How many people can a blimp carry

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.630
A helium-filled
blimp stays aloft

00:00:03.630 --> 00:00:07.020
since helium is less
dense than regular air.

00:00:07.020 --> 00:00:09.810
So helium will rise
above regular air.

00:00:09.810 --> 00:00:12.050
That was a strangely-phrased
first sentence,

00:00:12.050 --> 00:00:15.500
but they're just saying helium
is less dense, so helium

00:00:15.500 --> 00:00:18.350
will rise relative
to regular air.

00:00:18.350 --> 00:00:22.280
The density of helium
is about 1.1 kilograms

00:00:22.280 --> 00:00:26.860
per meter cubed less
than regular air.

00:00:26.860 --> 00:00:28.950
Which means a cubic
meter of helium

00:00:28.950 --> 00:00:32.640
can lift about 1.1 kilograms.

00:00:32.640 --> 00:00:38.630
Because the density of helium--
so for a cubic meter of helium,

00:00:38.630 --> 00:00:43.990
so if this is a cubic
meter of helium,

00:00:43.990 --> 00:00:47.310
and here is a cubic
meter of regular air,

00:00:47.310 --> 00:00:55.170
they're saying that the mass
of this cubic meter of helium

00:00:55.170 --> 00:00:58.630
is going to be
1.1 kilograms less

00:00:58.630 --> 00:01:02.830
than the mass of this
cubic meter of regular air.

00:01:02.830 --> 00:01:06.570
And that's what allows it,
because of the buoyancy, that's

00:01:06.570 --> 00:01:09.320
what allows a cubic
meter of helium

00:01:09.320 --> 00:01:13.580
to be able to lift
about 1.1 kilograms.

00:01:13.580 --> 00:01:18.350
A blimp with a mass of 2.11
times 10 to the fifth kilograms

00:01:18.350 --> 00:01:24.780
can carry as much as 2 times
10 to the fifth cubic meters

00:01:24.780 --> 00:01:26.250
of helium.

00:01:26.250 --> 00:01:28.870
If the average mass of
a person with luggage

00:01:28.870 --> 00:01:32.510
is 10 to the second
kilograms, what

00:01:32.510 --> 00:01:35.226
is the number of passengers
the blimp can carry?

00:01:35.226 --> 00:01:36.850
So I encourage you
to pause this video,

00:01:36.850 --> 00:01:38.900
and I'll give you a
little structure for this.

00:01:38.900 --> 00:01:42.400
Think about how
much mass 2 times 10

00:01:42.400 --> 00:01:46.280
to the fifth cubic meters
of helium can carry.

00:01:46.280 --> 00:01:49.160
And then just realize it
has to carry the blimp,

00:01:49.160 --> 00:01:51.910
and then, what's
left over, is what

00:01:51.910 --> 00:01:54.540
we can use to carry people.

00:01:54.540 --> 00:01:56.240
So I'm assuming you've
given a go at it.

00:01:56.240 --> 00:01:57.656
So the first step
is to say, look,

00:01:57.656 --> 00:02:04.170
we have 2 times 10 to the
fifth cubic meters of helium.

00:02:04.170 --> 00:02:12.070
And helium, a cubic meter of
helium, can lift 1.1 kilograms.

00:02:12.070 --> 00:02:19.370
So we could say that we can lift
1.1 kilograms per cubic meter.

00:02:22.000 --> 00:02:23.550
So what is this
going to give us?

00:02:23.550 --> 00:02:25.280
Well our units cancel nicely.

00:02:25.280 --> 00:02:29.740
This gives us the mass of
kilograms that we can lift.

00:02:29.740 --> 00:02:31.640
Meters cubed canceling
with meters cubed.

00:02:31.640 --> 00:02:33.640
You'd have a meter cubed
in the numerator, meter

00:02:33.640 --> 00:02:34.810
cubed in the denominator.

00:02:34.810 --> 00:02:37.430
Our dimensional analysis
allows our units

00:02:37.430 --> 00:02:39.940
to cancel just like
numbers would in fractions.

00:02:39.940 --> 00:02:43.550
And so we're left with 2 times
10 to the fifth times 1.1.

00:02:43.550 --> 00:02:46.400
Well 2 times 1.1 is 2.2.

00:02:46.400 --> 00:02:52.400
So this is going to give us 2.2
times 10 to the fifth kilograms

00:02:52.400 --> 00:02:54.150
is how much we can lift.

00:02:54.150 --> 00:03:00.550
So this is how much we can lift.

00:03:00.550 --> 00:03:02.990
Once again, and this is probably
the most important step,

00:03:02.990 --> 00:03:06.240
this is our mass of helium
in terms of cubic meters,

00:03:06.240 --> 00:03:10.300
and this is how many kilograms
we can lift per cubic meter.

00:03:10.300 --> 00:03:12.680
So you multiply
those two, we get

00:03:12.680 --> 00:03:16.930
in kilograms how much
mass we can lift.

00:03:16.930 --> 00:03:20.060
So now let's subtract out the
things that we need to lift.

00:03:20.060 --> 00:03:24.500
We need to subtract out the
mass of the actual blimp,

00:03:24.500 --> 00:03:27.600
the actual structure that
is holding the helium.

00:03:27.600 --> 00:03:28.840
So let's subtract that.

00:03:28.840 --> 00:03:34.790
So let's subtract 2.11 times
10 to the fifth kilograms.

00:03:34.790 --> 00:03:38.610
And so after lifting the blimp,
what do we have left over?

00:03:38.610 --> 00:03:39.820
Well let's see.

00:03:39.820 --> 00:03:44.542
2.2-- maybe I should, well let
me make it-- 2.2 minus 2.11.

00:03:44.542 --> 00:03:46.250
We might be able to
do that in our heads.

00:03:46.250 --> 00:03:50.880
That's the same thing
as 2.20 minus 2.11.

00:03:50.880 --> 00:03:52.630
20 minus 11 is 9.

00:03:52.630 --> 00:03:54.220
So we could think
of it that way.

00:03:54.220 --> 00:03:55.930
It's 0.09.

00:03:55.930 --> 00:03:58.340
Or if you want, you could
say well, 0 minus 1,

00:03:58.340 --> 00:04:00.830
well, let's do a little
bit of regrouping here.

00:04:00.830 --> 00:04:04.520
10 minus 1 is 9, 1 minus
1 is 0, 2 minus 2 is 0.

00:04:04.520 --> 00:04:13.380
So this gives us 0.09
kilograms of lifting capacity

00:04:13.380 --> 00:04:18.304
after we've netted out
the mass of the blimp.

00:04:18.304 --> 00:04:19.470
Actually, let me be careful.

00:04:19.470 --> 00:04:24.920
0.09 times 10 to
the fifth kilograms

00:04:24.920 --> 00:04:28.540
that we can lift after netting
out the mass of the blimp.

00:04:28.540 --> 00:04:31.490
This is what we can use to lift
the people and their luggage

00:04:31.490 --> 00:04:32.280
with.

00:04:32.280 --> 00:04:34.670
So they tell us that
the people and luggage--

00:04:34.670 --> 00:04:36.560
that each person plus
that person's luggage--

00:04:36.560 --> 00:04:39.310
is 10 to the second kilograms.

00:04:39.310 --> 00:04:41.280
So let's divide this
by 10 to the second.

00:04:44.810 --> 00:04:47.300
And I guess we could say
kilograms per person.

00:04:50.150 --> 00:04:52.290
And then what do we end up with?

00:04:52.290 --> 00:04:54.380
Well this is going
to be equal to 10

00:04:54.380 --> 00:04:57.700
to the fifth divided
by 10 squared is

00:04:57.700 --> 00:05:00.190
going to be 10 to the third.

00:05:00.190 --> 00:05:03.750
So this is going to
simplify to 10 to the third.

00:05:03.750 --> 00:05:08.610
So we get 0.09 times
10 to the third.

00:05:08.610 --> 00:05:12.080
And then kilograms divided
by kilograms per person.

00:05:12.080 --> 00:05:14.080
If we just think about
the dimensional analysis,

00:05:14.080 --> 00:05:16.060
this is the same
thing as kilograms

00:05:16.060 --> 00:05:24.420
times the reciprocal of this, of
people, or person per kilogram.

00:05:24.420 --> 00:05:26.400
Kilogram gets canceled
out, and you're just

00:05:26.400 --> 00:05:28.450
left with people, or persons.

00:05:28.450 --> 00:05:31.080
So this is how many people,
and when we say people,

00:05:31.080 --> 00:05:32.910
we're saying the people
plus their luggage.

00:05:32.910 --> 00:05:35.340
So what is 0.09 times
10 to the third?

00:05:35.340 --> 00:05:39.240
Well that's 0.09 times 1,000.

00:05:39.240 --> 00:05:40.490
So let's multiply it.

00:05:40.490 --> 00:05:44.630
If we start with 0.09 and
we multiply it by 10 once,

00:05:44.630 --> 00:05:46.280
we get to 0.9.

00:05:46.280 --> 00:05:48.510
Multiply it by 10
again, we get to 9.

00:05:48.510 --> 00:05:51.770
Multiply it by 10
again, we get to 90.

00:05:51.770 --> 00:05:56.550
So this right over here is
the same thing as 90 people.

00:05:56.550 --> 00:05:59.370
So we could lift,
the maximum load

00:05:59.370 --> 00:06:03.867
for this blimp is 90
people and their luggage.


Finding surface area: nets of polyhedra

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:02.510
Teddy knows that a
figure has a surface

00:00:02.510 --> 00:00:04.960
area of 40 square centimeters.

00:00:04.960 --> 00:00:10.790
The net below has 5 centimeter
and 2 centimeter edges.

00:00:10.790 --> 00:00:13.330
Could the net below
represent the figure?

00:00:13.330 --> 00:00:15.730
So let's just make sure we
understand what this here

00:00:15.730 --> 00:00:16.730
represents.

00:00:16.730 --> 00:00:19.030
So it tells us that it
has 5 centimeter edges.

00:00:19.030 --> 00:00:20.950
So this is one of
the 5 centimeter

00:00:20.950 --> 00:00:22.860
edges right over here.

00:00:22.860 --> 00:00:25.430
And we know that it has several
other 5 centimeter edges

00:00:25.430 --> 00:00:28.330
because any edge that
has this double hash mark

00:00:28.330 --> 00:00:31.020
right over here is also
going to be 5 centimeters.

00:00:31.020 --> 00:00:33.380
So this edge is
also 5 centimeters,

00:00:33.380 --> 00:00:38.544
this is also 5 centimeters,
this is also 5 centimeters,

00:00:38.544 --> 00:00:40.710
and then these two over
here are also 5 centimeters.

00:00:40.710 --> 00:00:43.710
So that's 5 centimeters,
and that's 5 centimeters.

00:00:43.710 --> 00:00:46.210
And then we have several
2 centimeter edges.

00:00:46.210 --> 00:00:47.710
So this one has 2 centimeters.

00:00:47.710 --> 00:00:50.290
And any other edge that has
the same number of hash marks,

00:00:50.290 --> 00:00:53.170
in this case one, is also
going to be 2 centimeters.

