not at a 90-degree angle,

00:07:40.100 --> 00:07:43.358
then we can drop a perpendicular
and find a closer point,

00:07:43.358 --> 00:07:45.266
which is a contradiction to the fact

00:07:45.266 --> 00:07:47.927
that A was supposed to
be the closer point.

00:07:47.927 --> 00:07:50.565
So this leads to a contradiction.

00:07:51.155 --> 00:07:53.457
Contradiction.

00:07:53.457 --> 00:07:56.911
Contradiction.

00:07:56.911 --> 00:07:58.927
Because you can actually
find that this is not

00:07:58.927 --> 00:08:01.860
the closest point, you can
always find a closer point,

00:08:01.860 --> 00:08:04.724
so therefore, the segment
connecting a point off the line

00:08:04.724 --> 00:08:08.872
to the closest point to the
line must be perpendicular.

00:08:08.872 --> 00:08:11.216
Must be perpendicular.

00:08:11.216 --> 00:08:15.039
So the segment connecting
a point off the line,

00:08:15.039 --> 00:08:17.676
to a closest point on the line,

00:08:17.676 --> 00:08:21.239
that must be perpendicular to the line.

00:08:21.239 --> 00:08:23.599
Must be perpendicular to the line.

00:08:23.599 --> 00:08:26.985
And just like that, we hopefully
feel good about the idea

00:08:26.985 --> 00:08:28.681
that if you have a radius,

00:08:28.681 --> 00:08:30.566
and the point at which it intersects

00:08:30.566 --> 00:08:32.890
a tangent line to the circle,

00:08:32.890 --> 00:08:35.884
that, that forms a 90-degree angle.

00:08:35.884 --> 00:08:38.999
The radius and the tangent line.


Noble gas configuration

WEBVTT
Kind: captions
Language: en

00:00:02.327 --> 00:00:04.277
- [Voiceover] In this video,
we're going to be talking about

00:00:04.277 --> 00:00:06.924
how you can write electron configurations

00:00:06.924 --> 00:00:09.455
using noble gas notation.

00:00:09.455 --> 00:00:11.498
And to be more specific,

00:00:11.498 --> 00:00:13.890
in this video we're also
going to be focusing

00:00:13.890 --> 00:00:18.890
on examples using main group elements.

00:00:19.741 --> 00:00:23.781
So, we're doing that because
the transition metals,

00:00:23.781 --> 00:00:25.453
right here, and the lanthanides,

00:00:25.453 --> 00:00:26.776
are a little bit more complicated,

00:00:26.776 --> 00:00:31.002
so we won't be doing that
in this particular video.

00:00:31.002 --> 00:00:32.233
So the rule for writing

00:00:32.233 --> 00:00:35.600
the noble gas notation for something

00:00:35.600 --> 00:00:39.077
is pretty simple.

00:00:39.593 --> 00:00:40.638
It is,

00:00:40.638 --> 00:00:44.005
you take the noble gas

00:00:44.005 --> 00:00:48.899
immediately preceding your atom or ion,

00:00:49.067 --> 00:00:52.224
so what do I mean by
immediately preceding?

00:00:52.224 --> 00:00:54.732
That means we want the
noble gas that comes

00:00:54.732 --> 00:00:56.636
right before it, so you go up a row,

00:00:56.636 --> 00:00:59.817
and you go over to here,

00:00:59.817 --> 00:01:03.898
where the noble gases live.

00:01:05.900 --> 00:01:08.431
And you put that in brackets
and then you write your other

00:01:08.431 --> 00:01:11.589
electrons, and you write
your other electrons

00:01:11.589 --> 00:01:14.050
using the same notation
that you normally write

00:01:14.050 --> 00:01:15.931
electron configurations in.

00:01:15.931 --> 00:01:17.417
If you don't remember what that is,

00:01:17.417 --> 00:01:19.251
we'll go through a couple
examples right now.

00:01:19.251 --> 00:01:21.945
So hopefully it will become more clear.

00:01:21.945 --> 00:01:23.895
So before we get started,

00:01:23.895 --> 00:01:27.215
I'm just going to remind you
why we call the noble gases

00:01:27.215 --> 00:01:28.260
noble,

00:01:28.260 --> 00:01:32.997
and that is because they have a full octet

00:01:32.997 --> 00:01:34.994
of electrons.

00:01:34.994 --> 00:01:37.943
And so, since they have a
full octet, they're not very

00:01:37.943 --> 00:01:40.938
reactive, they have all of the electrons

00:01:40.938 --> 00:01:42.911
they want,

00:01:42.911 --> 00:01:45.884
and as a result,

00:01:45.884 --> 00:01:50.365
you will see in chemistry
that a lot of chemistry,

00:01:50.365 --> 00:01:52.501
chemical reactions will be driven by other

00:01:52.501 --> 00:01:57.501
non-noble gases trying
to get a full octet.

00:01:59.072 --> 00:02:02.299
So the first example
we're going to talk about

00:02:02.299 --> 00:02:06.107
and write the configuration
for is silicon.

00:02:06.107 --> 00:02:10.635
Silicon is right below
carbon in the Periodic Table,

00:02:10.635 --> 00:02:15.635
and we can write its full
electron configuration,

00:02:15.766 --> 00:02:17.508
just for comparison,

00:02:17.508 --> 00:02:19.923
so if we start up here,

00:02:19.923 --> 00:02:23.591
with hydrogen, silicon has
the electron configuration

00:02:23.591 --> 00:02:26.795
1s2

00:02:26.795 --> 00:02:29.187
2s2

00:02:29.187 --> 00:02:32.066
2p6,

00:02:32.066 --> 00:02:34.736
so as a reminder, these
are our S electrons

00:02:34.736 --> 00:02:36.478
because they're in S orbitals.

00:02:36.478 --> 00:02:38.010
These are our P electrons

00:02:38.010 --> 00:02:39.984
because they're in P orbitals,

00:02:39.984 --> 00:02:42.143
and then once we're

00:02:42.143 --> 00:02:43.861
through our 2p6 electrons,

00:02:43.861 --> 00:02:47.623
we go to 3s2

00:02:47.623 --> 00:02:49.550
and we have two more electrons,

00:02:49.550 --> 00:02:53.637
so it's 3p2.

00:02:53.637 --> 00:02:58.637
So that's the electron
configuration for silicon.

00:02:59.604 --> 00:03:02.785
Now, we can write it out
using noble gas notation.

00:03:02.785 --> 00:03:04.201
And compare, so,

00:03:04.201 --> 00:03:06.941
the noble gas immediately
preceding silicon,

00:03:06.941 --> 00:03:09.890
if we go up a row and then move over,

00:03:09.890 --> 00:03:12.839
we see that it's neon.

00:03:12.839 --> 00:03:16.525
So we write neon in brackets.

00:03:16.972 --> 00:03:18.109
And then,

00:03:18.109 --> 00:03:20.571
the other electrons

00:03:20.571 --> 00:03:23.055
are the ones that come after neon.

00:03:23.055 --> 00:03:25.122
So we go down,

00:03:25.122 --> 00:03:28.929
and the electrons that
aren't included in neon

00:03:28.929 --> 00:03:30.648
would be our

00:03:30.648 --> 00:03:33.805
3s2 electrons, right here,

00:03:33.805 --> 00:03:38.165
and the 3p2 electrons.

00:03:38.867 --> 00:03:41.700
So if we compare these two ways of writing

00:03:41.700 --> 00:03:43.743
our electron configuration,

00:03:43.743 --> 00:03:46.413
you can see that

00:03:46.413 --> 00:03:48.991
these electrons right here,

00:03:48.991 --> 00:03:53.216
so everything up to 3s2 3p2,

00:03:53.216 --> 00:03:56.978
that gives us the electron
configuration for neon.

00:03:56.978 --> 00:03:58.139
And so we don't have to write that,

00:03:58.139 --> 00:03:59.253
we just write neon,

00:03:59.253 --> 00:04:02.504
and assume that whoever's reading it

00:04:02.504 --> 00:04:06.800
knows what the electron
configuration for neon is.

00:04:06.800 --> 00:04:09.841
The other thing you might notice is that

00:04:09.841 --> 00:04:12.210
these other electrons here,

00:04:12.210 --> 00:04:15.042
that we wrote outside of the brackets,

00:04:15.042 --> 00:04:18.966
these are our valence electrons.

00:04:18.966 --> 00:04:21.195
So, writing

00:04:21.195 --> 00:04:22.983
your electron configuration this way,

00:04:22.983 --> 00:04:24.307
using noble gas notation,

00:04:24.307 --> 00:04:27.325
makes it really obvious where
your valence electrons are

00:04:27.325 --> 00:04:28.347
and what kind they are,

00:04:28.347 --> 00:04:31.522
so that's helpful.

00:04:32.062 --> 00:04:34.779
We are going to go
through one more example

00:04:34.779 --> 00:04:38.354
and that will be calcium.

00:04:38.354 --> 00:04:42.348
So calcium is over here,

00:04:42.348 --> 00:04:45.366
and so, we're not going
to write the full electron

00:04:45.366 --> 00:04:46.388
configuration this time.

00:04:46.388 --> 00:04:49.290
We're just going to write
the noble gas notation.

00:04:49.290 --> 00:04:53.052
So calcium, the noble gas
immediately preceding it,

00:04:53.052 --> 00:04:55.977
we go up a row and then
over and we see that

00:04:55.977 --> 00:04:57.998
the noble gas is argon,

00:04:57.998 --> 00:05:02.688
so we write argon in our brackets,

00:05:02.688 --> 00:05:06.449
and then, what electrons are not in argon?

00:05:06.449 --> 00:05:09.166
So we move down and we
see that we have our

00:05:09.166 --> 00:05:14.166
4s electrons, and we have two of them.

00:05:14.344 --> 00:05:19.121
So this is the noble gas
configuration for calcium.

00:05:20.613 --> 00:05:24.932
So does calcium have a full octet?

00:05:24.932 --> 00:05:29.663
And we know that these
are the valence electrons

00:05:30.225 --> 00:05:35.225
and we also know that if
it did have a full octet,

00:05:35.659 --> 00:05:39.374
it would actually already have
a noble gas configuration,

00:05:39.374 --> 00:05:43.205
so it would either have krypton or argon

00:05:43.205 --> 00:05:46.665
and in this case, it
turns out that calcium

00:05:46.665 --> 00:05:50.914
prefers to lose two electrons,

00:05:50.914 --> 00:05:54.466
as opposed to gaining a bunch.

00:05:54.466 --> 00:05:57.554
It would be gaining 10 plus six.

00:05:57.554 --> 00:06:00.991
So, lose two electrons,
and when calcium loses two

00:06:00.991 --> 00:06:05.449
electrons, it becomes the
calcium 2-plus cation.

00:06:05.449 --> 00:06:07.817
And then we can write

00:06:07.817 --> 00:06:10.069
the electron configuration really short,

00:06:10.069 --> 00:06:11.579
or it'll be really short

00:06:11.579 --> 00:06:13.993
because we know it's going
to lose these two valence

00:06:13.993 --> 00:06:16.663
electrons and then,

00:06:16.663 --> 00:06:20.773
it will have the electron
configuration which is just

00:06:20.773 --> 00:06:22.236
the same as argon.

00:06:22.236 --> 00:06:25.092
And we can tell, really
easily, looking at this,

00:06:25.092 --> 00:06:29.289
that calcium 2-plus has a full octet.

00:06:30.339 --> 00:06:32.638
So, one of the

00:06:32.638 --> 00:06:35.262
many pros of writing things
in noble gas notation,

00:06:35.262 --> 00:06:36.957
besides the fact that it, you know,

00:06:36.957 --> 00:06:38.141
makes your hand less tired,

00:06:38.141 --> 00:06:40.161
is that it makes it
really easy to see where

00:06:40.161 --> 00:06:41.647
your valence electrons are,

00:06:41.647 --> 00:06:45.891
and whether or not your ion
or atom has a full octet.