00:00:53.170 --> 00:00:55.430
So all of these other
edges, pretty much

00:00:55.430 --> 00:00:58.759
all the rest of the edges,
are going to be 2 centimeters.

00:00:58.759 --> 00:01:00.800
Now, they don't ask us to
do this in the problem,

00:01:00.800 --> 00:01:03.460
but it's always fun to
start with a net like this

00:01:03.460 --> 00:01:07.080
and try to visualize the
polyhedron that it actually

00:01:07.080 --> 00:01:07.680
represents.

00:01:07.680 --> 00:01:09.138
It looks pretty
clear this is going

00:01:09.138 --> 00:01:10.300
to be a rectangular prism.

00:01:10.300 --> 00:01:11.760
But let's actually draw it.

00:01:11.760 --> 00:01:15.220
So if we were to-- we're
going to fold this in.

00:01:15.220 --> 00:01:18.060
We're going to
fold this that way.

00:01:18.060 --> 00:01:20.710
You could view this as
our base right over here.

00:01:20.710 --> 00:01:22.040
We're going to fold this in.

00:01:22.040 --> 00:01:23.410
We're going to fold that up.

00:01:23.410 --> 00:01:25.280
And then this is
going to be our top.

00:01:25.280 --> 00:01:27.180
This is the top right over here.

00:01:27.180 --> 00:01:30.260
This polyhedron is going to
look something like this.

00:01:30.260 --> 00:01:33.940
So you're going to
have your base that

00:01:33.940 --> 00:01:37.020
has a length of 5 centimeters.

00:01:37.020 --> 00:01:38.250
So this is our base.

00:01:38.250 --> 00:01:39.850
Let me do that in a new color.

00:01:39.850 --> 00:01:42.480
So this is our base
right over here.

00:01:42.480 --> 00:01:44.620
I'll do it in the same color.

00:01:44.620 --> 00:01:49.680
So that's our base, this
dimension right over here.

00:01:49.680 --> 00:01:51.790
I could put the double
hash marks if I want.

00:01:51.790 --> 00:01:53.670
5 centimeters, and
that's of course

00:01:53.670 --> 00:01:56.650
the same as that
dimension up there.

00:01:56.650 --> 00:02:01.360
Now, when we fold up this
side-- we'll do this in orange,

00:02:01.360 --> 00:02:04.030
actually-- when we
fold up that side, that

00:02:04.030 --> 00:02:07.670
could be this side
right over here,

00:02:07.670 --> 00:02:10.050
along this 2 centimeter edge.

00:02:10.050 --> 00:02:11.940
So that's that side
right over here.

00:02:11.940 --> 00:02:17.084
When you fold this side in right
over here, that could be that.

00:02:17.084 --> 00:02:20.370
That's that side
right over there.

00:02:20.370 --> 00:02:23.990
And then when of course we
fold this side in-- that's

00:02:23.990 --> 00:02:24.680
the same color.

00:02:24.680 --> 00:02:25.846
Let me do a different color.

00:02:25.846 --> 00:02:27.740
When we fold this
side in, that's

00:02:27.740 --> 00:02:30.210
the side that's kind of
facing us a little bit.

00:02:30.210 --> 00:02:32.710
So that's that right over there.

00:02:32.710 --> 00:02:34.590
That's that right over there.

00:02:34.590 --> 00:02:36.430
Color that in a
little bit better.

00:02:36.430 --> 00:02:39.960
And then we can
fold this side in,

00:02:39.960 --> 00:02:42.460
and that would be that side.

00:02:42.460 --> 00:02:43.960
And then, of course,
we have the top

00:02:43.960 --> 00:02:45.335
that's connected
right over here.

00:02:45.335 --> 00:02:49.894
So the top would go--
this would be the top,

00:02:49.894 --> 00:02:51.310
and then the top
would, of course,

00:02:51.310 --> 00:02:53.654
go on top of our
rectangular prism.

00:02:53.654 --> 00:02:55.570
So that's the figure
that we're talking about.

00:02:55.570 --> 00:02:57.750
It's 5 centimeters
in this dimension.

00:02:57.750 --> 00:03:05.420
It is 2 centimeters tall,
and it is 2 centimeters wide.

00:03:05.420 --> 00:03:07.830
But let's go back to
the original question.

00:03:07.830 --> 00:03:12.170
Is this thing's surface
area 40 square centimeters?

00:03:12.170 --> 00:03:13.990
Well, the good
thing about this net

00:03:13.990 --> 00:03:16.380
here is it's laid out all
of the surfaces for us,

00:03:16.380 --> 00:03:18.640
so we just have to figure
out the surface area of each

00:03:18.640 --> 00:03:21.660
of these sections and then
add them together, the surface

00:03:21.660 --> 00:03:24.040
area of each of these surfaces.

00:03:24.040 --> 00:03:26.190
So what is the surface
area of this one here?

00:03:26.190 --> 00:03:28.241
Well, it's going to be
5 centimeters times 2

00:03:28.241 --> 00:03:28.740
centimeters.

00:03:28.740 --> 00:03:30.921
So it's going to be
10 square centimeters.

00:03:30.921 --> 00:03:31.920
Same thing for this one.

00:03:31.920 --> 00:03:34.810
It's going to be 5 by 2, 5 by 2.

00:03:34.810 --> 00:03:36.820
This one is 5 by 2.

00:03:36.820 --> 00:03:38.520
So these are each 10
square centimeters,

00:03:38.520 --> 00:03:39.670
and so is this one.

00:03:39.670 --> 00:03:43.900
This is 5 long, 5 centimeters
long, 2 centimeters wide.

00:03:43.900 --> 00:03:47.030
So once again, that's
10 square centimeters.

00:03:47.030 --> 00:03:49.210
Now, these two sections
right over here, they're

00:03:49.210 --> 00:03:50.770
2 centimeters by 2 centimeters.

00:03:50.770 --> 00:03:54.940
So they're each going to
be 4 square centimeters.

00:03:54.940 --> 00:03:57.000
So what's the
total surface area?

00:03:57.000 --> 00:03:59.650
Well, 10 plus 10 plus
10 plus 10 is 40,

00:03:59.650 --> 00:04:04.480
plus 4 plus 4 gets us to
48 square centimeters,

00:04:04.480 --> 00:04:06.330
or centimeters squared.

00:04:06.330 --> 00:04:09.720
So could the net below
represent the figure

00:04:09.720 --> 00:04:12.380
that has a surface area
of 40 square centimeters?

00:04:12.380 --> 00:04:13.670
No.

00:04:13.670 --> 00:04:15.500
This represents a figure
that has a surface

00:04:15.500 --> 00:04:18.192
area of 48 square centimeters.


Nets of polyhedra

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.610
- [Instructor] What we're
gonna explore in this video

00:00:02.610 --> 00:00:05.733
are polyhedra, polyhedra,

00:00:07.020 --> 00:00:09.600
which is just the plural of a polyhedron.

00:00:09.600 --> 00:00:13.200
And a polyhedron is a
three-dimensional shape

00:00:13.200 --> 00:00:15.720
that has flat surfaces and straight edges.

00:00:15.720 --> 00:00:20.640
So for example, a cube is a polyhedron.

00:00:20.640 --> 00:00:22.860
A cube is a polyhedron.

00:00:22.860 --> 00:00:25.140
All the surfaces are flat.

00:00:25.140 --> 00:00:26.850
All the surfaces are flat.

00:00:26.850 --> 00:00:28.470
And all of the edges,

00:00:28.470 --> 00:00:31.080
all of the edges are straight.

00:00:31.080 --> 00:00:34.140
So this right over here is a polyhedron.

00:00:34.140 --> 00:00:35.640
Once again, polyhedra is plural.

00:00:35.640 --> 00:00:37.830
Polyhedron is when you have one of them.

00:00:37.830 --> 00:00:39.330
This is a polyhedron.

00:00:39.330 --> 00:00:42.060
A rectangular pyramid is a polyhedron.

00:00:42.060 --> 00:00:42.990
So let me draw that.

00:00:42.990 --> 00:00:45.090
I'll make this one a little
bit more transparent.

00:00:45.090 --> 00:00:47.670
And lemme do this in a
different color just for fun.

00:00:47.670 --> 00:00:51.630
I'll make it a magenta,
rectangular pyramid.

00:00:51.630 --> 00:00:55.560
So once again, here I
have one flat surface.

00:00:55.560 --> 00:00:58.500
And then I'm gonna have four,

00:00:58.500 --> 00:01:01.170
four triangular flat surfaces.

00:01:01.170 --> 00:01:02.370
So this right over here,

00:01:02.370 --> 00:01:04.920
this is a rectangular pyramid.

00:01:04.920 --> 00:01:06.510
Now, it clearly looks like a pyramid.

00:01:06.510 --> 00:01:08.220
Why is it called a rectangular pyramid?

00:01:08.220 --> 00:01:13.220
Because the base right
over here is a rectangle.

00:01:13.290 --> 00:01:18.150
So these are just a few
examples of polyhedra.

00:01:18.150 --> 00:01:21.270
Now, what I wanna think
about are nets of polyhedron.

00:01:21.270 --> 00:01:23.070
Actually, let me draw and
make this transparent too.

00:01:23.070 --> 00:01:25.860
So we get full appreciation
of the entire polyhedron.

00:01:25.860 --> 00:01:28.350
Polyhedron, this entire cube.

00:01:28.350 --> 00:01:32.370
So now let's think about
nets of polyhedron, nets.

00:01:32.370 --> 00:01:36.060
So what is a net of a polyhedron?

00:01:36.060 --> 00:01:37.410
Well, one way to think about it

00:01:37.410 --> 00:01:40.470
is if you kind of viewed
this as made up of cardboard

00:01:40.470 --> 00:01:42.390
and you were to unfold it in some way

00:01:42.390 --> 00:01:43.560
so it would become flat.

00:01:43.560 --> 00:01:44.730
Or another way of thinking about it

00:01:44.730 --> 00:01:47.040
is if you were to cut out
some cardboard or some paper

00:01:47.040 --> 00:01:49.620
and you wanted to fold it up
into one of these figures,

00:01:49.620 --> 00:01:51.720
how would you go about doing it?

00:01:51.720 --> 00:01:53.790
And each of these polyhedra

00:01:53.790 --> 00:01:56.880
has multiple different
nets that you could create.

00:01:56.880 --> 00:02:00.660
So that can be folded up into
this three dimensional figure.

00:02:00.660 --> 00:02:02.580
So let's take an example.

00:02:02.580 --> 00:02:05.820
And maybe the simplest example
would be a cube like this.

00:02:05.820 --> 00:02:06.840
And I'm gonna color code it.

00:02:06.840 --> 00:02:10.110
So let's say that the bottom of this cube,

00:02:10.110 --> 00:02:12.810
the bottom of this cube
was this green color.

00:02:12.810 --> 00:02:15.810
And so I can represent it like this.

00:02:15.810 --> 00:02:17.760
That's the bottom of the cube.

00:02:17.760 --> 00:02:19.800
It's that green color.

00:02:19.800 --> 00:02:24.450
Now let's say that this
back surface of the cube

00:02:24.450 --> 00:02:25.740
is orange.

00:02:25.740 --> 00:02:28.110
Well, I could represent it like this.

00:02:28.110 --> 00:02:31.500
And notice I've kind of folded it out.

00:02:31.500 --> 00:02:32.670
I'm folding it out.

00:02:32.670 --> 00:02:34.350
And so if I were to flatten it out,

00:02:34.350 --> 00:02:36.450
it would look like this.

00:02:36.450 --> 00:02:38.850
Would look like that.

00:02:38.850 --> 00:02:43.110
Now, this other backside,
I'll shade it in yellow.

00:02:43.110 --> 00:02:44.670
This other backside right over here,

00:02:44.670 --> 00:02:46.320
I could fold it backwards

00:02:46.320 --> 00:02:49.050
and keep it connected along this edge.

00:02:49.050 --> 00:02:51.000
Keep it connected along this edge,

00:02:51.000 --> 00:02:52.440
fold it backwards,

00:02:52.440 --> 00:02:54.450
it would look like this.

00:02:54.450 --> 00:02:55.980
It would look like that.

00:02:55.980 --> 00:02:57.540
I think you get the general idea here.

00:02:57.540 --> 00:02:59.310
And just to be clear,

00:02:59.310 --> 00:03:00.900
this edge right over here

00:03:00.900 --> 00:03:03.210
is this edge right over there.

00:03:03.210 --> 00:03:06.720
Now I have to worry about this top part.

00:03:06.720 --> 00:03:08.520
The top part of the cube.

00:03:08.520 --> 00:03:10.020
Maybe it is in,

00:03:10.020 --> 00:03:12.150
lemme do it in a pink color.

00:03:12.150 --> 00:03:15.540
This top part of the cube
is in this pink color,

00:03:15.540 --> 00:03:18.360
and it needs to be attached
to one of these sides.

00:03:18.360 --> 00:03:20.700
I could attach it to
this side or this side.

00:03:20.700 --> 00:03:22.200
Let's attach it over here.

00:03:22.200 --> 00:03:24.630
So let's say it's attached to
that yellow side back here.

00:03:24.630 --> 00:03:26.490
So then when we fold it out,

00:03:26.490 --> 00:03:28.830
when we really unpack the thing,

00:03:28.830 --> 00:03:30.480
so we folded that yellow part back,

00:03:30.480 --> 00:03:32.310
then we're folding this part back,

00:03:32.310 --> 00:03:34.053
then it would be right over here.

00:03:34.980 --> 00:03:37.320
It would be right over there.

00:03:37.320 --> 00:03:39.180
And then we could fold,

00:03:39.180 --> 00:03:41.460
we could fold this front face,

00:03:41.460 --> 00:03:43.050
this front face right over here,

00:03:43.050 --> 00:03:46.920
we could fold that out
and this along this edge

00:03:46.920 --> 00:03:49.770
and it would go right over there.

00:03:49.770 --> 00:03:50.940
It would go right over there.

00:03:50.940 --> 00:03:54.240
And then we have one
face of the cube left.

00:03:54.240 --> 00:03:57.060
We have this side right over here.

00:03:57.060 --> 00:04:00.210
And we could, well, we could
do actually several things.

00:04:00.210 --> 00:04:02.400
We could fold it out along this edge

00:04:02.400 --> 00:04:03.540
and then we would draw,

00:04:03.540 --> 00:04:06.254
we would draw the
surface right over there.

00:04:06.254 --> 00:04:09.180
Or if we wanna do something interesting,

00:04:09.180 --> 00:04:11.160
we could fold it out.

00:04:11.160 --> 00:04:13.140
We could fold this out

00:04:13.140 --> 00:04:15.240
along the edge that it
shares with the yellow,

00:04:15.240 --> 00:04:16.140
that backside.

00:04:16.140 --> 00:04:18.180
So we could fold it out like this.

00:04:18.180 --> 00:04:19.500
So if we folded it out like this,

00:04:19.500 --> 00:04:22.350
it would be connected to the
yellow square right over here.

00:04:22.350 --> 00:04:24.570
So you see that there's many, many ways.

00:04:24.570 --> 00:04:28.320
There's many, many ways to construct a net

00:04:28.320 --> 00:04:30.960
or a net that when you fold it all back up

00:04:30.960 --> 00:04:33.390
will turn into this polyhedron,

00:04:33.390 --> 00:04:34.950
in this case, a cube.

00:04:34.950 --> 00:04:35.910
Let's do one more example.

00:04:35.910 --> 00:04:37.098
Let's do the rectangular pyramid

00:04:37.098 --> 00:04:39.600
'cause all of these had rectangles,

00:04:39.600 --> 00:04:43.230
or in particular these had
squares as our surfaces.

00:04:43.230 --> 00:04:44.850
Now, the most obvious one

00:04:44.850 --> 00:04:48.510
might be to start with
your base right over here.

00:04:48.510 --> 00:04:50.340
Start with your base.

00:04:50.340 --> 00:04:52.980
And then take the different sides

00:04:52.980 --> 00:04:54.990
and then just fold them straight out.

00:04:54.990 --> 00:04:56.310
So for example,

00:04:56.310 --> 00:04:58.470
we could take this side right over here,

00:04:58.470 --> 00:04:59.610
fold it out,

00:04:59.610 --> 00:05:01.110
and it would look like,

00:05:01.110 --> 00:05:03.330
it would look like that.

00:05:03.330 --> 00:05:07.080
We could take this side back here.

00:05:07.080 --> 00:05:09.447
And once again, just fold it out.

00:05:09.447 --> 00:05:13.320
Fold it out, it'll look like that.

00:05:13.320 --> 00:05:15.090
It should be the same
size as that orange side,

00:05:15.090 --> 00:05:18.363
but I'm hand drawing it so
it's not gonna be perfect.

00:05:19.650 --> 00:05:22.350
So that's that right over there.

00:05:22.350 --> 00:05:25.390
And then you could take this front side

00:05:26.460 --> 00:05:27.660
right over here

00:05:27.660 --> 00:05:30.450
and once again, fold
it out along this edge

00:05:30.450 --> 00:05:32.253
so it would look like this.

00:05:33.600 --> 00:05:37.950
And then finally, you could take this side

00:05:37.950 --> 00:05:39.720
right over here

00:05:39.720 --> 00:05:43.230
and once again, fold
it out along this edge

00:05:43.230 --> 00:05:46.110
and it would go right there.

00:05:46.110 --> 00:05:49.170
But this isn't the only net
for this rectangular pyramid.

00:05:49.170 --> 00:05:50.820
There's other options, for example.

00:05:50.820 --> 00:05:52.320
And just to explore one of them,

00:05:52.320 --> 00:05:55.530
instead of folding that
green side out that way,

00:05:55.530 --> 00:05:57.990
instead, instead,

00:05:57.990 --> 00:06:01.380
we might have wanted to
fold it out along this edge,

00:06:01.380 --> 00:06:03.690
along this edge with the
yellow side that you can't see.

00:06:03.690 --> 00:06:05.040
Actually, let's make it
a little bit different.

00:06:05.040 --> 00:06:07.560
Let's fold it out along this
side since we can see the edge.

00:06:07.560 --> 00:06:09.870
And let me label, let me color the edge.

00:06:09.870 --> 00:06:12.060
So this is the edge right over here

00:06:12.060 --> 00:06:13.200
on the blue triangle.

00:06:13.200 --> 00:06:14.730
So this is the edge.

00:06:14.730 --> 00:06:17.760
And when you fold the green triangle out,

00:06:17.760 --> 00:06:19.320
it would look like this.

00:06:19.320 --> 00:06:21.810
If you fold the green triangle out,

00:06:21.810 --> 00:06:23.670
it would look like this.

00:06:23.670 --> 00:06:26.370
So hopefully this gives
you an appreciation.

00:06:26.370 --> 00:06:27.203
This gives you appreciation.

00:06:27.203 --> 00:06:28.410
There's multiple ways

00:06:28.410 --> 00:06:32.610
to unfold these three dimensional
figures, these polyhedra,

00:06:32.610 --> 00:06:33.840
or multiple ways to,

00:06:33.840 --> 00:06:35.550
if you wanted to do a cardboard cutout

00:06:35.550 --> 00:06:37.320
and then fold things back together

00:06:37.320 --> 00:06:38.370
to construct them.

00:06:38.370 --> 00:06:41.070
And these flattened versions
of them, these things,

00:06:41.070 --> 00:06:44.880
these unpacking of these polyhedra,

00:06:44.880 --> 00:06:46.773
we call nets.


Quadrilateral similarity by showing congruent angles

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.519
In the diagram below,
quadrilateral EFGH--

00:00:03.519 --> 00:00:05.990
so that's this one
here-- was obtained

00:00:05.990 --> 00:00:08.310
by performing a sequence
of transformations

00:00:08.310 --> 00:00:11.339
on quadrilateral ABCD.

00:00:11.339 --> 00:00:12.880
And then they tell
us the information

00:00:12.880 --> 00:00:14.046
that's already written here.

00:00:14.046 --> 00:00:15.700
Measure of angle
A is 69 degrees.

00:00:15.700 --> 00:00:18.150
Measure of angle
B is 102 degrees.

00:00:18.150 --> 00:00:21.280
Measure of angle
G is 145 degrees.

00:00:21.280 --> 00:00:24.290
And measure of angle
H is 44 degrees.

00:00:24.290 --> 00:00:26.500
Which of the following
additional facts

00:00:26.500 --> 00:00:30.790
is sufficient to conclude that
the only transformations used

00:00:30.790 --> 00:00:35.270
were translations, rotations,
reflections, and dilations,

00:00:35.270 --> 00:00:38.146
and not any other kind
of transformations?

00:00:38.146 --> 00:00:39.520
So this is another
way of saying,

00:00:39.520 --> 00:00:43.950
if we only used translations,
rotations, reflections,

00:00:43.950 --> 00:00:47.160
and dilations, then that means
that these two figures are

00:00:47.160 --> 00:00:48.134
similar.

00:00:48.134 --> 00:00:49.550
So if this is the
case, then we're

00:00:49.550 --> 00:00:53.960
dealing with similar
quadrilaterals.

00:00:53.960 --> 00:00:56.840
So another way of saying this
is, which of these facts--

00:00:56.840 --> 00:00:58.870
so either this first
fact, or this second fact,

00:00:58.870 --> 00:01:03.120
or both of them, or neither
of them-- which of these facts

00:01:03.120 --> 00:01:07.100
are sufficient in order to
determine that these two

00:01:07.100 --> 00:01:09.550
quadrilaterals are similar,
which is another way of saying

00:01:09.550 --> 00:01:11.720
that you could go from
one to another only

00:01:11.720 --> 00:01:15.600
through translations, rotations,
reflections, and dilations?

00:01:15.600 --> 00:01:20.290
So let's look at this first
set of facts right over here.