Inscribed quadrilaterals proof

WEBVTT
Kind: captions
Language: en

00:00:00.160 --> 00:00:03.930
- [Voiceover] So I have a
arbitrary inscribed quadrilateral

00:00:03.930 --> 00:00:06.740
in this circle and what I wanna prove

00:00:06.740 --> 00:00:09.960
is that for any inscribed quadrilateral,

00:00:09.960 --> 00:00:13.542
that opposite angles are supplementary.

00:00:13.542 --> 00:00:15.857
So when I say they're supplementary,

00:00:15.857 --> 00:00:18.080
the measure of this angle
plus the measure of this angle

00:00:18.080 --> 00:00:19.970
need to be 180 degrees.

00:00:19.970 --> 00:00:22.380
The measure of this angle
plus the measure of this angle

00:00:22.380 --> 00:00:24.412
need to be 180 degrees.

00:00:24.412 --> 00:00:26.650
And the way I'm gonna prove
it is we're gonna assume

00:00:26.650 --> 00:00:29.120
that this, the measure of
this angle right over here,

00:00:29.120 --> 00:00:31.461
that this is x degrees.

00:00:31.461 --> 00:00:33.410
And so from that, if we
can prove that the measure

00:00:33.410 --> 00:00:36.810
of this opposite angle
is 180 minus x degrees,

00:00:36.810 --> 00:00:38.970
then we've proven that opposite angles

00:00:38.970 --> 00:00:41.590
for an arbitrary quadrilateral
that's inscribed in a circle

00:00:41.590 --> 00:00:44.260
are supplementary, 'cause
if this is 180 minus x,

00:00:44.260 --> 00:00:48.380
180 minus x plus x is
going to be 180 degrees.

00:00:48.380 --> 00:00:50.610
So I encourage you to pause
the video and see if you can do

00:00:50.610 --> 00:00:53.110
that proof and I'll give
you a little bit of a hint.

00:00:53.110 --> 00:00:57.390
It's going to involve
the measure of the arcs

00:00:57.390 --> 00:01:00.246
that the various angles intercept.

00:01:00.246 --> 00:01:01.770
So let's think about it a little bit.

00:01:01.770 --> 00:01:06.770
This angle that has a
measure of x degrees,

00:01:06.770 --> 00:01:10.782
it intercepts this arc, so
we see one side of the angle

00:01:10.782 --> 00:01:13.070
goes and intercepts the circle there.

00:01:13.070 --> 00:01:15.630
The other side right over there.

00:01:15.630 --> 00:01:18.280
And so the arc that it intercepts,

00:01:18.280 --> 00:01:20.640
I am highlighting in yellow.

00:01:20.640 --> 00:01:23.818
I am highlighting it in yellow.

00:01:23.818 --> 00:01:26.490
Trying to color it in, so there you go.

00:01:26.490 --> 00:01:28.220
Not a great job at coloring it in,

00:01:28.220 --> 00:01:29.020
but you get the point.

00:01:29.020 --> 00:01:31.360
That's the arc that it intercepts

00:01:31.360 --> 00:01:34.150
and we've already learned
in previous videos

00:01:34.150 --> 00:01:39.150
that the relationship
between an inscribed angle,

00:01:39.150 --> 00:01:41.970
the vertex of this angle
sits on the circle,

00:01:41.970 --> 00:01:44.090
the relationship between
and inscribed angle

00:01:44.090 --> 00:01:47.460
and the measure of the
arc that it intercepts

00:01:47.460 --> 00:01:49.080
is that the measure of the inscribed angle

00:01:49.080 --> 00:01:52.200
is half the measure of the
arc that it intercepts.

00:01:52.200 --> 00:01:54.930
So if this angle measure is x degrees,

00:01:54.930 --> 00:01:58.521
then the measure of this
arc is going to be 2x,

00:01:58.521 --> 00:02:01.255
2x degrees.

00:02:01.255 --> 00:02:03.080
All right, well that's
kind of interesting,

00:02:03.080 --> 00:02:04.320
but let's keep going.

00:02:04.320 --> 00:02:06.600
If the measure of that arc is 2x degrees,

00:02:06.600 --> 00:02:10.410
what is the measure of
this arc right over here?

00:02:10.410 --> 00:02:12.727
The arc that completes the circle.

00:02:12.727 --> 00:02:14.370
Well, if you go all the
way around the circle,

00:02:14.370 --> 00:02:16.428
that's 360 degrees.

00:02:16.428 --> 00:02:19.710
So this blue arc that I'm
showing you right now,

00:02:19.710 --> 00:02:22.840
that's going to have a measure of 360

00:02:28.420 --> 00:02:31.011
minus 2x degrees.

00:02:31.011 --> 00:02:32.640
360's all the way around.

00:02:32.640 --> 00:02:36.900
The blue is all the way
around minus the yellow arc.

00:02:36.900 --> 00:02:39.360
What you have left over if you
subtract out the yellow arc

00:02:39.360 --> 00:02:41.483
is you have this blue arc.

00:02:41.483 --> 00:02:44.160
Now, what's the angle that intercepts

00:02:44.160 --> 00:02:45.693
this blue arc?

00:02:45.693 --> 00:02:47.530
What's the inscribed angle

00:02:47.530 --> 00:02:51.776
that intercepts this
blue arc right over here?

00:02:51.776 --> 00:02:53.399
Well, it's this angle.

00:02:53.399 --> 00:02:55.960
It's the angle that we
wanted to figure out

00:02:55.960 --> 00:02:57.280
in terms of x.

00:02:59.640 --> 00:03:01.730
Wow, I'm having trouble changing colors.

00:03:01.730 --> 00:03:04.960
It is that angle right over there.

00:03:04.960 --> 00:03:08.174
Notice, the two sides of this angle,

00:03:08.174 --> 00:03:11.842
they intercept, this
angle intercepts that arc.

00:03:11.842 --> 00:03:14.990
So, once again, the measure
of an inscribed angle

00:03:14.990 --> 00:03:17.060
is gonna half the measure of the arc

00:03:17.060 --> 00:03:18.500
that it intercepts.

00:03:18.500 --> 00:03:22.230
So what's 1/2, what is 1/2

00:03:22.230 --> 00:03:25.891
times 360 minus 2x?

00:03:25.891 --> 00:03:29.511
Well, one 1/2 times 360 is 180.

00:03:29.511 --> 00:03:32.607
1/2 times 2x is x.

00:03:32.607 --> 00:03:34.240
So the measure of this angle is gonna be

00:03:34.240 --> 00:03:36.774
180 minus x degrees.

00:03:36.774 --> 00:03:41.240
180 minus x degrees, and just like that

00:03:41.240 --> 00:03:43.170
we've proven that these opposite sides

00:03:43.170 --> 00:03:46.200
for this arbitrary
inscribed quadrilateral,

00:03:46.200 --> 00:03:47.650
that they are supplementary.

00:03:47.650 --> 00:03:50.420
You add these together,
x plus 180 minus x,

00:03:50.420 --> 00:03:52.000
you're going to get 180 degrees.

00:03:52.000 --> 00:03:54.150
So they are supplementary.


Solving inscribed quadrilaterals

WEBVTT
Kind: captions
Language: en

00:00:00.048 --> 00:00:01.808
- [Voiceover] What I wanna
do in this video is see

00:00:01.808 --> 00:00:05.668
if we can find the measure of angle D,

00:00:05.668 --> 00:00:07.336
if we could find the measure of angle D

00:00:07.336 --> 00:00:09.196
and like always, pause this video,

00:00:09.196 --> 00:00:11.216
and see if you can figure it out.

00:00:11.216 --> 00:00:12.996
And I'll give you a little bit of a hint.

00:00:12.996 --> 00:00:17.966
It'll involve thinking about
how an inscribed angle relates

00:00:17.966 --> 00:00:20.690
to the corresponding to
the measure of the arc

00:00:20.690 --> 00:00:22.630
that it intercepts.

00:00:22.630 --> 00:00:24.970
So, think about it like that.

00:00:24.970 --> 00:00:27.490
All right, so, let's work
on this a little bit.

00:00:27.490 --> 00:00:31.010
So, what do we know, what do we know?

00:00:31.010 --> 00:00:35.430
Well, angle D, angle D intercepts an arc.

00:00:35.430 --> 00:00:39.170
It intercepts this fairly
large arc that I'm going

00:00:39.170 --> 00:00:42.430
to highlight right now
in this purple color.

00:00:42.430 --> 00:00:44.690
So, it intercepts that arc.

00:00:44.690 --> 00:00:46.450
We don't know the measure of that arc

00:00:46.450 --> 00:00:48.810
or at least we don't know
the measure of that arc yet.

00:00:48.810 --> 00:00:50.990
If we did know the measure of this arc

00:00:50.990 --> 00:00:53.368
that I'm highlighting, then
we know that the measure

00:00:53.368 --> 00:00:56.178
of angle D would just be
half that because the measure

00:00:56.178 --> 00:01:00.518
of an inscribed angle is half the measure

00:01:00.518 --> 00:01:02.378
of the arc that it intercepts.

00:01:02.378 --> 00:01:03.918
We've seen that multiple times.

00:01:03.918 --> 00:01:05.618
So, if we knew the measure of this arc,

00:01:05.618 --> 00:01:07.018
we would be able to figure out

00:01:07.018 --> 00:01:09.298
what the measure of angle D is.

00:01:09.298 --> 00:01:12.738
But we do know, we don't
know the measure of that arc,

00:01:12.738 --> 00:01:15.418
but we do know the measure of another arc.

00:01:15.418 --> 00:01:19.658
We do know the measure of the
arc that completes the circle.

00:01:19.658 --> 00:01:22.358
So, we do know the measure of this arc.

00:01:22.358 --> 00:01:24.102
You might be saying,
hey wait, how do we know

00:01:24.102 --> 00:01:25.872
that measure, it's not labeled.

00:01:25.872 --> 00:01:28.252
Well, the reason why we
know the measure of this arc

00:01:28.252 --> 00:01:31.192
that I've just highlighted
in this teal color is

00:01:31.192 --> 00:01:34.012
because the inscribed
angle that intercepts it,

00:01:34.012 --> 00:01:35.390
they gave us the information,

00:01:35.390 --> 00:01:38.190
they said this is a 45 degree angle.

00:01:38.190 --> 00:01:40.330
So, this is a 45 degree angle.

00:01:40.330 --> 00:01:44.810
Then this over here is a
90 degree, 90 degree arc.

00:01:44.810 --> 00:01:48.530
The measure of this arc is 90 degrees.

00:01:48.530 --> 00:01:50.560
The measure of arc, I guess you could say

00:01:50.560 --> 00:01:53.768
this is the measure of arc,
I'ma write it this way.

00:01:53.768 --> 00:01:58.598
The measure of arc, WL.

00:01:58.598 --> 00:02:03.270
WL is equal to 90 degrees,

00:02:03.270 --> 00:02:07.390
it's twice that, the inscribed
angle that intercepts it.

00:02:07.390 --> 00:02:09.130
Now, why is that helpful?

00:02:09.130 --> 00:02:10.890
Well, if you go all the
way around the circle,

00:02:10.890 --> 00:02:12.710
you're 360 degrees.

00:02:12.710 --> 00:02:16.220
So, this purple arc that we
cared about, that we said hey,

00:02:16.220 --> 00:02:17.730
if we could figure out
the measure of that,

00:02:17.730 --> 00:02:19.890
we're gonna be able to figure
out the measure of angle D.

00:02:19.890 --> 00:02:23.378
That plus arc, WL,

00:02:23.378 --> 00:02:27.008
they are going to add up to 360 degrees.

00:02:27.008 --> 00:02:28.108
Let me write that down.

00:02:28.108 --> 00:02:32.548
So, the measure of arc, let's see,

00:02:32.548 --> 00:02:36.684
and this is going to be a
major arc right over here.