00:01:20.290 --> 00:01:24.060
So they give us the measure
of angle C is 145 degrees.

00:01:24.060 --> 00:01:28.120
So they tell us that right
over there is 145 degrees.

00:01:32.582 --> 00:01:34.415
And they tell us that
the measure of angle E

00:01:34.415 --> 00:01:36.040
is 69 degrees.

00:01:36.040 --> 00:01:39.340
So they tell us that
this is 69 degrees.

00:01:39.340 --> 00:01:43.280
So with that piece of
information right over here,

00:01:43.280 --> 00:01:47.420
in each of these quadrilaterals,
we know three of the angles.

00:01:47.420 --> 00:01:50.240
And because we know three,
we can figure out the fourth.

00:01:50.240 --> 00:01:51.850
We can figure it
out because the sum

00:01:51.850 --> 00:01:54.580
of the angles in a
quadrilateral are

00:01:54.580 --> 00:01:57.460
going to add up to 360 degrees.

00:01:57.460 --> 00:02:01.590
So if we wanted to figure
out the measure of angle F

00:02:01.590 --> 00:02:03.150
right over here--
let's say that is

00:02:03.150 --> 00:02:09.410
x-- we could say that x
plus 69 degrees plus 145

00:02:09.410 --> 00:02:17.610
degrees plus 44 degrees
is equal to 360 degrees.

00:02:17.610 --> 00:02:19.620
So let's get the calculator
out right over here.

00:02:19.620 --> 00:02:22.330
Our x would essentially
be 360 minus these things.

00:02:22.330 --> 00:02:37.380
So 360 minus 69 minus 145
minus 44 is 102 degrees.

00:02:37.380 --> 00:02:41.360
So this angle right over
here is 102 degrees.

00:02:41.360 --> 00:02:44.400
Now notice, these two already
have three angles in common.

00:02:44.400 --> 00:02:47.380
They have a 69 degree--
let me color code

00:02:47.380 --> 00:02:50.270
it-- so this angle
right over here

00:02:50.270 --> 00:02:52.550
has a measure of 69 degrees.

00:02:52.550 --> 00:02:54.150
So does this one.

00:02:54.150 --> 00:02:55.660
Then you move one over.

00:02:55.660 --> 00:03:03.080
This angle right over here is
102 degrees, so is this one.

00:03:03.080 --> 00:03:11.640
Then you move to the next
angle is 145 degrees, well,

00:03:11.640 --> 00:03:14.371
that would correspond
to this one.

00:03:14.371 --> 00:03:15.870
And if you know
three of the angles,

00:03:15.870 --> 00:03:17.244
you know that this
last one right

00:03:17.244 --> 00:03:20.920
over here, this fourth one, was
going to have to be 44 degrees.

00:03:20.920 --> 00:03:25.470
You could take 360 and from
it subtract 145, 102, and 69,

00:03:25.470 --> 00:03:27.815
and you will get 44 degrees.

00:03:32.370 --> 00:03:36.600
So you have two quadrilaterals
where all of these angles

00:03:36.600 --> 00:03:38.580
are congruent to each other.

00:03:38.580 --> 00:03:41.000
So if you have these
two shapes and you

00:03:41.000 --> 00:03:43.040
have all the
corresponding angles

00:03:43.040 --> 00:03:46.265
are congruent to
each other, then you

00:03:46.265 --> 00:03:47.630
know that they are similar.

00:03:47.630 --> 00:03:49.730
And if you know that
they are similar,

00:03:49.730 --> 00:03:51.570
you know that you
can go from one

00:03:51.570 --> 00:03:56.345
to another purely through some
combination of translations,

00:03:56.345 --> 00:03:58.470
which is really just shifting
it in two dimensions,

00:03:58.470 --> 00:04:01.590
rotating it around some
points, reflecting it, which

00:04:01.590 --> 00:04:04.710
you can kind of use flipping
it over around some line,

00:04:04.710 --> 00:04:07.450
and dilations, which is
scaling it up or down.

00:04:07.450 --> 00:04:11.520
So the first one
is definitely good.

00:04:11.520 --> 00:04:15.260
Let's see if the second one
could also independently help

00:04:15.260 --> 00:04:18.010
us determine that these
two things are similar.

00:04:18.010 --> 00:04:21.360
So what I'm going to
do is clear this out

00:04:21.360 --> 00:04:24.150
so I have some
space to work with.

00:04:24.150 --> 00:04:26.100
Let me clear it out.

00:04:26.100 --> 00:04:29.230
Let me clear this
right over here.

00:04:29.230 --> 00:04:32.320
And now let's think about
what the second statement is

00:04:32.320 --> 00:04:33.720
telling us.

00:04:33.720 --> 00:04:35.860
The second statement tells
us the measure of angle

00:04:35.860 --> 00:04:37.190
D is 44 degrees.

00:04:37.190 --> 00:04:39.570
So they're saying
that is 44 degrees.

00:04:39.570 --> 00:04:42.775
And that the measure of
angle F is 102 degrees.

00:04:46.270 --> 00:04:50.970
So once again, it's given us
one more angle in each of them.

00:04:50.970 --> 00:04:54.460
And from that, we can
figure out the last angle.

00:04:54.460 --> 00:04:59.237
And so if you know all of the
angles in two quadrilaterals,

00:04:59.237 --> 00:05:00.820
you can determine
if they are similar.

00:05:00.820 --> 00:05:06.750
They're going to be
similar only if you

00:05:06.750 --> 00:05:08.900
have four corresponding
angles that

00:05:08.900 --> 00:05:10.250
are congruent to each other.

00:05:13.120 --> 00:05:15.130
Let me just show
it to you just so I

00:05:15.130 --> 00:05:16.870
don't say that it's enough.

00:05:16.870 --> 00:05:19.200
Let's calculate this last
angle right over here.

00:05:19.200 --> 00:05:21.250
We can get our
calculator back out.

00:05:21.250 --> 00:05:28.970
It's going to be 360
minus 102 minus 145

00:05:28.970 --> 00:05:34.940
minus 44, which is
equal to 69 degrees.

00:05:34.940 --> 00:05:37.740
So this right over
here is 69 degrees.

00:05:37.740 --> 00:05:43.950
So notice, 102 degrees, then
69 degrees, then 44 degrees.

00:05:48.560 --> 00:05:53.190
And so this last one has
to be 145 degrees, just

00:05:53.190 --> 00:05:54.140
like this one.

00:05:54.140 --> 00:05:55.640
If three of the
angles are the same,

00:05:55.640 --> 00:05:57.700
that fourth one
has to be the same.

00:05:57.700 --> 00:06:02.890
So this angle corresponds
to this angle.

00:06:02.890 --> 00:06:06.930
This angle corresponds
to that angle.

00:06:06.930 --> 00:06:09.870
This angle corresponds
to that angle.

00:06:09.870 --> 00:06:14.750
And then finally, this angle
corresponds to this angle.

00:06:14.750 --> 00:06:16.440
You have two quadrilaterals.

00:06:16.440 --> 00:06:19.680
The interior four angles
are all congruent.

00:06:19.680 --> 00:06:20.930
They correspond to each other.

00:06:20.930 --> 00:06:22.207
So they must be similar.

00:06:22.207 --> 00:06:23.790
And so you can get
from one to another

00:06:23.790 --> 00:06:26.630
through translations, rotations,
reflections, and dilations.

00:06:26.630 --> 00:06:29.100
So this one is also good.

00:06:29.100 --> 00:06:31.980
So actually, either of
these are sufficient

00:06:31.980 --> 00:06:34.530
in order to conclude that the
only transformations used were

00:06:34.530 --> 00:06:35.340
these.

00:06:35.340 --> 00:06:39.800
So I would click, both
answers are correct.


Testing similarity through transformations

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:07.250
These two quadrilaterals, EFGH
and ABCD, these are similar.

00:00:07.250 --> 00:00:09.720
If through some combination
of translations, rotations,

00:00:09.720 --> 00:00:12.080
reflections, and
dilations, I can

00:00:12.080 --> 00:00:14.240
make them sit on
top of each other.

00:00:14.240 --> 00:00:17.160
And similarity allows us to use
dilations, which essentially

00:00:17.160 --> 00:00:19.770
means scale up one of
the figures up or down.

00:00:19.770 --> 00:00:21.320
If we're talking
about congruence,

00:00:21.320 --> 00:00:23.444
then we wouldn't be scaling
the figures up or down.

00:00:23.444 --> 00:00:25.910
We would only be translating,
rotating, and reflecting.

00:00:25.910 --> 00:00:27.160
But let's see if we can do it.

00:00:27.160 --> 00:00:29.410
So let's first translate it.

00:00:29.410 --> 00:00:31.340
Let's first translate
this figure.

00:00:31.340 --> 00:00:34.330
So let's make that point
on top of that point.

00:00:34.330 --> 00:00:37.300
And now let's rotate it.

00:00:37.300 --> 00:00:40.800
I'm going to rotate it around
point E right over here.

00:00:40.800 --> 00:00:42.880
So I get right over there.

00:00:42.880 --> 00:00:45.340
So I got two of the sides
to kind of match up.

00:00:45.340 --> 00:00:46.700
But now let me dilate it down.

00:00:46.700 --> 00:00:49.180
Let me scale it down.

00:00:49.180 --> 00:00:51.970
And so this is-- let
me put it on what

00:00:51.970 --> 00:00:53.500
I want to not
change-- and then let

00:00:53.500 --> 00:00:57.700
me see if I can scale this down.

00:00:57.700 --> 00:00:59.200
And I was able to.

00:00:59.200 --> 00:01:01.555
So through just purely
translations, rotations--

00:01:01.555 --> 00:01:03.930
I didn't even have to use the
reflections-- translations,

00:01:03.930 --> 00:01:06.360
rotations, and dilations,
I was able to make

00:01:06.360 --> 00:01:08.360
these sit on top of each other.

00:01:08.360 --> 00:01:12.420
So these two
figures are similar.

00:01:12.420 --> 00:01:14.500
So yes, the rectangles
are similar.

00:01:14.500 --> 00:01:17.380
Let's do one more of these.

00:01:17.380 --> 00:01:21.570
So these two triangles--
and just eyeballing it,

00:01:21.570 --> 00:01:23.710
this one looks kind of
taller than this one does

00:01:23.710 --> 00:01:24.380
right over here.

00:01:24.380 --> 00:01:28.370
So they don't feel similar,
but let me at least try.

00:01:28.370 --> 00:01:33.450
So let me translate, maybe make
this point C might correspond

00:01:33.450 --> 00:01:36.360
to point F, although it already
looks pretty clear that it

00:01:36.360 --> 00:01:37.040
won't.

00:01:37.040 --> 00:01:41.736
But now let me dilate it.

00:01:41.736 --> 00:01:43.425
So I'll keep that
point where it is.

00:01:43.425 --> 00:01:46.550
And let me try to scale this up.

00:01:46.550 --> 00:01:48.920
And then we see
pretty clearly we

00:01:48.920 --> 00:01:53.710
made segment CB sit on top
of, after scaling it up,

00:01:53.710 --> 00:01:55.520
sit on top of segment FE.