00:02:36.684 --> 00:02:39.242
This is so LIW,

00:02:39.242 --> 00:02:43.280
the measure of arc LIW

00:02:43.280 --> 00:02:47.090
plus the measure of arc WL

00:02:47.090 --> 00:02:50.426
plus the measure

00:02:50.426 --> 00:02:54.744
of arc WL plus this right over here.

00:02:54.744 --> 00:02:58.104
That's going to be equal to 360 degrees.

00:02:58.104 --> 00:03:02.464
This is going to be equal to 360 degrees.

00:03:02.464 --> 00:03:05.444
Now, we already know
that this is 90 degrees.

00:03:05.444 --> 00:03:08.064
We already know WL is 90 degrees.

00:03:08.064 --> 00:03:10.604
So, if you subtract 90
degrees from both sides,

00:03:10.604 --> 00:03:15.284
you get that the measure of
this large arc right over here,

00:03:15.284 --> 00:03:18.530
measure of arc LIW is going

00:03:18.530 --> 00:03:21.450
to be equal to 270 degrees.

00:03:21.450 --> 00:03:23.846
I just took 300, I went all
the way around the circle,

00:03:23.846 --> 00:03:25.426
I subtracted out this 90 degrees

00:03:25.426 --> 00:03:28.046
and I'm left with 270 degrees.

00:03:28.046 --> 00:03:29.086
So, let me write that down.

00:03:29.086 --> 00:03:32.106
This is the measure of this
arc in purple is 270 degrees.

00:03:32.106 --> 00:03:34.802
And now, we can figure out
the measure of angle D.

00:03:34.802 --> 00:03:37.700
It's an inscribed angle
that intercepts that arc

00:03:37.700 --> 00:03:40.028
so it's going to have half the measure,

00:03:40.028 --> 00:03:42.168
the angle's going to
have half the measure.

00:03:42.168 --> 00:03:45.156
So, half of 270

00:03:45.156 --> 00:03:49.106
is 135 degrees and we're done,

00:03:49.106 --> 00:03:51.986
and you might notice
something interesting,

00:03:51.986 --> 00:03:56.926
that if you add 135
degrees plus 45 degrees,

00:03:56.926 --> 00:04:00.646
that they add up to 180 degrees.

00:04:00.646 --> 00:04:03.866
So, it looks like at least for
this case that these angles,

00:04:03.866 --> 00:04:07.606
these opposite angles of
this inscribed quadrilateral,

00:04:07.606 --> 00:04:12.426
it looks like they are supplementary.

00:04:12.426 --> 00:04:13.864
So, an interesting question is

00:04:13.864 --> 00:04:15.704
are they always going to be supplementary?

00:04:15.704 --> 00:04:19.064
If you have a quadrilateral,
an arbitrary quadrilateral

00:04:19.064 --> 00:04:21.344
inscribed in a circle,
so each of the vertices

00:04:21.344 --> 00:04:24.944
of the quadrilateral sit on the circle.

00:04:24.944 --> 00:04:28.544
If you have that, are opposite
angles of that quadrilateral,

00:04:28.544 --> 00:04:30.464
are they always supplementary?

00:04:30.464 --> 00:04:32.784
Do they always add up to 180 degrees?

00:04:32.784 --> 00:04:35.550
So, I encourage you to think
about that and even prove it

00:04:35.550 --> 00:04:38.150
if you get a chance, and
the proof is very close

00:04:38.150 --> 00:04:39.550
to what we just did here.

00:04:39.550 --> 00:04:41.110
In order to prove it, you
would just have to do it

00:04:41.110 --> 00:04:42.776
with more general numbers like, you know,

00:04:42.776 --> 00:04:45.494
instead of saying 45 degrees,
you could call this X,

00:04:45.494 --> 00:04:47.970
and then you would want to
prove that this right over here

00:04:47.970 --> 00:04:50.330
would have to be 180 minus X.

00:04:50.330 --> 00:04:51.900
So, I encourage you to do that on your own

00:04:51.900 --> 00:04:53.198
but I'm gonna do it in a video as well

00:04:53.198 --> 00:04:56.008
so you can check if our
reasoning is similar.


Inscribed shapes: find inscribed angle

WEBVTT
Kind: captions
Language: en

00:00:00.084 --> 00:00:01.767
- So what I would like you to do

00:00:01.767 --> 00:00:05.673
is see if you can figure out
the measure of angle DEG here.

00:00:05.673 --> 00:00:08.436
So try to figure out the
measure of this angle.

00:00:08.436 --> 00:00:09.980
I encourage you to pause the video now

00:00:09.980 --> 00:00:11.508
and try it on your own.

00:00:11.733 --> 00:00:13.416
All right, now let's work
through this together

00:00:13.416 --> 00:00:16.749
and the key realization
here is to think about

00:00:16.749 --> 00:00:19.586
this angle, it is an inscribed angle,

00:00:19.586 --> 00:00:24.586
we see it's vertexes sitting
on the circle itself.

00:00:24.590 --> 00:00:27.032
And then think about the
arc that it intercepts.

00:00:27.032 --> 00:00:30.631
And we see, we see that it intercepts,

00:00:30.631 --> 00:00:35.490
so let me draw these
two sides of the angle,

00:00:35.681 --> 00:00:38.978
we see that it intercepts arc CD.

00:00:38.978 --> 00:00:42.444
It intercepts arc CD.

00:00:42.577 --> 00:00:44.330
And so the measure of this angle,

00:00:44.330 --> 00:00:46.536
since it's an inscribed angle,

00:00:46.536 --> 00:00:49.960
is going to be half the measure of arc CD.

00:00:49.960 --> 00:00:54.290
So if we could figure out
the measure of arc CD,

00:00:54.290 --> 00:00:57.193
then we're going to be in good shape.

00:00:57.193 --> 00:00:59.514
Because if we figure out
the measure of arc CD,

00:00:59.514 --> 00:01:00.536
then we take half of that

00:01:00.536 --> 00:01:02.469
and we'll figure out what we care about.

00:01:02.753 --> 00:01:05.888
Well, what you might notice,
is that there's another

00:01:05.888 --> 00:01:09.893
inscribed angle that
also intercepts arc CD.

00:01:09.893 --> 00:01:13.132
We have this angle right over here.

00:01:13.132 --> 00:01:16.150
It also intercepts arc CD.

00:01:16.150 --> 00:01:19.353
So you could call this angle C ... Whoops.

00:01:19.353 --> 00:01:23.393
You could call this angle CFD.

00:01:23.393 --> 00:01:26.678
This also intercepts the same arc.

00:01:26.678 --> 00:01:28.757
So there's two ways you
could think about it.

00:01:28.757 --> 00:01:32.994
Two inscribed angles that
intercept the same arc

00:01:32.994 --> 00:01:34.549
are going to have the same angle measure

00:01:34.549 --> 00:01:37.498
so just off of that you could
say that this is going to be,

00:01:37.498 --> 00:01:39.518
that these two angles
have the same measure,

00:01:39.518 --> 00:01:41.550
so you could say this is
going to be 50 degrees.

00:01:41.550 --> 00:01:43.918
Or you could go, you could actually solve

00:01:43.918 --> 00:01:46.112
what the measure of arc CD is.

00:01:46.112 --> 00:01:50.187
It's going to be twice the
measure of the inscribed angle

00:01:50.187 --> 00:01:51.278
that intercepts it.

00:01:51.278 --> 00:01:54.437
So the measure of arc CD
is going to be 100 degrees.

00:01:54.437 --> 00:01:56.130
Twice the 50 degrees.

00:01:56.130 --> 00:01:57.326
And then you use that and you say,

00:01:57.326 --> 00:02:00.275
Well, if the measure of
that arc is 100 degrees,

00:02:00.275 --> 00:02:02.597
then an inscribed angle that intercepts it

00:02:02.597 --> 00:02:03.746
is going to have half its measure,

00:02:03.746 --> 00:02:05.453
it's going to be 50 degrees.

00:02:05.453 --> 00:02:08.686
So either way we get to 50 degrees.


Inscribed shapes: angle subtended by diameter

WEBVTT
Kind: captions
Language: en

00:00:00.388 --> 00:00:04.102
- [Voiceover] "On circle O
below", so this is circle O,

00:00:04.102 --> 00:00:06.227
"segment SE is a diameter."

00:00:06.227 --> 00:00:09.392
So this is SE, let me color that in.

00:00:09.392 --> 00:00:12.757
So this is a diameter, that's
what they're telling us.

00:00:12.757 --> 00:00:14.175
And they say, "What is the measure of

00:00:14.175 --> 00:00:16.423
"angle ISE?"

00:00:16.423 --> 00:00:19.151
So what we care about, the measure of

00:00:19.151 --> 00:00:20.521
angle ISE.

00:00:20.521 --> 00:00:22.889
I, S, E.

00:00:22.889 --> 00:00:25.107
So we're trying to figure out this angle

00:00:25.107 --> 00:00:26.163
right over there.

00:00:26.163 --> 00:00:27.870
Now like always, I encourage
you to pause the video

00:00:27.870 --> 00:00:30.551
and see if you can work
through it yourself.

00:00:30.551 --> 00:00:32.629
So there's a bunch of
ways that we can actually

00:00:32.629 --> 00:00:34.707
tackle this problem.

00:00:34.707 --> 00:00:37.122
The first one that jumps out at me is

00:00:37.122 --> 00:00:39.165
there's a bunch of
triangles here, and we can

00:00:39.165 --> 00:00:42.787
use the fact that the
angles, interior angles

00:00:42.787 --> 00:00:45.283
of a triangle add up to 180 degrees.

00:00:45.283 --> 00:00:48.267
So we could look at this
triangle right over here.

00:00:48.267 --> 00:00:50.263
And so we know one of the angles already.

00:00:50.263 --> 00:00:53.688
We know this angle has a
measurement of 27 degrees.

00:00:53.688 --> 00:00:56.207
If we could figure out
this angle right over here,

00:00:56.207 --> 00:01:00.526
this would be angle
SIE, then if we know two

00:01:01.556 --> 00:01:03.012
interior angles of a triangle, we can

00:01:03.012 --> 00:01:04.649
figure out the third.

00:01:04.649 --> 00:01:08.178
And this one, SIE, we can figure it out

00:01:08.178 --> 00:01:12.300
because it's supplementary
to this 61-degree angle.

00:01:12.300 --> 00:01:15.922
So this angle right over
here is just going to be

00:01:15.922 --> 00:01:18.058
this angle plus the 61-degree angle

00:01:18.058 --> 00:01:20.925
is going to be equal to 180 degrees,

00:01:20.925 --> 00:01:23.038
because they are supplementary.

00:01:23.038 --> 00:01:25.197
Or we could say that this angle over here

00:01:25.197 --> 00:01:27.856
is going to be 180 minus 61.

00:01:27.856 --> 00:01:29.342
So what is that going to be?

00:01:29.342 --> 00:01:31.164
180 minus 60 would be 120,

00:01:31.164 --> 00:01:34.217
and then minus one gives us 119.

00:01:34.217 --> 00:01:37.040
119 degrees.

00:01:37.040 --> 00:01:40.593
And so this angle that we
are trying to figure out,

00:01:40.593 --> 00:01:43.657
this angle plus the 119 degrees, plus

00:01:43.657 --> 00:01:45.573
the 27 degrees, is going to be equal to

00:01:45.573 --> 00:01:47.233
180 degrees.

00:01:47.233 --> 00:01:49.694
Or we could say this angle is going to be

00:01:49.694 --> 00:01:53.397
180 minus 119,

00:01:53.397 --> 00:01:55.475
minus 27.

00:01:55.475 --> 00:01:57.797
Which is going to be equal to,

00:01:57.797 --> 00:02:01.489
so let's see, 180 minus 119

00:02:01.489 --> 00:02:05.239
is 61.

00:02:05.239 --> 00:02:09.116
And then 61 minus 27

00:02:09.116 --> 00:02:12.250
is going to be 34.

00:02:12.250 --> 00:02:13.307
So there you have it.