00:01:55.520 --> 00:01:58.050
But then everything
else is not matching up.

00:01:58.050 --> 00:01:59.810
Point D is in a
very different place

00:01:59.810 --> 00:02:03.370
than where point A is
once I try to scale things

00:02:03.370 --> 00:02:04.570
up a little bit.

00:02:04.570 --> 00:02:09.650
So these two triangles are
clearly, clearly not similar.


Another example of rigid transformations for congruence

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.150
Perform transformations on
the movable quadrilateral

00:00:04.150 --> 00:00:07.170
until it matches
quadrilateral NEUT,

00:00:07.170 --> 00:00:09.040
quadrilateral NEWT
right over here.

00:00:09.040 --> 00:00:12.110
Are the two figures congruent?

00:00:12.110 --> 00:00:13.530
So we have our
little tools here.

00:00:13.530 --> 00:00:15.571
But I'm actually going to
do it on my scratch pad

00:00:15.571 --> 00:00:19.119
first to think about how we
might want to transform it.

00:00:19.119 --> 00:00:21.660
So the first thing-- it looks
like this point right over here

00:00:21.660 --> 00:00:27.050
would correspond to point E
once we shift it over and then

00:00:27.050 --> 00:00:29.530
maybe dilate it down and
then reflect it over.

00:00:29.530 --> 00:00:30.620
So let's try to do that.

00:00:30.620 --> 00:00:36.270
So the first thing, if
we were to translate,

00:00:36.270 --> 00:00:38.310
so how much are we
going to translate by?

00:00:38.310 --> 00:00:41.100
So we're going to move
to the right by 1.

00:00:41.100 --> 00:00:43.790
So in the x direction, we're
going to increase by 1.

00:00:43.790 --> 00:00:45.285
And in the y
direction, we're going

00:00:45.285 --> 00:00:50.010
to go up by-- we're going
to go from negative 4 to 1.

00:00:50.010 --> 00:00:54.200
So we're going to
translate by 1, 5.

00:00:54.200 --> 00:01:01.500
So that's going to put
our figure like this.

00:01:01.500 --> 00:01:03.800
So that point is going
to be right over here.

00:01:03.800 --> 00:01:06.610
Everything is going to go
in the x direction by 1,

00:01:06.610 --> 00:01:10.240
and then 1, 2, 3, 4,
5 in the y direction.

00:01:10.240 --> 00:01:16.100
So this line is going
to look like that.

00:01:16.100 --> 00:01:21.480
So 1, and then 1, 2, 3, 4, 5 is
going to go right over there.

00:01:21.480 --> 00:01:26.256
So that's going to
go right over there.

00:01:26.256 --> 00:01:27.890
I'll try my best to draw it.

00:01:27.890 --> 00:01:30.080
And then 1 in the x
direction, and then

00:01:30.080 --> 00:01:33.860
1, 2, 3, 4, 5 in
the y direction.

00:01:33.860 --> 00:01:36.419
So it's going to look
something like this.

00:01:36.419 --> 00:01:37.960
Or it's going to
look like this right

00:01:37.960 --> 00:01:40.080
after that transformation.

00:01:40.080 --> 00:01:43.160
And now, what we
could do-- let's see,

00:01:43.160 --> 00:01:45.740
we could either
reflect first, or we

00:01:45.740 --> 00:01:47.140
could try to scale
it down first.

00:01:47.140 --> 00:01:48.390
So let's try to scale it down.

00:01:48.390 --> 00:01:49.470
Let's try to dilate it.

00:01:49.470 --> 00:01:50.590
So let's dilate it.

00:01:53.290 --> 00:01:55.470
And let's make our center.

00:01:55.470 --> 00:01:57.920
I don't want to change
this point right over here.

00:01:57.920 --> 00:01:59.420
I like it sitting
there because that

00:01:59.420 --> 00:02:00.960
seems to be the point
that it corresponds to.

00:02:00.960 --> 00:02:02.084
Let's put the center there.

00:02:02.084 --> 00:02:07.330
So let's put the center
at the point 5 comma 1.

00:02:07.330 --> 00:02:09.460
So the center is going
to be right over there.

00:02:09.460 --> 00:02:14.420
And we essentially want this
line, which has length 6,

00:02:14.420 --> 00:02:17.710
to be scaled down
to have a length 3.

00:02:17.710 --> 00:02:19.470
Or essentially, we
want this point right

00:02:19.470 --> 00:02:22.830
over here, which is 6 to
the left of this point--

00:02:22.830 --> 00:02:25.270
we want it to be 3 to
the left of this point.

00:02:25.270 --> 00:02:27.510
So we want it to
get half as far.

00:02:27.510 --> 00:02:31.894
So we want to scale by 1/2.

00:02:31.894 --> 00:02:33.560
So then that's going
to get the figure--

00:02:33.560 --> 00:02:36.020
this point is going
to be over here.

00:02:36.020 --> 00:02:41.680
This point, which is in
the y direction, 2 away,

00:02:41.680 --> 00:02:44.080
is now going to only be 1 away.

00:02:49.086 --> 00:02:49.960
Let me make it clear.

00:02:49.960 --> 00:02:53.000
In the x direction, it sits
at negative 3 relative to 5.

00:02:53.000 --> 00:02:54.430
So it's 8 away.

00:02:54.430 --> 00:02:56.970
So now it's only going to be
4 away after a scale of 1/2.

00:02:56.970 --> 00:02:58.800
In the y direction, it's 2 away.

00:02:58.800 --> 00:03:01.660
So now, it's only
going to be 1 away.

00:03:01.660 --> 00:03:04.470
So that point is going
to go right over there.

00:03:04.470 --> 00:03:07.269
And then this
point-- or actually,

00:03:07.269 --> 00:03:09.060
let me focus on this
point right over here.

00:03:09.060 --> 00:03:11.000
We're not looking
up there just yet.

00:03:11.000 --> 00:03:15.060
This point right over here,
in the x direction, is 4 away

00:03:15.060 --> 00:03:17.050
and, in the y
direction, is 4 away.

00:03:17.050 --> 00:03:19.320
So at a scale of 1/2
in the x direction,

00:03:19.320 --> 00:03:20.531
it's now going to be 2 away.

00:03:20.531 --> 00:03:22.530
And in the y direction,
it's going to be 2 away.

00:03:22.530 --> 00:03:26.461
So it's going to look like
this after the scaling.

00:03:26.461 --> 00:03:27.710
And now it looks pretty clear.

00:03:27.710 --> 00:03:31.150
We just have to do a
reflection around this line.

00:03:31.150 --> 00:03:36.630
We just have to do a reflection
around the line y equals 1.

00:03:36.630 --> 00:03:42.590
Reflect around the
line y is equal to 1.

00:03:42.590 --> 00:03:45.650
So let's do all of that
on this actual tool now.

00:03:45.650 --> 00:03:47.890
So first, I did the translation.

00:03:47.890 --> 00:03:50.120
I'm going to translate
in the x direction

00:03:50.120 --> 00:03:53.420
by 1 and the y direction by 5.

00:03:53.420 --> 00:03:55.500
That got me right over there.

00:03:55.500 --> 00:03:58.820
And then I wanted
to dilate it down.

00:03:58.820 --> 00:04:01.010
So let me scroll
down a little bit.

00:04:01.010 --> 00:04:05.540
So dilate it about
the point 5 comma 1.

00:04:05.540 --> 00:04:09.130
And then by a scale of 1/2.

00:04:09.130 --> 00:04:10.830
So that scaled me down.

00:04:10.830 --> 00:04:15.720
And now I'm going to
reflect over y equals 1.

00:04:15.720 --> 00:04:18.500
So I'm going to
reflect over the line.

00:04:18.500 --> 00:04:21.182
Well, the line y equals 1,
it goes from the point 0--

00:04:21.182 --> 00:04:22.640
I'm just picking
two points on it--

00:04:22.640 --> 00:04:27.405
so it goes from the
point 0, 1 to 1, 1.

00:04:27.405 --> 00:04:28.280
And there we have it.

00:04:28.280 --> 00:04:30.390
We've gotten to that point.

00:04:30.390 --> 00:04:32.290
Now, we have to
answer their question.

00:04:32.290 --> 00:04:34.190
Are the figures congruent?

00:04:34.190 --> 00:04:34.720
Well, no.

00:04:34.720 --> 00:04:36.890
The first figure was much
bigger than this one.

00:04:36.890 --> 00:04:39.630
And the way that we got it to
be able to squeeze into this one

00:04:39.630 --> 00:04:41.930
is that we dilated it down.

00:04:41.930 --> 00:04:43.600
These figures would
have been congruent

00:04:43.600 --> 00:04:45.770
if all of my
transformations were just

00:04:45.770 --> 00:04:48.910
translations, reflections,
and rotations.

00:04:48.910 --> 00:04:50.030
But I had to dilate it.

00:04:50.030 --> 00:04:51.150
I had to scale the figure.

00:04:51.150 --> 00:04:52.890
Because I had to
scale the figure,

00:04:52.890 --> 00:04:57.730
these figures are not congruent.

00:04:57.730 --> 00:05:01.300
No, the figures
are not congruent.


Example of rigid transformation and congruence

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.980
Starting with the figure
shown, perform the following

00:00:02.980 --> 00:00:05.190
transformations--
reflection over the line y

00:00:05.190 --> 00:00:11.010
equals x minus 2, translation
by 3 in the x direction.

00:00:11.010 --> 00:00:13.130
So we're going to
shift to the right by 3

00:00:13.130 --> 00:00:15.690
and then shift down by 9.

00:00:15.690 --> 00:00:17.802
Is a transformed
figure congruent

00:00:17.802 --> 00:00:18.760
to the original figure?

00:00:18.760 --> 00:00:19.960
And actually, we could
answer this question

00:00:19.960 --> 00:00:21.830
before we even do
the transformation,

00:00:21.830 --> 00:00:23.710
because this transformation
only involves

00:00:23.710 --> 00:00:25.770
reflections and translations.

00:00:25.770 --> 00:00:28.510
As long as we're only
reflecting, translating,

00:00:28.510 --> 00:00:31.310
and rotating, we're going to
get to a congruent figure.

00:00:31.310 --> 00:00:34.050
The thing that will make it
not necessarily congruent

00:00:34.050 --> 00:00:35.820
is when we dilate it.

00:00:35.820 --> 00:00:38.496
So dilating it is going to
actually scale it up or down.

00:00:38.496 --> 00:00:39.870
The figure will
still be similar,

00:00:39.870 --> 00:00:41.930
but it won't necessarily
be congruent anymore.

00:00:41.930 --> 00:00:44.300
So I could already say yes,
the figures are congruent.

00:00:44.300 --> 00:00:48.380
But let's actually
perform the transformation

00:00:48.380 --> 00:00:49.810
that they want us to.

00:00:49.810 --> 00:00:54.110
So a reflection over the
line y equals x minus 2.

00:00:54.110 --> 00:00:57.770
And so let's click on translate,
and this isn't popping up--

00:00:57.770 --> 00:01:00.510
this isn't allowing us
just to do it by hand,

00:01:00.510 --> 00:01:02.659
or do it with our mouse.

00:01:02.659 --> 00:01:05.200
They want us to think about how
much-- actually we don't want

00:01:05.200 --> 00:01:08.280
to translate first, we
want to reflect first.

00:01:08.280 --> 00:01:12.390
So once again, the reflection
line tool doesn't show up.

00:01:12.390 --> 00:01:15.120
Instead, for this
exercise, they want

00:01:15.120 --> 00:01:18.700
us to specify two
points on the line

00:01:18.700 --> 00:01:22.600
that we want to reflect around--
because two points define

00:01:22.600 --> 00:01:23.100
a line.

00:01:23.100 --> 00:01:27.160
So what are two points that are
on the line y equals x minus 2?

00:01:27.160 --> 00:01:30.020
Well when x is equal
to 0, y is negative 2.

00:01:35.780 --> 00:01:39.270
And then another
point on it that

00:01:39.270 --> 00:01:41.930
might jump out at you--
when x is 2, y is 0.

00:01:45.690 --> 00:01:48.060
So we just need any two
different points on that line.

00:01:48.060 --> 00:01:50.330
And then notice,
y equals x minus 2

00:01:50.330 --> 00:01:51.740
is a line that looks like this.

00:01:51.740 --> 00:01:53.340
I'm just going to
kind of trace it.

00:01:53.340 --> 00:01:55.370
I don't have my drawing
tool out right now,

00:01:55.370 --> 00:01:59.600
so it's a line that looks
something like that.

00:01:59.600 --> 00:02:01.290
Actually, let me do
it, just because I

00:02:01.290 --> 00:02:06.180
think it'll be
interesting to look at.

00:02:06.180 --> 00:02:09.300
Let me copy and paste
this right over here.

00:02:09.300 --> 00:02:15.280
And then let me put it
onto my actual scratch pad.

00:02:15.280 --> 00:02:17.520
So that's from a
previous problem.

00:02:17.520 --> 00:02:19.900
So let me clear all
of this business out.

00:02:22.830 --> 00:02:24.850
So let me paste it
right over here.

00:02:24.850 --> 00:02:27.550
And notice, we just
reflected around the line y

00:02:27.550 --> 00:02:30.870
equals x minus 2.
y equals x minus 2.

00:02:30.870 --> 00:02:33.450
Let's see, it has its
y-intercept right over there,

00:02:33.450 --> 00:02:35.040
it has a slope of 1.

00:02:35.040 --> 00:02:38.990
So this is the line that
we just reflected around.

00:02:38.990 --> 00:02:41.790
And it looks pretty clear
that that is what happened.

00:02:41.790 --> 00:02:46.400
So just like that, I know
my line isn't as straight

00:02:46.400 --> 00:02:48.980
as it could be, but you
get the general idea.

00:02:48.980 --> 00:02:53.260
That is the line y is
equal to x minus 2.

00:02:53.260 --> 00:02:59.634
Notice each of these points--
you drop a perpendicular

00:02:59.634 --> 00:03:01.050
from that point,
whatever distance

00:03:01.050 --> 00:03:03.260
that is, go the same
distance on the other side,

00:03:03.260 --> 00:03:04.970
and we have reflected over.

00:03:04.970 --> 00:03:07.630
Drop a perpendicular, same
distance on the other side.

00:03:11.870 --> 00:03:14.240
And so that is our
reflection about the line y

00:03:14.240 --> 00:03:16.310
equals x minus 2.

00:03:16.310 --> 00:03:18.110
But we are not done yet.

00:03:18.110 --> 00:03:24.225
We now have to perform the
translation by 3, negative 9.

00:03:26.800 --> 00:03:28.210
So we're going to translate.

00:03:28.210 --> 00:03:29.810
So when we do it
by 3, it's going

00:03:29.810 --> 00:03:32.560
to shift this to the right by 3.

00:03:32.560 --> 00:03:36.600
And then negative 9, which is
going to shift it down by 9.

00:03:36.600 --> 00:03:38.704
And we see that it did.

00:03:38.704 --> 00:03:40.870
And we already said, these
are definitely congruent.

00:03:40.870 --> 00:03:42.800
We haven't scaled
this up or down,

00:03:42.800 --> 00:03:45.579
we've just translated it,
rotated it-- actually,

00:03:45.579 --> 00:03:46.620
we didn't even rotate it.

00:03:46.620 --> 00:03:49.740
We've just translated it, and
reflected-- or reflected it,

00:03:49.740 --> 00:03:50.840
and translated it.

00:03:50.840 --> 00:03:54.360
But let's check our answer, just
to feel good about ourselves.

00:03:54.360 --> 00:03:56.370
And we got it right.


Another congruence by transformation example

WEBVTT
Kind: captions
Language: en

00:00:00.324 --> 00:00:01.740
So we have another
situation where

00:00:01.740 --> 00:00:04.871
we want to see whether these
two figures are congruent.

00:00:04.871 --> 00:00:06.370
And the way we're
going to test that

00:00:06.370 --> 00:00:10.000
is by trying to transform
this figure by translating it,

00:00:10.000 --> 00:00:11.620
rotating it, and reflecting it.

00:00:11.620 --> 00:00:14.020
So the first thing that
might-- let me translate it.

00:00:14.020 --> 00:00:17.070
So if these two things are
congruent, looks like point E

00:00:17.070 --> 00:00:19.080
and the point that I'm
touching right now,

00:00:19.080 --> 00:00:21.210
those would correspond
to each other.

00:00:21.210 --> 00:00:23.430
Let me try to rotate
it a little bit.

00:00:23.430 --> 00:00:26.272
So let me rotate-- whoops, I
don't want to rotate there,

00:00:26.272 --> 00:00:28.230
I want to rotate around
point E since I already

00:00:28.230 --> 00:00:30.300
have those on top of each other.

00:00:30.300 --> 00:00:31.510
So just like that.

00:00:31.510 --> 00:00:37.980
And it looks like now, if I
reflect it across that line,

00:00:37.980 --> 00:00:41.010
I'm going to be-- oh
no, I'm not there.

00:00:41.010 --> 00:00:42.700
You see, this is tricky.

00:00:42.700 --> 00:00:46.200
See, when I reflect it, this
point, this point, this point,

00:00:46.200 --> 00:00:48.440
and this point seem to be
in the exact same place,

00:00:48.440 --> 00:00:50.870
but point C does not
correspond with that point

00:00:50.870 --> 00:00:52.130
right over there.

00:00:52.130 --> 00:00:55.560
So these two polygons
are not congruent.

00:00:55.560 --> 00:00:57.790
And this is why it's
important to do this,

00:00:57.790 --> 00:01:00.600
to make sure the
rotations work out.

00:01:00.600 --> 00:01:01.940
So these two are not congruent.

00:01:01.940 --> 00:01:04.980
I'm not able to get them to
exactly overlap each other just

00:01:04.980 --> 00:01:07.900
through translations,
rotations, and reflections.

00:01:07.900 --> 00:01:09.930
So are these polygons congruent?

00:01:09.930 --> 00:01:11.480
No.


Testing congruence by transformations example

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.720
Perform a sequence of
transformations to try to map

00:00:03.720 --> 00:00:06.980
the movable figure-- that's
this figure right over here--

00:00:06.980 --> 00:00:09.590
onto C-O-R-A-L or CORAL.

00:00:09.590 --> 00:00:13.882
So C-O-R-A-L is this
polygon right over here.

00:00:13.882 --> 00:00:16.090
And then we have to figure
out, are these two figures

00:00:16.090 --> 00:00:17.120
congruent?

00:00:17.120 --> 00:00:19.570
So they're congruent if,
through some combination

00:00:19.570 --> 00:00:22.670
of translations,
rotations, and reflections,

00:00:22.670 --> 00:00:27.690
I can make this figure sit right
on top of figure C-O-R-A-L.

00:00:27.690 --> 00:00:30.510
So let's use this tool
right over here to do that--

00:00:30.510 --> 00:00:33.621
to do some translations,
rotations, and reflections.

00:00:33.621 --> 00:00:35.120
And so the first
thing I want to do,

00:00:35.120 --> 00:00:37.370
let me translate it so they
get close to each other.

00:00:37.370 --> 00:00:41.250
And let me see if I can
get one point in common.

00:00:41.250 --> 00:00:43.675
So just like that,
I've been able to get

00:00:43.675 --> 00:00:44.550
this point in common.

00:00:44.550 --> 00:00:46.800
And it seems like, if these
are going to be congruent,

00:00:46.800 --> 00:00:48.874
that those would
correspond to each other.

00:00:48.874 --> 00:00:50.540
And now it looks
pretty clear that these

00:00:50.540 --> 00:00:52.180
are reflections of each other.

00:00:52.180 --> 00:00:56.450
So let me reflect them.

00:00:56.450 --> 00:01:00.370
So it looks like if I were
to put my point right there

00:01:00.370 --> 00:01:04.312
and if I go right
in between, it looks

00:01:04.312 --> 00:01:06.270
like if I reflect right
along that, actually, I

00:01:06.270 --> 00:01:08.150
might end up being done.

00:01:08.150 --> 00:01:09.460
And I am done.

00:01:09.460 --> 00:01:12.460
So just with that little
translation followed

00:01:12.460 --> 00:01:14.430
by reflection, I was
able to make these two

00:01:14.430 --> 00:01:16.450
figures sit on
top of each other.

00:01:16.450 --> 00:01:21.960
So these are
definitely congruent.


Dilating from an arbitrary point example

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.500
Perform a dilation on
the coordinate plane.

00:00:02.500 --> 00:00:05.300
The dilation should be
centered at 9, negative 9,

00:00:05.300 --> 00:00:07.780
and have a scale factor of 3.

00:00:07.780 --> 00:00:09.180
So we get our dilation tool out.

00:00:09.180 --> 00:00:12.400
We'll center it-- actually, so
it's already actually centered

00:00:12.400 --> 00:00:13.222
at 9, negative 9.

00:00:13.222 --> 00:00:14.680
We could put this
wherever we want,

00:00:14.680 --> 00:00:18.490
but let's center it
at 9, negative 9.

00:00:18.490 --> 00:00:20.880
And we want to
scale this up by 3.

00:00:20.880 --> 00:00:23.960
So one way to think about it
is, pick any of these points

00:00:23.960 --> 00:00:28.210
right over here, and they're
going to have to get 3 times

00:00:28.210 --> 00:00:31.520
further away from our
center of dilation.

00:00:31.520 --> 00:00:34.047
So for example, this
point C-- actually

00:00:34.047 --> 00:00:36.130
let's think about these
points where they actually

00:00:36.130 --> 00:00:37.960
want us to fill something in.

00:00:37.960 --> 00:00:44.600
So point A right over here, it
is at the point 4, negative 3.

00:00:44.600 --> 00:00:52.810
So in the x direction,
it is 5 less than 9.

00:00:55.900 --> 00:00:58.770
We want it to be 3
times further than 9.