00:02:13.307 --> 00:02:17.591
The measure of angle ISE is 34 degrees.

00:02:17.591 --> 00:02:19.390
Now I mentioned that
there is multiple ways

00:02:19.390 --> 00:02:20.353
that we could figure this out.

00:02:20.353 --> 00:02:21.793
Let me do it one more way.

00:02:21.793 --> 00:02:23.708
So let me unwind everything that I

00:02:23.708 --> 00:02:24.428
just wrote.

00:02:24.428 --> 00:02:25.496
We already figured out the answer,

00:02:25.496 --> 00:02:26.471
but I want to show you that there's

00:02:26.471 --> 00:02:30.372
multiple ways that we can tackle this.

00:02:30.372 --> 00:02:31.986
So ISE is still the thing that we

00:02:31.986 --> 00:02:32.833
want to figure out.

00:02:32.833 --> 00:02:34.307
Another way that we could approach it

00:02:34.307 --> 00:02:38.034
is we know we have some angles

00:02:38.034 --> 00:02:40.274
inscribed angles on this circle,

00:02:40.274 --> 00:02:42.585
and we know that if an inscribed angle

00:02:42.585 --> 00:02:45.533
intercepts a diameter,
then it's going to be

00:02:45.533 --> 00:02:46.306
a right angle.

00:02:46.306 --> 00:02:48.267
It's going to be a 90 degree angle.

00:02:48.267 --> 00:02:50.868
So this angle right over here is

00:02:50.868 --> 00:02:52.499
a 90 degree angle.

00:02:52.499 --> 00:02:54.960
And we can use that information to

00:02:54.960 --> 00:02:57.038
figure out this angle,

00:02:57.038 --> 00:02:58.907
and we can also use that information

00:02:58.907 --> 00:03:03.005
if we look at this triangle,

00:03:03.005 --> 00:03:06.802
we could use 90 plus 61 plus this angle

00:03:06.802 --> 00:03:09.878
is going to be equal to 180 degrees.

00:03:09.878 --> 00:03:12.385
So this angle right over here, another way

00:03:12.385 --> 00:03:15.915
to think about it, it's
going to be 180 minus 90

00:03:15.915 --> 00:03:17.621
minus 61.

00:03:17.621 --> 00:03:21.301
Which is equal to, 180 minus 90 is 90,

00:03:21.301 --> 00:03:24.459
minus 61 is 29 degrees.

00:03:24.459 --> 00:03:27.538
So this one right over here is 29 degrees.

00:03:29.766 --> 00:03:32.751
And then we could look
at this larger triangle.

00:03:35.821 --> 00:03:38.069
To figure out this entire angle.

00:03:39.758 --> 00:03:41.940
If we know this entire
angle, you subtract 29

00:03:41.940 --> 00:03:44.738
then you figure out angle ISE.

00:03:44.738 --> 00:03:47.861
And so this large, or what I've depicted,

00:03:47.861 --> 00:03:51.506
this kind of magenta
measure right over here

00:03:51.506 --> 00:03:54.559
of that angle, plus 90 degrees,

00:03:54.559 --> 00:03:57.647
plus 27 degrees, is
going to be equal to 180

00:03:57.647 --> 00:04:02.024
because they're the interior
angles of triangle SLE.

00:04:02.024 --> 00:04:05.518
So this angle right
over here is going to be

00:04:05.518 --> 00:04:09.302
180 minus 61

00:04:09.302 --> 00:04:11.543
minus 27.

00:04:11.543 --> 00:04:13.675
Sorry, not minus 61, minus 90.

00:04:15.434 --> 00:04:19.508
It's 180 minus 90, minus 27

00:04:19.508 --> 00:04:21.168
is going to give us this
angle right over here

00:04:21.168 --> 00:04:23.572
because the three angles add up to 180.

00:04:23.572 --> 00:04:27.588
So minus 90, minus 27.

00:04:27.588 --> 00:04:31.326
Which is equal to, so 180 minus 90 is 90.

00:04:31.326 --> 00:04:34.056
90 minus 27

00:04:35.953 --> 00:04:37.753
is 63.

00:04:37.753 --> 00:04:39.448
63 degrees.

00:04:39.448 --> 00:04:43.569
So this large one over here is 63 degrees,

00:04:43.569 --> 00:04:45.926
and then the smaller one is 29 degrees.

00:04:45.926 --> 00:04:49.594
And so angle ISE, which
we set out to figure out,

00:04:49.594 --> 00:04:53.380
is going to be 63 degrees
minus the 29 degrees.

00:04:53.380 --> 00:04:58.163
So 63 minus 29 is once again

00:04:58.163 --> 00:05:00.554
equal to 34 degrees.

00:05:00.554 --> 00:05:03.027
So the way I did it just
now, a little bit harder.

00:05:03.027 --> 00:05:05.023
It really depends what jumps out at you.

00:05:05.023 --> 00:05:07.833
The first way I tackled it,
it does seem a little bit

00:05:07.833 --> 00:05:09.354
easier, a little bit clearer.

00:05:09.354 --> 00:05:10.747
But it's good to see
these different things.

00:05:10.747 --> 00:05:13.243
And at least here, we used this idea

00:05:13.243 --> 00:05:17.108
of an inscribed angle that
intercepts a diameter.

00:05:17.108 --> 00:05:18.664
And if you say, "Hey, how do we know?"

00:05:18.664 --> 00:05:20.429
Well, we've proven in other videos,

00:05:20.429 --> 00:05:22.577
but it comes straight out of the idea that

00:05:24.637 --> 00:05:26.305
the measure of an inscribed angle

00:05:26.305 --> 00:05:29.357
is going to be half of
the measure of the arc

00:05:29.357 --> 00:05:30.727
that it intercepts.

00:05:30.727 --> 00:05:33.525
And notice, it's intercepting an arc

00:05:33.525 --> 00:05:36.485
that has a measure of 180 degrees.

00:05:36.485 --> 00:05:38.389
It's intercepting an arc that has

00:05:38.389 --> 00:05:40.304
a measure of 180 degrees.

00:05:40.304 --> 00:05:42.220
And so this angle's
going to be half of that

00:05:42.220 --> 00:05:45.203
since it's an inscribed
angle, not a central angle.

00:05:45.203 --> 00:05:47.497
So it's going to be a 90 degree angle.


Finding arc measures with equations

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:02.768
- So we're told Circle P is below,

00:00:02.768 --> 00:00:04.834
this is Circle P right over here.

00:00:04.834 --> 00:00:09.714
What is the arc measure
of arc BC in degrees?

00:00:09.714 --> 00:00:13.527
So this is point B, this is point C,

00:00:13.527 --> 00:00:16.290
let me pick a different
color so you can see the arc.

00:00:16.290 --> 00:00:17.915
And since they only gave us two letters,

00:00:17.915 --> 00:00:19.889
we really wanna find the minor arc,

00:00:19.889 --> 00:00:22.490
so we want to find the
shorter arc between B and C.

00:00:22.490 --> 00:00:24.765
So the major arc would
be the long way around,

00:00:24.765 --> 00:00:26.530
and if they wanted to
specify the major arc

00:00:26.530 --> 00:00:28.201
they would've had to give us three letters

00:00:28.201 --> 00:00:29.989
to force us to go the long way around,

00:00:29.989 --> 00:00:33.286
so if they said arc BAC or BDC,

00:00:33.286 --> 00:00:34.610
that would go the long way around,

00:00:34.610 --> 00:00:36.839
but since they just gave us just B and C

00:00:36.839 --> 00:00:39.486
we assume it's going to be the minor arc,

00:00:39.486 --> 00:00:42.713
so we wanna find that arc
measure right over there.

00:00:42.713 --> 00:00:44.409
Now, the arc measure is going to be

00:00:44.409 --> 00:00:46.405
the exact same measure in degrees

00:00:46.405 --> 00:00:49.865
as the measure of the
angle, the central angle,

00:00:49.865 --> 00:00:51.768
that intercepts that
arc, so it's going to be

00:00:51.768 --> 00:00:54.793
the same thing as the measure
of this central angle,

00:00:54.793 --> 00:00:58.752
which is 4k + 159 degrees.

00:00:58.752 --> 00:01:00.706
So if we can figure out what
K is, we're gonna know what

00:01:00.706 --> 00:01:03.168
this central angle measure
is, and then that's going

00:01:03.168 --> 00:01:05.744
to be the same thing as this arc measure.

00:01:05.744 --> 00:01:07.718
So how do we figure that out?

00:01:07.718 --> 00:01:10.729
Well, what might jump out
at you is that this angle,

00:01:10.729 --> 00:01:13.822
angle BPC that we care about,

00:01:13.822 --> 00:01:16.792
is vertical to angle APD.

00:01:16.792 --> 00:01:20.418
These are vertical angles,
and vertical angles

00:01:20.418 --> 00:01:23.274
are going to have the same measure,

00:01:23.274 --> 00:01:27.314
they are, they're going to be congruent.

00:01:27.314 --> 00:01:31.617
So let's set these two
measures equal to each other.

00:01:31.617 --> 00:01:36.399
So we know that 4k + 159

00:01:37.016 --> 00:01:41.996
is going to be equal to 2k + 153,

00:01:41.996 --> 00:01:45.137
so let's get all of our K
terms on the left-hand side,

00:01:45.137 --> 00:01:48.504
and all of the non-K terms
on the right-hand side.

00:01:48.504 --> 00:01:51.314
So let's subtract 2k from both sides,

00:01:51.314 --> 00:01:54.053
so we can subtract 2k from both sides.

00:01:54.053 --> 00:01:56.143
And let's subtract, well
let me just do that first,

00:01:56.143 --> 00:01:57.825
I don't want to skip steps.

00:01:58.487 --> 00:02:01.174
And so I got rid of the
K's on the right-hand side

00:02:01.174 --> 00:02:03.832
so it's just gonna be left with the 153.

00:02:03.832 --> 00:02:07.269
And on the left-hand side, 4k - 2k is 2k,

00:02:07.269 --> 00:02:10.427
and I still have + 159.

00:02:10.427 --> 00:02:13.050
Now let's get rid of this
159 on the left-hand side

00:02:13.050 --> 00:02:14.513
so let's subtract it.

00:02:14.513 --> 00:02:15.720
But if I do it on the left-hand side

00:02:15.720 --> 00:02:17.764
I need to do it on the
right-hand side as well,

00:02:17.764 --> 00:02:20.527
so subtract 159 from both sides.

00:02:20.527 --> 00:02:24.339
And I'm left with 2k

00:02:24.339 --> 00:02:28.594
is equal to 153 - 159 is negative 6,

00:02:28.594 --> 00:02:32.379
so K is equal to, just
divide both sides by 2,

00:02:32.379 --> 00:02:34.688
K is going to be equal to negative 3.

00:02:35.467 --> 00:02:37.224
Now you might be tempted
to say oh, negative 3,

00:02:37.224 --> 00:02:39.639
but we're not just trying to solve for K,

00:02:39.639 --> 00:02:42.193
we're trying to figure
out this angle measure

00:02:42.193 --> 00:02:44.074
which is going to be the
same as the arc measure

00:02:44.074 --> 00:02:45.133
that we care about.

00:02:45.133 --> 00:02:47.112
And that's just expressed in terms of K,

00:02:47.112 --> 00:02:51.972
so it's 4 times K + 159, so
that's going to be 4 times

00:02:51.972 --> 00:02:55.987
- 3 + 159

00:02:55.987 --> 00:03:00.462
well what's that going
to be? 4 times -3 is -12.

00:03:00.462 --> 00:03:04.970
Plus 159 is going to be 147.

00:03:04.970 --> 00:03:09.097
So this angle right over here
has a measure of 147 degrees

00:03:09.097 --> 00:03:11.860
and you can calculate, that's
the same thing as over here.

00:03:11.860 --> 00:03:16.503
2 times -3 is -6, plus 153 is 147 degrees,

00:03:16.503 --> 00:03:21.503
these two are the same,
and so 147 degrees.