00:00:58.770 --> 00:01:01.700
So we want it to
be 15 less than 9.

00:01:01.700 --> 00:01:06.840
So we want the x-coordinate of
A, 9 minus 15 is negative 6.

00:01:06.840 --> 00:01:09.730
We want it to go to negative 6.

00:01:09.730 --> 00:01:12.300
And likewise, we want its
y-coordinate to be 3 times

00:01:12.300 --> 00:01:13.020
further.

00:01:13.020 --> 00:01:15.380
So right now, let's
see, it is at negative 3

00:01:15.380 --> 00:01:20.450
relative to negative 9, so it
is 6 more on the y direction.

00:01:20.450 --> 00:01:22.570
We want it to be 18 more.

00:01:22.570 --> 00:01:24.740
18 more than negative
9 would be positive 9.

00:01:24.740 --> 00:01:28.360
So point A should map
to negative 6 comma 9.

00:01:28.360 --> 00:01:30.120
And that should give
us enough information

00:01:30.120 --> 00:01:33.160
to just make sure that we are
dilating up by a factor of 3.

00:01:33.160 --> 00:01:35.900
So let's see.

00:01:35.900 --> 00:01:37.950
Let's dilate up
by a factor of 3.

00:01:37.950 --> 00:01:41.850
So we want to get the image of
point A to the point negative 6

00:01:41.850 --> 00:01:42.920
comma 9.

00:01:42.920 --> 00:01:44.882
So we are there.

00:01:44.882 --> 00:01:45.790
There we go.

00:01:45.790 --> 00:01:47.336
We have dilated it up.

00:01:47.336 --> 00:01:48.710
And then we could
even look where

00:01:48.710 --> 00:01:51.000
the point that corresponds
to E has mapped to.

00:01:51.000 --> 00:01:53.330
And you can look at each
direction, it's 3 times

00:01:53.330 --> 00:01:53.830
further.

00:01:53.830 --> 00:02:01.460
E is now at negative
6 comma negative 3.

00:02:04.600 --> 00:02:08.820
The images of point A
and E are 3 times as far

00:02:08.820 --> 00:02:11.690
as the original points.

00:02:11.690 --> 00:02:16.220
3 times far apart, I should
say, as the original points.

00:02:16.220 --> 00:02:20.544
And they're 3 times further
from our center of dilation

00:02:20.544 --> 00:02:21.210
right over here.

00:02:21.210 --> 00:02:25.680
You see, for example, point
E has the x-coordinate of 4,

00:02:25.680 --> 00:02:27.860
which is 5 less.

00:02:27.860 --> 00:02:29.710
Now it is at
negative 6, which is

00:02:29.710 --> 00:02:33.370
15 less than our
center of dilation.

00:02:33.370 --> 00:02:37.600
And the same thing true,
its y-coordinate is 2 more.

00:02:37.600 --> 00:02:40.340
And now after we mapped
it, its y-coordinate

00:02:40.340 --> 00:02:44.670
is 6 more than our
center of dilation.

00:02:44.670 --> 00:02:46.470
Got it right.


Thinking about dilations

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.410
The graph below contains
the rectangle ABCP.

00:00:06.390 --> 00:00:11.600
Draw the image of ABCP under a
dilation whose center is at P

00:00:11.600 --> 00:00:14.950
and a scale factor is 1 and 2/3.

00:00:14.950 --> 00:00:19.880
What are the lengths of
the side AB and its image?

00:00:19.880 --> 00:00:23.010
So we're going to do a
dilation centered at P.

00:00:23.010 --> 00:00:26.720
So if we're centering
a dilation at P

00:00:26.720 --> 00:00:28.440
and its scale
factor is 1 and 2/3,

00:00:28.440 --> 00:00:33.120
that means once we
perform the dilation,

00:00:33.120 --> 00:00:38.440
every point is going to be 1 and
2/3 times as far away from P.

00:00:38.440 --> 00:00:42.520
Well P is 0 away from
P, so its image is still

00:00:42.520 --> 00:00:46.260
going to be at P. So let's put
that point right over there.

00:00:46.260 --> 00:00:50.205
Now point C is going
to be 1 and 2/3 times

00:00:50.205 --> 00:00:52.580
as far as it is right now.

00:00:52.580 --> 00:00:55.180
So let's see, right
now it is 6 away.

00:00:55.180 --> 00:00:56.550
It's at negative 3.

00:00:56.550 --> 00:01:01.320
And P, its x-coordinate is the
same, but in the y direction,

00:01:01.320 --> 00:01:02.080
P is at 3.

00:01:02.080 --> 00:01:03.320
C is at negative 3.

00:01:03.320 --> 00:01:04.640
So it's 6 less.

00:01:04.640 --> 00:01:07.730
We want to be 1 and
2/3 times as far away.

00:01:07.730 --> 00:01:10.170
So what's 1 and 2/3 of 6?

00:01:10.170 --> 00:01:15.940
Well, 2/3 of 6 is 4, so
it's going to be 6 plus 4.

00:01:15.940 --> 00:01:17.830
You're going to be 10 away.

00:01:17.830 --> 00:01:21.870
So 3 minus 10, that
gets us to negative 7.

00:01:21.870 --> 00:01:24.070
So that gets us
right over there.

00:01:24.070 --> 00:01:28.500
Now point A, right
now it is 3 more

00:01:28.500 --> 00:01:33.260
in the horizontal direction
than point P's x-coordinate.

00:01:33.260 --> 00:01:35.920
So we want to go
1 and 2/3 as far.

00:01:35.920 --> 00:01:38.670
So what is 1 and 2/3 times 3?

00:01:38.670 --> 00:01:42.480
Well that's going to be 3 plus
2/3 of 3, which is another 2.

00:01:42.480 --> 00:01:43.976
So that's going to be 5.

00:01:43.976 --> 00:01:45.600
So we're going to
get right over there.

00:01:45.600 --> 00:01:47.650
Then we could complete
the rectangle.

00:01:47.650 --> 00:01:51.710
And notice point B is
now 1 and 2/3 times

00:01:51.710 --> 00:01:53.890
as far in the
horizontal direction.

00:01:53.890 --> 00:01:56.650
It was 3 away in the
horizontal direction,

00:01:56.650 --> 00:01:59.920
now it is 5 away from
P's x-coordinate.

00:01:59.920 --> 00:02:03.080
And in the vertical
direction, in the y direction,

00:02:03.080 --> 00:02:06.350
it was 6 below P's y-coordinate.

00:02:06.350 --> 00:02:08.430
Now it is 1 and
2/3 times as far.

00:02:08.430 --> 00:02:13.060
It is 10 below P's y-coordinate.

00:02:13.060 --> 00:02:14.750
So then let's answer
these questions.

00:02:14.750 --> 00:02:19.370
The length of segment AB--
well, we already saw that.

00:02:19.370 --> 00:02:21.380
That is, we're going
from 3 to negative 3.

00:02:21.380 --> 00:02:24.310
That is 6 units long.

00:02:24.310 --> 00:02:27.227
And its image, well
it's 1 and 2/3 as long.

00:02:27.227 --> 00:02:28.060
We see it over here.

00:02:28.060 --> 00:02:29.850
We're going from
3 to negative 7.

00:02:29.850 --> 00:02:33.220
3 minus negative 7 is 10.

00:02:33.220 --> 00:02:36.940
It is 10 units long.

00:02:36.940 --> 00:02:38.930
We got it right.


Comparing side lengths after dilation

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:05.070
The graph below contains
triangle ABC and the point P.

00:00:05.070 --> 00:00:07.700
Draw the image of triangle
ABC under a dilation whose

00:00:07.700 --> 00:00:11.970
center is P and a
scale factor of 2.

00:00:11.970 --> 00:00:14.060
So essentially, we
want to scale this

00:00:14.060 --> 00:00:18.380
so that every point is going
to be twice as far away from P.

00:00:18.380 --> 00:00:20.230
So for example,
B right over here

00:00:20.230 --> 00:00:23.770
has the same y-coordinate
as P, but its x-coordinate

00:00:23.770 --> 00:00:26.900
is three more.

00:00:26.900 --> 00:00:28.590
So we want to be twice as far.

00:00:28.590 --> 00:00:33.090
So if this maps to point B, we
just want to go twice as far.

00:00:33.090 --> 00:00:36.580
So we're at 3 away,
we want to go 6 away.

00:00:36.580 --> 00:00:39.870
So point P's x-coordinate
is at 3, now we're at 9.

00:00:39.870 --> 00:00:43.220
Likewise, point C
is 3 below P. Well

00:00:43.220 --> 00:00:46.090
we want to go twice as
far, so we'll go 3 more.

00:00:46.090 --> 00:00:50.400
And point A is 4 above P.
Well we want to go 4 more.

00:00:50.400 --> 00:00:55.570
We want to go twice as
far-- one, two, three, four.

00:00:55.570 --> 00:00:56.937
And we get right over there.

00:00:56.937 --> 00:00:58.520
Then they ask us,
what are the lengths

00:00:58.520 --> 00:01:01.350
of side AB and its image?

00:01:01.350 --> 00:01:03.320
AB right over
here, let's see, we

00:01:03.320 --> 00:01:06.230
might have to apply
the distance formula.

00:01:06.230 --> 00:01:09.780
Let's see, it's the
base right over here.

00:01:09.780 --> 00:01:16.090
The change in x between the two
is 3 and the change in y is 4,

00:01:16.090 --> 00:01:19.630
so this is actually a
3, 4, 5 right triangle.

00:01:19.630 --> 00:01:23.770
3 squared plus 4 squared
is equal to 5 squared.

00:01:23.770 --> 00:01:25.447
So AB is 5 units long.

00:01:25.447 --> 00:01:27.030
Essentially just
using the Pythagorean

00:01:27.030 --> 00:01:28.760
theorem to figure that out.

00:01:28.760 --> 00:01:32.740
And its image, well it's
image should be twice as long.

00:01:32.740 --> 00:01:36.240
And let's see whether
that actually is the case.

00:01:36.240 --> 00:01:40.640
So this is a base right over
here that's of length 6.

00:01:40.640 --> 00:01:43.515
This has a height, or this
change in y, I could say.

00:01:43.515 --> 00:01:45.890
Because I'm really just trying
to figure out this length,

00:01:45.890 --> 00:01:48.390
which is the hypotenuse
of this right triangle.

00:01:48.390 --> 00:01:50.990
I don't have my drawing
tool, so I apologize.

00:01:50.990 --> 00:01:52.940
But this height right here is 8.

00:01:52.940 --> 00:01:56.510
So 8 squared is 64,
plus 6 squared is 36,

00:01:56.510 --> 00:02:00.280
that's 100, which is 10 squared.

00:02:00.280 --> 00:02:03.620
So notice, our scale factor
of 2, the corresponding side

00:02:03.620 --> 00:02:04.990
got twice as long.

00:02:04.990 --> 00:02:08.009
Each of these points
got twice as far away

00:02:08.009 --> 00:02:10.860
from our center of dilation.