00:03:21.519 --> 00:03:25.290
This angle measures the same
as the measure of arc BC.

00:03:25.290 --> 00:03:27.096
Let's do one more of these.

00:03:27.960 --> 00:03:30.208
Circle P is below.

00:03:30.208 --> 00:03:34.355
What is the arc measure of BC in degrees?

00:03:34.898 --> 00:03:37.050
Now since once again they
only gave us two letters,

00:03:37.050 --> 00:03:39.465
we can assume it is the minor arc.

00:03:39.465 --> 00:03:44.046
So we care about BC,

00:03:44.693 --> 00:03:47.731
we care about this right over here.

00:03:47.731 --> 00:03:50.030
And so what is the measure of this arc

00:03:50.030 --> 00:03:52.003
is going to be the same
thing as the measure

00:03:52.003 --> 00:03:55.045
of the central angle
that intercepts that arc,

00:03:55.045 --> 00:03:58.249
and that measure is going to
be the sum of these two angles

00:03:59.709 --> 00:04:03.941
so it's going to be 4y + 6

00:04:03.941 --> 00:04:07.984
+ 7y - 7.

00:04:08.816 --> 00:04:13.816
4y + 7y, we can combine the
y terms, is going to be 11y.

00:04:14.120 --> 00:04:17.465
And then 6 - 7 is going to be negative 1.

00:04:17.465 --> 00:04:22.176
So it's going to be 11y - 1,
and how do we figure that out?

00:04:22.176 --> 00:04:23.920
Or how do we figure out what Y is?

00:04:23.920 --> 00:04:24.918
We need to figure out what Y is

00:04:24.918 --> 00:04:27.751
in order to figure out what 11y - 1 is.

00:04:27.751 --> 00:04:29.632
Well, we know, let me write this down.

00:04:29.632 --> 00:04:32.801
So the angle that we care about is

00:04:32.801 --> 00:04:36.847
11y - 1, 11y - 1.

00:04:36.847 --> 00:04:40.444
We know that that angle,
plus this big angle

00:04:40.444 --> 00:04:44.719
that I'm going to show in blue,
that if we add them together

00:04:44.719 --> 00:04:46.597
that it's going to be 360 degrees,

00:04:46.597 --> 00:04:49.221
'cause we would've gone all
the way around the circle.

00:04:49.221 --> 00:04:54.123
So we know that 11y - 1 + 20y

00:04:54.123 --> 00:04:58.836
- 11 is going to be equal to 360 degrees.

00:04:59.855 --> 00:05:02.246
And so now we can just solve for Y.

00:05:02.246 --> 00:05:06.351
What is, let me get some
new colors involved,

00:05:06.351 --> 00:05:09.607
what is 11y + 20y?

00:05:09.607 --> 00:05:14.607
Well that's going to be
31y, and then if I have

00:05:15.637 --> 00:05:19.465
- 1 and -11

00:05:19.465 --> 00:05:21.674
that's going to be negative,

00:05:21.674 --> 00:05:25.528
let me do this in a different
color, so that's going to be,

00:05:27.667 --> 00:05:31.134
- 1 and -11, that's -12,

00:05:31.134 --> 00:05:35.426
and that's going to be
equal to 360 degrees, 360.

00:05:35.986 --> 00:05:39.457
So let's see, we can add 12 to
both sides to get rid of that

00:05:39.457 --> 00:05:44.034
- 12 right over there, and
that's going to leave us with 31y

00:05:44.034 --> 00:05:49.034
31y is equal to 372

00:05:50.545 --> 00:05:54.098
and so if we divide both sides by 31,

00:05:54.098 --> 00:05:57.441
it looks like 12, yep,
it'll go exactly 12 times

00:05:57.441 --> 00:06:01.760
so Y is equal to 12, which is equal to 12.

00:06:01.760 --> 00:06:04.105
And remember, we weren't
trying to solve for Y,

00:06:04.105 --> 00:06:08.145
we were trying to solve for 11y - 1,

00:06:08.145 --> 00:06:12.806
so what is 11 times 12?
We know that Y is 12.

00:06:12.806 --> 00:06:16.207
11 times 12 - 1, let's see.

00:06:16.207 --> 00:06:21.125
11 times 12 is 121.

00:06:22.755 --> 00:06:27.008
And then 121 - 1

00:06:27.948 --> 00:06:30.897
is going to be, oh sorry no,

00:06:30.897 --> 00:06:33.970
my multiplication tables are off,

00:06:33.970 --> 00:06:35.906
it's been a long day.

00:06:35.906 --> 00:06:40.906
11 times 12 is going to be 132,

00:06:41.022 --> 00:06:46.022
132 - 1 is going to be 131,

00:06:47.041 --> 00:06:48.364
and it's going to be in degrees.

00:06:48.364 --> 00:06:52.242
So 131 degrees, that's
the measure of this angle

00:06:52.242 --> 00:06:56.354
which is going to be the
measure of minor arc BC.


Finding arc measures

WEBVTT
Kind: captions
Language: en

00:00:00.496 --> 00:00:02.307
- So I have some example questions here

00:00:02.307 --> 00:00:03.998
from Khan Academy on arc measure.

00:00:03.998 --> 00:00:05.839
And like always, I encourage
you to pause the video

00:00:05.839 --> 00:00:07.167
after you see each of these questions,

00:00:07.167 --> 00:00:09.333
and try to solve them before I do.

00:00:10.051 --> 00:00:11.918
So this first question says
what is the arc measure,

00:00:11.918 --> 00:00:15.651
in degrees, of arc AC on circle P below.

00:00:16.181 --> 00:00:17.965
So this is point A,

00:00:17.965 --> 00:00:19.887
that is point C,

00:00:19.887 --> 00:00:21.921
and when they're talking about arc AC,

00:00:21.921 --> 00:00:23.509
since they only have two letters here,

00:00:23.509 --> 00:00:25.543
we can assume that it's
going to be the minor arc.

00:00:25.543 --> 00:00:26.574
When we talk about the minor arc.

00:00:26.574 --> 00:00:30.055
There's two potential arcs that
connect point A and point C.

00:00:30.529 --> 00:00:32.591
There's the one here on the left,

00:00:32.591 --> 00:00:34.764
and then there's the one,

00:00:34.764 --> 00:00:36.881
there is the one on the right.

00:00:37.160 --> 00:00:41.450
And since C isn't exactly
straight down from A,

00:00:41.450 --> 00:00:42.843
it's a little bit to the right,

00:00:42.843 --> 00:00:44.682
the shorter arc,

00:00:44.682 --> 00:00:46.772
the arc with the smaller length,

00:00:46.772 --> 00:00:48.443
or the minor arc is going to be this one

00:00:48.443 --> 00:00:51.006
that I'm depicting here
right on the right.

00:00:51.898 --> 00:00:54.294
So what is this arc measure going to be?

00:00:54.991 --> 00:00:57.220
Well, the measure of
this arc is going to be

00:00:57.220 --> 00:01:00.425
exactly the same thing as, in degrees,

00:01:00.425 --> 00:01:03.767
as the measure of the central
angle that intercepts the arc.

00:01:04.353 --> 00:01:06.859
So that central angle, let me
do it in a different color,

00:01:06.859 --> 00:01:08.753
I'll do it in this blue color,

00:01:09.227 --> 00:01:13.294
that central angle is angle C, P, A.

00:01:14.187 --> 00:01:17.084
Angle C, P, A, and the
measure of that central angle

00:01:17.084 --> 00:01:20.537
is going to be 70
degrees plus 104 degrees.

00:01:20.537 --> 00:01:22.962
It's going to be this whole
thing right over there.

00:01:22.962 --> 00:01:25.273
So it's going to be 174 degrees.

00:01:25.887 --> 00:01:29.090
One hundred and seventy four degrees,

00:01:29.090 --> 00:01:32.098
that's the arc measure,
in degrees, of arc AC.

00:01:32.517 --> 00:01:33.853
Let's keep doing these.

00:01:33.853 --> 00:01:35.160
So let me do another one.

00:01:35.599 --> 00:01:36.780
So,

00:01:37.526 --> 00:01:40.488
this next one asks us,

00:01:40.488 --> 00:01:42.378
in the figure below,

00:01:42.378 --> 00:01:45.188
in the figure below, segment AD--

00:01:45.188 --> 00:01:47.052
so this is point A,

00:01:47.052 --> 00:01:48.815
this is point D,

00:01:48.815 --> 00:01:51.711
so segment AD is this one right over here.

00:01:51.711 --> 00:01:54.103
Let me see if I can draw that.

00:01:54.103 --> 00:01:56.137
That's AD right over there,

00:01:56.137 --> 00:01:59.925
AD and CE are diameters of the circle.

00:02:00.093 --> 00:02:01.486
So let me draw CE,

00:02:01.486 --> 00:02:04.022
so CE is,

00:02:04.022 --> 00:02:07.170
we're going to connect point C and E.

00:02:07.170 --> 00:02:08.898
These are diameters.

00:02:08.898 --> 00:02:10.681
So, let me,

00:02:10.681 --> 00:02:11.850
so they go straight.

00:02:11.850 --> 00:02:14.107
Whoops, I'm using the
wrong tool, let me...

00:02:14.107 --> 00:02:16.280
So those are, somehow I should,

00:02:16.280 --> 00:02:17.537
alright.

00:02:17.537 --> 00:02:21.546
So, those are di--

00:02:21.546 --> 00:02:23.859
whoops, how did that happen?

00:02:23.859 --> 00:02:26.976
So let me, somehow my pen got

00:02:26.976 --> 00:02:29.205
really big, alright.

00:02:29.205 --> 00:02:31.880
That'll be almost there, ok.

00:02:31.880 --> 00:02:35.670
So CE, there you go.

00:02:35.670 --> 00:02:39.235
So those are both
diameters of the circle P.

00:02:39.821 --> 00:02:44.307
What is the arc measure of
AB, of arc AB in degrees?

00:02:44.307 --> 00:02:47.288
So arc AB, once again
there's two potential arcs

00:02:47.288 --> 00:02:50.128
that connect point A and B.

00:02:50.965 --> 00:02:52.107
There's the minor arc,

00:02:52.107 --> 00:02:53.361
and since this only has two letters

00:02:53.361 --> 00:02:54.615
we'll assume it's the minor arc.

00:02:54.615 --> 00:02:56.398
It's going to be this one over here.

00:02:56.398 --> 00:02:57.792
There's a major arc,

00:02:57.792 --> 00:02:59.324
but to not the major arc
they would've said something

00:02:59.324 --> 00:03:02.835
like A, E, B or A, D, B

00:03:02.835 --> 00:03:06.150
or arc A, C, B to make us go this kind of,

00:03:06.150 --> 00:03:08.352
this long way around.

00:03:08.352 --> 00:03:10.274
But this is arc AB, so we,

00:03:10.274 --> 00:03:11.918
in order to find the arc measure,

00:03:11.918 --> 00:03:13.540
we just really have to find the

00:03:13.540 --> 00:03:15.021
measure of this central angle.

00:03:15.021 --> 00:03:17.933
This is the central angle
that intercepts that arc,

00:03:17.933 --> 00:03:20.538
or you can even say it
defines that arc in some way.

00:03:20.538 --> 00:03:22.403
So how can we figure out this angle?

00:03:22.403 --> 00:03:23.839
And this one's a little bit trickier.

00:03:24.764 --> 00:03:25.811
Well, the key to,

00:03:25.811 --> 00:03:29.954
the key here is to realize
that this 93 degree angle,

00:03:30.531 --> 00:03:33.871
it is vertical to this
whole angle right over here.

00:03:34.771 --> 00:03:37.121
And we know from geometry,

00:03:37.121 --> 00:03:39.649
which we're still learning as
we do this example problem,

00:03:39.649 --> 00:03:42.658
that vertical angles are going
to have the same measure.

00:03:42.969 --> 00:03:44.859
So if this one on, this one is 93 degrees,

00:03:44.859 --> 00:03:46.681
then this entire blue one right over here

00:03:46.681 --> 00:03:48.794
is also gonna be, let me write it,

00:03:48.794 --> 00:03:52.076
this is also gonna be 93 degrees.

00:03:52.076 --> 00:03:55.949
So 93 degrees, that's gonna
be made up of this red angle,

00:03:55.949 --> 00:03:58.512
that we care about, and the 38 degrees.

00:03:58.512 --> 00:04:02.553
So this red one, which is the
measure of the central angle,

00:04:02.553 --> 00:04:05.339
it's also the arc measure of arc AB,

00:04:05.339 --> 00:04:07.456
is going to be 93 minus,

00:04:07.456 --> 00:04:10.299
93 degrees minus 38 degrees.

00:04:10.299 --> 00:04:11.942
So what is that going to be?

00:04:11.942 --> 00:04:14.365
Let's see, 93,

00:04:14.365 --> 00:04:15.591
I can write degrees there,

00:04:15.591 --> 00:04:18.517
minus 38 degrees,

00:04:18.517 --> 00:04:20.161
that is going to be equal to,

00:04:20.161 --> 00:04:22.640
let's see if it was 93 minus 40

00:04:22.640 --> 00:04:25.148
it would be, it would be 53,

00:04:25.148 --> 00:04:26.179
it's gonna be two more,

00:04:26.179 --> 00:04:27.737
it's gonna be 55 degrees.

00:04:28.769 --> 00:04:30.829
Fifty five degrees,

00:04:30.829 --> 00:04:32.081
and we are done.

00:04:32.081 --> 00:04:34.561
This angle right here is 55 degrees.

00:04:34.561 --> 00:04:36.567
If you were to add this angle meausre,

00:04:36.567 --> 00:04:39.047
plus 38 degrees, you would get 93 degrees,

00:04:39.047 --> 00:04:41.271
and that has the same
measure because it's vertical

00:04:41.271 --> 00:04:42.834
with this angle right over here,

00:04:42.834 --> 00:04:44.924
with angle D, P, E.

00:04:44.924 --> 00:04:46.816
Alright, let's do one more of these.

00:04:48.042 --> 00:04:50.327
So, we have

00:04:50.327 --> 00:04:52.027
in the figure below,

00:04:52.027 --> 00:04:53.281
and it doesn't quite fit on the page,

00:04:53.281 --> 00:04:55.064
but we'll scroll down in a second,

00:04:55.064 --> 00:04:59.073
AB is the diameter of circle P,

00:04:59.520 --> 00:05:01.525
is the diameter of circle P.

00:05:01.972 --> 00:05:04.535
Alright, so AB is the diameter,

00:05:04.535 --> 00:05:05.871
let me label that.

00:05:06.429 --> 00:05:09.158
So AB is the diameter.

00:05:09.158 --> 00:05:10.802
It's going straight across,

00:05:10.802 --> 00:05:13.588
straight across the circle.

00:05:13.908 --> 00:05:18.825
What is the arc measure
of A, B, C in degrees?

00:05:19.211 --> 00:05:21.661
So A, B, C.

00:05:21.661 --> 00:05:23.556
So they're making us
go the long way around.

00:05:23.556 --> 00:05:25.534
This is a major arc they're talking about.

00:05:25.534 --> 00:05:26.760
Let me draw it.

00:05:26.760 --> 00:05:29.853
Arc A, what is the arc measure of

00:05:29.853 --> 00:05:31.776
arc A, B, C.

00:05:31.776 --> 00:05:33.475
So we're going the long way around.

00:05:33.475 --> 00:05:35.314
So it's a major arc.

00:05:35.314 --> 00:05:37.070
So what is that going to be.

00:05:37.070 --> 00:05:39.048
Well it's going to be in degrees,

00:05:39.048 --> 00:05:41.528
the same measure as the angle,

00:05:41.528 --> 00:05:43.896
as the central angle that intercepts it.

00:05:43.896 --> 00:05:45.428
So it's going to be the same thing

00:05:45.428 --> 00:05:48.270
as this central angle right over here.

00:05:48.270 --> 00:05:50.945
Well, what is that
central angle going to be?

00:05:50.945 --> 00:05:52.924
Well, since we know that
this is the diameter,

00:05:52.924 --> 00:05:54.345
since AB is the diameter,

00:05:54.345 --> 00:05:57.322
we know that this part of it

00:05:58.660 --> 00:06:00.527
is going to 180 degrees.

00:06:00.527 --> 00:06:02.645
We're going halfway around the circle.

00:06:02.645 --> 00:06:04.177
One hundred eighty degrees.

00:06:04.177 --> 00:06:07.103
And so if we wanna look
at this whole angle,

00:06:07.103 --> 00:06:11.309
the angle that intercepts
the major arc A, B, C,

00:06:11.309 --> 00:06:15.851
is going to be 180
degrees plus 69 degrees.

00:06:15.851 --> 00:06:19.027
So we're going to have 180 degrees,

00:06:19.027 --> 00:06:21.451
plus 69 degrees

00:06:21.451 --> 00:06:23.680
which is equal to, what is that,

00:06:23.680 --> 00:06:27.156
249,

00:06:27.156 --> 00:06:29.996
249 degrees.

00:06:29.996 --> 00:06:34.008
That's the arc measure of
this major arc A, B, C.


Intro to arc measure

WEBVTT
Kind: captions
Language: en

00:00:00.075 --> 00:00:01.981
- [Voiceover] What I want
to talk about in this video,

00:00:01.981 --> 00:00:04.788
is the notion of Arc Measure,

00:00:04.788 --> 00:00:06.946
when we're dealing with circles.

00:00:06.946 --> 00:00:09.175
As we'll see, sometimes when you see

00:00:09.175 --> 00:00:10.499
something like arc measure,

00:00:10.499 --> 00:00:12.356
you might think it's the length of an arc,

00:00:12.356 --> 00:00:14.771
but arc length is
actually a different idea.

00:00:14.771 --> 00:00:16.884
So we will compare these two things.

00:00:16.884 --> 00:00:20.297
Arc length to arc measure.

00:00:20.297 --> 00:00:23.269
So arc measure, all that is

00:00:23.269 --> 00:00:24.685
is just a fancy way of saying,

00:00:24.685 --> 00:00:27.077
if I have a circle right over here,

00:00:27.077 --> 00:00:29.933
this is my best attempt
at drawing a circle.

00:00:29.933 --> 00:00:32.557
I have a circle here.

00:00:32.557 --> 00:00:37.131
The center of the circle,
let's call that point O,

00:00:37.131 --> 00:00:39.731
and let me put some
other points over here.

00:00:39.731 --> 00:00:43.725
So let's say that this is point A,

00:00:43.725 --> 00:00:46.837
let's say this is point B,

00:00:46.837 --> 00:00:51.178
and let's say this is
point C right over here.

00:00:51.178 --> 00:00:53.222
And let's say that I have,

00:00:53.222 --> 00:00:57.053
let's say the central
angle, right over here,

00:00:57.053 --> 00:00:59.166
cause it includes the
center of the circle,

00:00:59.166 --> 00:01:02.114
so the central angle, angle AOB.

00:01:02.114 --> 00:01:07.078
Let's say it has a measure of 120 degrees.

00:01:09.638 --> 00:01:11.077
And if someone were to say,

00:01:11.077 --> 00:01:14.345
what is the measure of arc AB?

00:01:14.769 --> 00:01:16.324
So, let me write that down.

00:01:16.324 --> 00:01:17.600
The measure.

00:01:17.600 --> 00:01:22.476
So, if someone were to say
what is the measure of arc AB,

00:01:22.476 --> 00:01:23.916
and they'd write it like this,

00:01:23.916 --> 00:01:27.444
so that's referring to
arc AB right over here.

00:01:27.444 --> 00:01:31.414
It's the minor arc, so there's
two ways to connect AB,

00:01:31.414 --> 00:01:33.481
you could connect it right over here,

00:01:33.481 --> 00:01:35.408
this is the shorter distance,

00:01:35.408 --> 00:01:36.942
or you can go the other way around,

00:01:36.942 --> 00:01:39.843
which would be what you'd
consider the major arc.

00:01:39.843 --> 00:01:41.840
Now, if someone's
referring to the major arc,

00:01:41.840 --> 00:01:44.556
they would say mark ACB.

00:01:44.556 --> 00:01:46.809
So when you're given just two letters,

00:01:46.809 --> 00:01:49.480
you assume it's the shortest
distance between the two.

00:01:49.480 --> 00:01:51.661
You assume that it is the minor arc.

00:01:51.661 --> 00:01:53.473
In order to specify the major arc,

00:01:53.473 --> 00:01:55.423
you would give the third letter,

00:01:55.423 --> 00:01:57.373
to go the long way around.

00:01:57.373 --> 00:02:00.646
So the measure of arc AB, and sometimes

00:02:00.646 --> 00:02:02.639
you'll see it with
parenthesis right over here,

00:02:02.639 --> 00:02:05.797
all this is, this is the
same thing as the measure

00:02:05.797 --> 00:02:10.185
of the central angle
that intercepts that arc.

00:02:10.185 --> 00:02:12.670
Well, the central angle
that intercepts that arc

00:02:12.670 --> 00:02:16.315
has a measure of 120 degrees.

00:02:16.315 --> 00:02:20.425
So this is just going to be 120 degrees.

00:02:20.425 --> 00:02:22.004
Now, some of y'all might be saying,

00:02:22.004 --> 00:02:23.514
well, what about the major arc?

00:02:23.514 --> 00:02:24.813
Well, let's write that.

00:02:24.813 --> 00:02:27.274
So if we're talking about arc ACB,

00:02:27.274 --> 00:02:29.271
so we're going the other way around,

00:02:29.271 --> 00:02:31.548
so this is major arc.

00:02:31.548 --> 00:02:36.504
So what is the measure of arc ACB,

00:02:36.864 --> 00:02:38.652
once again we're using three letters,

00:02:38.652 --> 00:02:41.067
so that we're specifying the major arc.

00:02:41.067 --> 00:02:45.130
Well, this angle, this
central angle right over here,

00:02:45.130 --> 00:02:48.868
to go all the way around
the circle is 360 degrees.

00:02:48.868 --> 00:02:52.955
So this is going to be
the 360 minus the 120

00:02:52.955 --> 00:02:54.627
that we're not including.

00:02:54.627 --> 00:02:57.273
So 360 degrees minus 120 is going to be,

00:02:57.273 --> 00:03:01.592
is going to be 240 degrees.

00:03:01.592 --> 00:03:06.556
So the measure of this angle
right over here is 240 degrees,

00:03:06.974 --> 00:03:11.803
so the measure of this arc,

00:03:11.803 --> 00:03:14.055
I have to be careful not
to say length of that arc,

00:03:14.055 --> 00:03:16.192
the measure of this arc
is going to be the same

00:03:16.192 --> 00:03:18.026
as the measure of the central angle.

00:03:18.026 --> 00:03:21.694
It's going to be 240 degrees.

00:03:24.424 --> 00:03:27.476
These arc measures are
going to be the case

00:03:27.476 --> 00:03:29.659
regardless of the size of the circle,

00:03:29.659 --> 00:03:31.376
and that's where the
difference starts to be

00:03:31.376 --> 00:03:33.513
from arc measure to arc length.

00:03:33.513 --> 00:03:36.764
So, I could have two circles,

00:03:36.764 --> 00:03:38.900
so this circle right over here

00:03:38.900 --> 00:03:42.033
and that circle right over here,

00:03:42.033 --> 00:03:45.609
and as long as the central
angle that intercepts the arc

00:03:45.609 --> 00:03:47.885
has the same degree measure,

00:03:47.885 --> 00:03:52.885
so let's say that that degree
measure is the same as,

00:03:53.025 --> 00:03:54.357
these are central angles,

00:03:54.357 --> 00:03:55.844
so we're assuming the vertex of the angle

00:03:55.844 --> 00:03:57.284
is the center of the circle.

00:03:57.284 --> 00:03:59.559
As long as these two are the same,

00:03:59.559 --> 00:04:02.253
these two central angles
have the same degree measure,

00:04:02.253 --> 00:04:05.457
then the arc measures, then
the corresponding arc measures

00:04:05.457 --> 00:04:07.012
are going to be the same.

00:04:07.012 --> 00:04:10.402
But clearly, these two
arc lengths are different.

00:04:10.402 --> 00:04:12.260
The arc length is not going to depend

00:04:12.260 --> 00:04:15.163
only on the measure of the central angle,

00:04:15.163 --> 00:04:16.974
the arc length is going to depend

00:04:16.974 --> 00:04:20.132
on the size of the actual circle.

00:04:20.132 --> 00:04:22.825
Arc measure is only dependent

00:04:22.825 --> 00:04:26.052
on the measure of the central angle

00:04:26.052 --> 00:04:28.885
that intercepts that arc.

00:04:28.885 --> 00:04:33.087
So your maximum arc measure
is going to be 360 degrees.

00:04:33.087 --> 00:04:36.733
Your minimum arc measure is
going to be zero degrees.

00:04:36.733 --> 00:04:39.984
It's measured in degrees,
not in units of length

00:04:39.984 --> 00:04:42.561
that arc length would be measured in.

00:04:42.561 --> 00:04:44.278
So, let me write this down.

00:04:44.278 --> 00:04:46.275
This only depends...

00:04:46.275 --> 00:04:49.387
So, this is what's going to drive this

00:04:49.387 --> 00:04:54.387
is the measure of central
angle, central angle,

00:04:57.798 --> 00:05:00.045
that intercepts the arc.

00:05:00.747 --> 00:05:05.747
That intercepts, intercepts the arc.

00:05:06.757 --> 00:05:08.684
When you talk about arc length,

00:05:08.684 --> 00:05:10.797
yes, it's going to be
dependent on the angle,

00:05:10.797 --> 00:05:12.747
but it's also dependent,
it's going to be dependent

00:05:12.747 --> 00:05:14.233
on the measure of that central angle

00:05:14.233 --> 00:05:16.875
plus the size of the circle.

00:05:16.875 --> 00:05:19.452
Size of the circle.

00:05:19.452 --> 00:05:21.519
You're actually talking
about a length now,

00:05:21.519 --> 00:05:22.935
when you're talking about arc length.

00:05:22.935 --> 00:05:25.298
While here, you're talking
about a degree measure.


Inscribed angles

WEBVTT
Kind: captions
Language: en

00:00:00.356 --> 00:00:02.252
- [Voiceover] A circle
is centered on point B.

00:00:02.252 --> 00:00:03.391
We see that right over there.

00:00:03.391 --> 00:00:06.135
That's the center of
this big, blue circle.

00:00:06.364 --> 00:00:08.728
Points A, C, and D lie
on its circumference.

00:00:08.728 --> 00:00:09.383
We see that.

00:00:09.383 --> 00:00:13.216
Points A, C, and D lie
on the circumference.

00:00:13.857 --> 00:00:16.035
If angle ABC...

00:00:16.055 --> 00:00:20.391
So ABC; So that's this
central angle right over here;

00:00:20.391 --> 00:00:23.275
measures 132 degrees.

00:00:23.430 --> 00:00:25.305
Alright, so this is 132 degrees.

00:00:25.727 --> 00:00:28.402
What does angle ADC measure?

00:00:28.629 --> 00:00:30.811
A, D, C.

00:00:30.991 --> 00:00:32.841
So let's think about
how these are related.

00:00:33.023 --> 00:00:35.607
ABC is a central angle.

00:00:35.853 --> 00:00:39.212
ADC is an inscribed angle.

00:00:39.492 --> 00:00:42.043
And they intercept the same arc.

00:00:42.120 --> 00:00:43.397
The arc AC.

00:00:43.397 --> 00:00:46.773
They both intercept this
arc right over here.

00:00:47.121 --> 00:00:51.538
And we know from the
inscribed angle theorem

00:00:51.723 --> 00:00:54.784
that an inscribed angle
that intercepts the same arc

00:00:54.784 --> 00:00:58.208
as a central angle is going to
have half the angle measure.

00:00:58.392 --> 00:01:00.744
And it even looks that
way right over here.

00:01:00.900 --> 00:01:01.629
So if ABC-

00:01:01.629 --> 00:01:04.083
if the central angle is 132 degrees,

00:01:04.083 --> 00:01:06.674
then the inscribed angle
that intercepts the same arc

00:01:06.776 --> 00:01:07.859
is going to be half of that.

00:01:07.859 --> 00:01:11.128
So half of 132 degrees is what?

00:01:11.128 --> 00:01:13.276
It is 66 degrees.

00:01:13.992 --> 00:01:17.978
We can check our answer,
and we got it right.


Proof: all circles are similar

WEBVTT
Kind: captions
Language: en

00:00:00.341 --> 00:00:02.245
- [Voiceover] We're asked
to translate and dilate

00:00:02.245 --> 00:00:05.178
the unit circle to map
it onto each circle.

00:00:05.178 --> 00:00:07.367
So, this is the unit
circle right over here.

00:00:07.367 --> 00:00:09.771
It's centered at zero comma zero.

00:00:09.771 --> 00:00:10.990
It has a radius of one.

00:00:10.990 --> 00:00:12.673
That's why we call it a unit circle.

00:00:12.673 --> 00:00:15.123
When they say translate,
they say move it around.

00:00:15.349 --> 00:00:17.685
So, that would be a translation of it.

00:00:18.269 --> 00:00:20.366
Dilating it means making it larger.

00:00:20.366 --> 00:00:22.978
So, dilating that unit
circle would be doing...

00:00:22.978 --> 00:00:24.580
Whoops, I just translated it.

00:00:24.580 --> 00:00:26.159
Would be doing something like that.

00:00:26.159 --> 00:00:29.328
So, we're gonna translate
and dilate this unit circle

00:00:29.328 --> 00:00:30.716
to map it onto each circle.

00:00:31.059 --> 00:00:33.491
For example, I could translate
it so that the center

00:00:33.491 --> 00:00:36.441
is translated to the center
of that magenta circle,

00:00:36.441 --> 00:00:39.668
and I can dilate it so
that it has been mapped

00:00:39.668 --> 00:00:41.781
onto that larger magenta circle.

00:00:41.781 --> 00:00:43.788
I can do that, let me
do that for a few more.

00:00:43.788 --> 00:00:44.760
I'm not gonna do that for all of them

00:00:44.760 --> 00:00:47.316
Just to give you the idea of
what they're talking about.

00:00:47.705 --> 00:00:50.584
So, now, I'm translating the center of my,

00:00:50.584 --> 00:00:52.185
it's no longer a unit circle.

00:00:52.185 --> 00:00:53.776
I am translating the center of my circle

00:00:53.776 --> 00:00:55.680
to the center of the purple circle.

00:00:55.680 --> 00:00:57.225
Now, I'm gonna dilate it

00:00:57.579 --> 00:00:59.302
so it has the same radius.

00:00:59.313 --> 00:01:00.934
And notice, I can map it.

00:01:00.934 --> 00:01:04.416
So, if you can map one shape to another

00:01:04.416 --> 00:01:06.234
through translation and dilation,

00:01:06.234 --> 00:01:08.056
then the things are, by definition,

00:01:08.056 --> 00:01:10.354
they are going to be similar.

00:01:10.354 --> 00:01:12.874
This is really just an exercise in seeing

00:01:12.874 --> 00:01:14.626
that all circles are similar.

00:01:14.626 --> 00:01:16.367
If you just take any circle,

00:01:16.367 --> 00:01:19.352
and you make it have the same
center as another circle,

00:01:19.352 --> 00:01:23.752
then you can just scale it up or down

00:01:23.752 --> 00:01:27.299
to match the circle that you
moved it to the center of.

00:01:27.594 --> 00:01:28.488
So, there you have it.

00:01:28.488 --> 00:01:29.590
Hopefully, this gives you a sense

00:01:29.590 --> 00:01:31.977
that all circles are similar.


Analyzing a resistor circuit with two batteries

WEBVTT
Kind: captions
Language: en

00:00:01.026 --> 00:00:02.163
- [Voiceover] Hi, my name's Willy.

00:00:02.163 --> 00:00:04.578
I'm the Khan Academy
Electrical Engineering

00:00:04.578 --> 00:00:06.110
Content Fellow,

00:00:06.110 --> 00:00:08.386
and we got a Tweet in from Jane,

00:00:08.386 --> 00:00:10.708
and Jane asks, "Could you take a video,

00:00:10.708 --> 00:00:12.658
"could you make a video
on solving complex circuit

00:00:12.658 --> 00:00:14.005
"problems in physics?

00:00:14.005 --> 00:00:17.790
"For example, something
similar to this question."

00:00:17.790 --> 00:00:19.786
So, Jane sent us a circuit here,

00:00:19.786 --> 00:00:21.760
here it is over here,

00:00:21.760 --> 00:00:23.014
and it's an interesting looking circuit.

00:00:23.014 --> 00:00:24.895
We're going to take a
shot at solving it here,

00:00:24.895 --> 00:00:27.217
and I'll tell you how
I would approach this.

00:00:27.217 --> 00:00:28.633
Let's take a quick look at it.

00:00:28.633 --> 00:00:32.093
We have two batteries in
it, which is a bit unusual.

00:00:32.093 --> 00:00:33.857
There's two resistors here.

00:00:33.857 --> 00:00:35.668
They happen to have the same value,

00:00:35.668 --> 00:00:37.433
1.4 ohms.

00:00:37.433 --> 00:00:39.615
There's a third resistor connected up,

00:00:39.615 --> 00:00:41.659
and the question we're
being asked is to find,

00:00:41.659 --> 00:00:44.677
what's the current in
this resistor right here?

00:00:44.677 --> 00:00:47.603
So that's what we're looking for.

00:00:47.603 --> 00:00:49.530
So the way I approach
these kind of questions,

00:00:49.530 --> 00:00:52.711
first, I sort of just live with
this schematic a little bit.

00:00:52.711 --> 00:00:55.590
I look at it and see if I
can imagine how the currents

00:00:55.590 --> 00:00:58.121
are flowing and what's going on,

00:00:58.121 --> 00:01:01.256
and then a really good
way to do that is to,

00:01:01.256 --> 00:01:03.415
I'm going to draw this schematic myself,

00:01:03.415 --> 00:01:05.040
I'm just going to draw it
over and let me own hand

00:01:05.040 --> 00:01:08.477
kind of learn this circuit as we go,

00:01:08.477 --> 00:01:09.800
so I'll draw the battery.

00:01:09.800 --> 00:01:12.633
Here's the two batteries.

00:01:12.633 --> 00:01:16.459
And I see that they're connected together.

00:01:17.277 --> 00:01:19.575
Okay, that's interesting.

00:01:19.575 --> 00:01:23.732
Each battery then goes to

00:01:23.732 --> 00:01:25.682
a resistor,

00:01:25.682 --> 00:01:28.538
this goes to a resistor,

00:01:28.538 --> 00:01:30.465
and those are connected, let me check,

00:01:30.465 --> 00:01:32.950
yeah, those are connected together,

00:01:32.950 --> 00:01:37.408
and then they go to a
third resistor like this.

00:01:37.408 --> 00:01:40.333
And now if you notice what I'm
doing is I'm taking all the

00:01:40.333 --> 00:01:43.073
funny corners and dots and
curves and things like that

00:01:43.073 --> 00:01:45.209
and I'm just going to
draw a square schematic

00:01:45.209 --> 00:01:47.160
that I can understand.

00:01:47.160 --> 00:01:48.855
Now, I'm going to label
it just to make sure

00:01:48.855 --> 00:01:50.317
I got everything right,

00:01:50.317 --> 00:01:53.568
so, this is 11 volts.

00:01:53.568 --> 00:01:54.891
And I know from the battery symbol,

00:01:54.891 --> 00:01:58.072
that the short line is the minus side,

00:01:58.072 --> 00:02:00.441
and the long line is the plus side,

00:02:00.441 --> 00:02:02.948
and same here.

00:02:02.948 --> 00:02:07.244
So, these two batteries
are hooked up exactly

00:02:07.244 --> 00:02:08.753
the same way in the same direction.

00:02:08.753 --> 00:02:12.166
This one's also 11 volts.

00:02:12.166 --> 00:02:13.815
And that's going to prove
out to be interesting.

00:02:13.815 --> 00:02:16.021
We'll call this one R1

00:02:16.021 --> 00:02:19.666
because the problem had i1 here.

00:02:19.666 --> 00:02:22.963
We'll call this one R2.

00:02:22.963 --> 00:02:27.963
And we'll call this one R3, of course.

00:02:28.420 --> 00:02:30.974
And the current that we're
looking for is this current

00:02:30.974 --> 00:02:33.342
right here, what's that?

00:02:33.342 --> 00:02:36.012
Okay.

00:02:36.012 --> 00:02:37.800
So, before I start doing
mathematics on this,

00:02:37.800 --> 00:02:40.679
what I'm going to do is
see if I can simplify

00:02:40.679 --> 00:02:44.603
the diagram, and what
I'm going to notice here,

00:02:44.603 --> 00:02:48.783
this is kind of a trick
that was done here.

00:02:48.783 --> 00:02:51.012
Let's look at these batteries.

00:02:51.012 --> 00:02:53.473
These two ends of the
batteries are connected

00:02:53.473 --> 00:02:56.422
together, so they are at the same voltage.

00:02:56.422 --> 00:02:57.978
We don't know what that
voltage is, but we know

00:02:57.978 --> 00:03:02.227
it's the same, and we know
that they're both 11 volts

00:03:02.227 --> 00:03:04.432
and that means that this node here,

00:03:04.432 --> 00:03:08.890
this node here, and this node here

00:03:08.890 --> 00:03:13.232
are at exactly the same voltage

00:03:13.232 --> 00:03:16.831
with respect to this node here.

00:03:16.831 --> 00:03:20.709
This one is 11 volts lower in voltage

00:03:20.709 --> 00:03:25.376
and this node is 11
volts lower in voltage,

00:03:25.376 --> 00:03:27.187
so what I'm going to imagine in my head,

00:03:27.187 --> 00:03:29.973
if I connected a wire
between these two nodes,

00:03:29.973 --> 00:03:31.529
this end would be at 11 volts,

00:03:31.529 --> 00:03:33.549
this end would be at 11 volts.

00:03:33.549 --> 00:03:36.451
So no current would
flow through that wire.

00:03:36.451 --> 00:03:38.332
It would have no effect on the circuit.

00:03:38.332 --> 00:03:38.959
It wouldn't,

00:03:38.959 --> 00:03:41.560
there would be no current
like diverted from here

00:03:41.560 --> 00:03:42.535
over to here.

00:03:42.535 --> 00:03:45.367
I'm actually going to draw that wire in.

00:03:45.367 --> 00:03:48.084
I'm going to draw that wire in,

00:03:48.084 --> 00:03:50.778
and I didn't change the
circuit when I did that.

00:03:50.778 --> 00:03:53.262
The same current goes
through this wire as before

00:03:53.262 --> 00:03:55.212
because zero goes this way,

00:03:55.212 --> 00:03:56.907
the same current goes through this wire

00:03:56.907 --> 00:03:58.858
because zero current goes this way

00:03:58.858 --> 00:04:00.646
because I know they're both at 11,

00:04:00.646 --> 00:04:04.430
at the same potential above here.

00:04:04.430 --> 00:04:06.938
Right, so this is a good
time to redraw this again.

00:04:06.938 --> 00:04:10.351
If you look here, I have two batteries

00:04:10.351 --> 00:04:11.512
that are hooked up,

00:04:11.512 --> 00:04:12.743
their inputs,

00:04:12.743 --> 00:04:14.368
their positive side is hooked up together

00:04:14.368 --> 00:04:16.435
and their negative side
is hooked up together,

00:04:16.435 --> 00:04:20.219
so they're actually just
acting like one, big battery,

00:04:20.219 --> 00:04:21.218
so let me draw that.

00:04:21.218 --> 00:04:26.218
I'm going to draw the circuit
again so it looks like this.

00:04:26.233 --> 00:04:30.895
Here's my combined big battery.

00:04:31.666 --> 00:04:35.236
And it goes to...

00:04:51.379 --> 00:04:55.553
Relabel these again so
we don't get mixed up.

00:04:55.791 --> 00:04:56.905
Okay.

00:04:56.905 --> 00:04:59.459
This is R1.

00:04:59.459 --> 00:05:02.269
This is R2.

00:05:02.269 --> 00:05:06.170
And this is R3.

00:05:06.170 --> 00:05:09.490
So this circuit looks a little simpler,

00:05:09.490 --> 00:05:10.419
and I'm gonna look at it again,

00:05:10.419 --> 00:05:12.323
see if I can do any more simplification,

00:05:12.323 --> 00:05:14.877
so what I recognize right
here, right in this area right

00:05:14.877 --> 00:05:18.476
here, R1 and R2 are in parallel.

00:05:18.476 --> 00:05:22.144
They have the same voltage
on their terminals.

00:05:22.144 --> 00:05:24.373
That means they're in parallel.

00:05:24.373 --> 00:05:26.579
I know how to simplify parallel resistors.

00:05:26.579 --> 00:05:29.528
We'll just use the
parallel resistor equation

00:05:29.528 --> 00:05:31.803
that I have in my head,

00:05:31.803 --> 00:05:33.405
and that looks like this,

00:05:33.405 --> 00:05:37.306
let's go to this color here.

00:05:37.306 --> 00:05:41.509
Okay, so parallel resistors, R1

00:05:41.509 --> 00:05:43.993
in parallel with R2.

00:05:43.993 --> 00:05:46.687
I made up this symbol, two vertical lines,

00:05:46.687 --> 00:05:48.219
that means they're in parallel,

00:05:48.219 --> 00:05:51.656
and the formula for two
parallel resistors is R1

00:05:51.656 --> 00:05:54.233
times R2,

00:05:54.233 --> 00:05:59.233
over R1 plus R2.

00:06:00.014 --> 00:06:01.013
Now I'll plug in the values.

00:06:01.013 --> 00:06:03.195
We know that R1 and R2
are actually, if we look

00:06:03.195 --> 00:06:04.798
over here at our schematic,

00:06:04.798 --> 00:06:06.516
they're the same value,

00:06:06.516 --> 00:06:10.022
and that has a special thing
when in parallel resistors

00:06:10.022 --> 00:06:13.923
so it's actually R R

00:06:13.923 --> 00:06:17.888
over 2R.

00:06:18.218 --> 00:06:20.447
Because those resistors are the same.

00:06:20.447 --> 00:06:23.675
And you can see I can cancel
that and I can cancel that

00:06:23.675 --> 00:06:25.114
and two parallel resistors,

00:06:25.114 --> 00:06:27.181
if the resistors are equal,

00:06:27.181 --> 00:06:29.990
is equal to half the resistance.

00:06:29.990 --> 00:06:32.451
And let's plug in the real values,

00:06:32.451 --> 00:06:37.451
1.4 ohms over 2

00:06:37.583 --> 00:06:40.346
equals

00:06:40.346 --> 00:06:42.877
0.7 ohms.

00:06:42.877 --> 00:06:46.058
That's the equivalent resistance
of these two resistors

00:06:46.058 --> 00:06:49.024
in parallel.

00:06:50.168 --> 00:06:52.188
So this is a good time to
redraw this circuit again.

00:06:52.188 --> 00:06:53.604
Let's do it again.

00:06:53.604 --> 00:06:56.965
Here's our battery.

00:06:57.133 --> 00:07:00.477
This time I'm going to draw
the equivalent resistance.

00:07:00.477 --> 00:07:01.986
Then we have R3,

00:07:01.986 --> 00:07:04.656
we'll put in R3 down here.

00:07:04.656 --> 00:07:06.374
Nothing's changed there,

00:07:06.374 --> 00:07:11.157
and that goes back to our voltage source.

00:07:11.157 --> 00:07:13.108
Again, do some labels,

00:07:13.108 --> 00:07:18.108
so this is R1 in parallel with R2.

00:07:19.232 --> 00:07:22.070
This is R3.

00:07:22.070 --> 00:07:25.042
This is the current we
want to know, right there,

00:07:25.042 --> 00:07:28.014
and this is 11 volts,

00:07:28.014 --> 00:07:32.124
that's the plus side and
that's the minus side.

00:07:32.124 --> 00:07:34.191
Okay, we got a pretty
simple circuit here now.

00:07:34.191 --> 00:07:35.375
Let's fill, well,

00:07:35.375 --> 00:07:36.373
let me fill in the values

00:07:36.373 --> 00:07:38.486
for

00:07:38.486 --> 00:07:40.344
our resistor here.

00:07:40.344 --> 00:07:44.988
R1 in parallel with R2 is 0.7 ohms.

00:07:44.988 --> 00:07:47.426
R3, let's go look that
up, remind ourselves,

00:07:47.426 --> 00:07:50.810
2.7 ohms.

00:07:52.975 --> 00:07:54.298
Now I can almost do this in my head.

00:07:54.298 --> 00:07:57.479
These are both now just
two series resistors.

00:07:57.479 --> 00:07:59.871
So, Ohm's law equals

00:07:59.871 --> 00:08:02.541
V equals iR.

00:08:02.541 --> 00:08:05.165
This is the form of it I always memorized.

00:08:05.165 --> 00:08:07.185
If I want to know i, I'm
solving for a current,

00:08:07.185 --> 00:08:12.079
so i equals V over R.

00:08:12.572 --> 00:08:14.708
We can start filling in values,

00:08:14.708 --> 00:08:19.708
i equals 11 volts over,

00:08:19.839 --> 00:08:20.629
what's the resistance?

00:08:20.629 --> 00:08:22.672
It's the series resistance here,

00:08:22.672 --> 00:08:26.642
so it's the sum of these two resistors,

00:08:26.642 --> 00:08:29.522
which is 0.7

00:08:29.522 --> 00:08:33.371
plus 2.7.

00:08:35.280 --> 00:08:38.949
If I work that out,

00:08:38.949 --> 00:08:43.949
it equals 11 over 3.4

00:08:45.798 --> 00:08:47.470
equals,

00:08:47.470 --> 00:08:48.724
okay let me actually work that out.

00:08:48.724 --> 00:08:51.812
It's 11 volts

00:08:51.812 --> 00:08:56.812
divided by 3.4 ohms

00:08:57.292 --> 00:08:59.404
and that equals

00:08:59.404 --> 00:09:04.404
3.23 blah-blah-blah

00:09:05.929 --> 00:09:08.135
something something something.

00:09:08.135 --> 00:09:11.594
And that will be in Amps.

00:09:11.594 --> 00:09:15.031
So, this right here is our answer.

00:09:15.031 --> 00:09:18.096
Right there.

00:09:18.096 --> 00:09:19.860
So find the current i,

00:09:19.860 --> 00:09:24.860
i equals 3.2 Amperes.

00:09:27.128 --> 00:09:28.985
And now I'm going to check,

00:09:28.985 --> 00:09:31.307
actually can check to
see if that's the answer.

00:09:31.307 --> 00:09:33.536
Let's see what Jane sent.

00:09:33.536 --> 00:09:35.046
She actually sent in the answer.

00:09:35.046 --> 00:09:37.368
So thank you, Jane, for doing that,

00:09:37.368 --> 00:09:40.618
and, there it is.

00:09:40.618 --> 00:09:42.336
So that's it.

00:09:42.336 --> 00:09:43.840
Thank you very much for listening.