Dilating one line onto another

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.670
Define a dilation that
maps line a onto line b

00:00:03.670 --> 00:00:06.180
by choosing a center
and a scale factor.

00:00:06.180 --> 00:00:09.260
So let me get my scratch out to
think about this a little bit.

00:00:09.260 --> 00:00:13.650
So let's, just for
fun, let's imagine

00:00:13.650 --> 00:00:18.410
that we pick a point
that sits on line a

00:00:18.410 --> 00:00:19.515
as our center of dilation.

00:00:19.515 --> 00:00:21.830
And for simplicity,
let's pick the origin.

00:00:21.830 --> 00:00:26.224
So let's imagine that our
center of dilation is at 0, 0.

00:00:26.224 --> 00:00:28.140
And let's just pick an
arbitrary scale factor.

00:00:28.140 --> 00:00:30.070
Let's say our scale factor is 2.

00:00:30.070 --> 00:00:31.830
And let's think
about what happens.

00:00:31.830 --> 00:00:34.680
This means that each of
the points on point a

00:00:34.680 --> 00:00:38.070
are going to get twice as far
from our center of dilation

00:00:38.070 --> 00:00:40.900
after the dilation than they
were before the dilation.

00:00:40.900 --> 00:00:42.060
Twice as far.

00:00:42.060 --> 00:00:45.440
So for example, this
point right over here,

00:00:45.440 --> 00:00:48.560
it's 3 away in the x
direction before dilation.

00:00:48.560 --> 00:00:51.750
So it's going to be 6 away in
the x direction after dilation.

00:00:51.750 --> 00:00:55.996
Likewise, it's 3 away in the
y direction before dilation.

00:00:55.996 --> 00:00:58.120
So it's going to be twice
as far in the y direction

00:00:58.120 --> 00:00:59.540
after dilation.

00:00:59.540 --> 00:01:03.440
So it's going to go from the point (3,3) to the point (6,6).

00:01:03.440 --> 00:01:08.465
Notice, it got twice as far away
from the center of dilation.

00:01:10.990 --> 00:01:14.110
It essentially got pushed
out along the line.

00:01:14.110 --> 00:01:15.570
Same thing for this point.

00:01:15.570 --> 00:01:20.800
This point is 3 less in the
x direction than our center,

00:01:20.800 --> 00:01:23.240
and 3 less in the y
direction than our center.

00:01:23.240 --> 00:01:25.330
So after a scaling,
after dilation

00:01:25.330 --> 00:01:27.630
centered at the origin,
with a scale factor of 2,

00:01:27.630 --> 00:01:29.190
it's going to be
twice as far away.

00:01:29.190 --> 00:01:31.550
So it's going to be 6
less in the x direction,

00:01:31.550 --> 00:01:33.050
and 6 less in the y direction.

00:01:33.050 --> 00:01:35.800
Once again, just pushed out.

00:01:35.800 --> 00:01:39.460
So notice when we picked
a center of dilation

00:01:39.460 --> 00:01:43.270
that sits on the line,
it really doesn't matter

00:01:43.270 --> 00:01:45.380
what scale factor we pick,
we'll just essentially

00:01:45.380 --> 00:01:48.650
be stretching and shrinking
the points along the same line.

00:01:48.650 --> 00:01:51.650
And since this line just keeps
going on and on and on forever

00:01:51.650 --> 00:01:55.370
in both directions, it's really
just going to map onto itself.

00:01:55.370 --> 00:01:58.530
If this was a line segment,
we would be mapping it

00:01:58.530 --> 00:02:02.320
to a different line segment
that would have the same slope,

00:02:02.320 --> 00:02:03.810
but its length would change.

00:02:03.810 --> 00:02:06.060
But a line has an
infinite length.

00:02:06.060 --> 00:02:07.440
So as you stretch
and squeeze it,

00:02:07.440 --> 00:02:09.356
it still has an infinite
length, and it's just

00:02:09.356 --> 00:02:11.180
mapping it onto itself.

00:02:11.180 --> 00:02:13.705
So this dilation
right over here,

00:02:13.705 --> 00:02:16.250
a, is going to map onto itself.

00:02:16.250 --> 00:02:19.040
It's just going to map onto a.

00:02:19.040 --> 00:02:21.350
So that's not going
to suit our purposes.

00:02:21.350 --> 00:02:24.550
But what can suit our purpose is
if we pick another point that's

00:02:24.550 --> 00:02:26.710
not on a, and not on b.

00:02:26.710 --> 00:02:31.810
So let's say we pick this
point right over here.

00:02:31.810 --> 00:02:36.410
And I like this point, and so
let me pick that as my center.

00:02:36.410 --> 00:02:38.392
I'll tell you why I
like it in a second.

00:02:38.392 --> 00:02:39.350
That is the point 3, 2.

00:02:42.460 --> 00:02:44.690
The reason why I
like that, is we

00:02:44.690 --> 00:02:49.230
can see that this point on
point a right now is 1 away,

00:02:49.230 --> 00:02:53.320
and if I want to map it to the
corresponding point on point b,

00:02:53.320 --> 00:02:58.550
well that's going to be 3 away
from our center of dilation.

00:02:58.550 --> 00:03:01.190
So I could have a
scale factor of 3.

00:03:01.190 --> 00:03:03.590
If I have a scale factor of
3, it's going to go 3 times

00:03:03.590 --> 00:03:04.440
as far away.

00:03:08.150 --> 00:03:09.900
And that'll work for
other points as well.

00:03:09.900 --> 00:03:11.870
For example, this
point right over here.

00:03:11.870 --> 00:03:14.170
It's 1 to the left of
our center of dilation.

00:03:14.170 --> 00:03:16.900
If we have a scale factor of 3,
it's going to be 3 to the left.

00:03:16.900 --> 00:03:18.080
So it's going to map.

00:03:18.080 --> 00:03:20.200
If we think of it that
way, you could think of it

00:03:20.200 --> 00:03:24.760
as going right over there.

00:03:24.760 --> 00:03:29.894
So I'm going to go with center
at 3, 2, scale factor of 3.

00:03:29.894 --> 00:03:32.060
And you could have picked
many, many other-- in fact

00:03:32.060 --> 00:03:33.445
an infinite number
of other centers,

00:03:33.445 --> 00:03:35.580
and then you would have had to
calculate the appropriate scale

00:03:35.580 --> 00:03:36.621
factor for each of those.

00:03:36.621 --> 00:03:39.460
But I like this one because
it fit nicely on the grid,

00:03:39.460 --> 00:03:43.160
and I was able to figure
out how much to scale it.

00:03:43.160 --> 00:03:46.160
So I picked my center,
so I want to dilate,

00:03:46.160 --> 00:03:49.195
I want to pick my
center at 3, 2.

00:03:51.860 --> 00:03:55.030
And I want to scale it by 3.

00:03:55.030 --> 00:03:57.040
So let's check our answer.

00:03:57.040 --> 00:03:58.940
We got it right.


Scaling down a triangle by half

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:05.060
Plot the images of points
D, E, and F after a dilation

00:00:05.060 --> 00:00:08.760
centered at the origin
with a scale factor of 1/2.

00:00:08.760 --> 00:00:11.060
So we're going to center
around the origin.

00:00:11.060 --> 00:00:13.270
We want to scale this
thing down by 1/2.

00:00:13.270 --> 00:00:15.940
So one way to think
about it is the points

00:00:15.940 --> 00:00:19.820
that will correspond to
points D, E, and F are going

00:00:19.820 --> 00:00:23.000
to be half as far away from
the origin, because our scale

00:00:23.000 --> 00:00:25.720
factor is 1/2 in
either direction.

00:00:25.720 --> 00:00:28.950
So for example, let's
think about point D first.

00:00:28.950 --> 00:00:31.760
Point D is at negative 8.

00:00:31.760 --> 00:00:37.280
So if we have a scale factor of
1/2, what point D will map to

00:00:37.280 --> 00:00:40.560
is going to be at negative
4 on the x direction.

00:00:40.560 --> 00:00:43.730
And on the y direction,
D is at negative 9,

00:00:43.730 --> 00:00:46.120
so this is going to
be at negative 4.5.

00:00:46.120 --> 00:00:47.124
Half of that.

00:00:47.124 --> 00:00:48.790
So that is going to
be right over there.

00:00:48.790 --> 00:00:51.350
That's where point D is going
to be, or the image of point

00:00:51.350 --> 00:00:54.070
D after the scaling.

00:00:54.070 --> 00:00:56.940
Now let's think
about point E. E is

00:00:56.940 --> 00:00:59.450
2 more than the origin
in the x direction.

00:00:59.450 --> 00:01:02.760
So it's only going to be 1
more once we scale it by 1/2.

00:01:02.760 --> 00:01:05.076
And it's 7 more in
the y direction,

00:01:05.076 --> 00:01:06.450
so it's going to
be at 3 and 1/2.

00:01:06.450 --> 00:01:08.070
7 times 1/2 is 3 and 1/2.

00:01:08.070 --> 00:01:10.050
So we're going to stick
it right over there.

00:01:10.050 --> 00:01:15.160
And then finally F, its
x-coordinate is 6 more

00:01:15.160 --> 00:01:15.840
than the origin.

00:01:15.840 --> 00:01:18.070
Its y-coordinate is 6 less.

00:01:18.070 --> 00:01:19.860
So its image after
scaling is going

00:01:19.860 --> 00:01:23.960
to be 3 more in the x direction
and 3 less in the y direction.

00:01:23.960 --> 00:01:25.780
So it's going to be
right over there.

00:01:25.780 --> 00:01:27.686
So we've plotted the
images of the points.

00:01:27.686 --> 00:01:29.310
So if you were to
connect these points,

00:01:29.310 --> 00:01:35.200
you would essentially
have dilated down DEF,

00:01:35.200 --> 00:01:37.597
and your center of dilation
would be the origin.

00:01:37.597 --> 00:01:39.180
So let's just write
these coordinates.

00:01:39.180 --> 00:01:42.780
Point D-- and point D, remember,
was the point negative 8,

00:01:42.780 --> 00:01:43.960
negative 9.

00:01:43.960 --> 00:01:46.770
That's going to map to--
take 1/2 of each of those.

00:01:46.770 --> 00:01:51.790
So negative 4 and negative 4.5.

00:01:51.790 --> 00:01:56.170
Point E maps to--
well, E was at 2, 7.

00:01:56.170 --> 00:02:00.910
So it maps to 1, 3.5.

00:02:00.910 --> 00:02:03.930
And then finally, point
F was at 6, negative 6,

00:02:03.930 --> 00:02:07.915
so it maps to 3, negative 3.

00:02:10.850 --> 00:02:12.440
So the important
thing to recognize

00:02:12.440 --> 00:02:15.330
is the center of our
dilation was the origin.

00:02:15.330 --> 00:02:17.550
So in each dimension,
in the x direction

00:02:17.550 --> 00:02:20.140
or in the y direction, we
just halved the distance

00:02:20.140 --> 00:02:24.520
from the origin, because
the scale factor was 1/2.

00:02:24.520 --> 00:02:26.450
We got it right.


Constructing a shape by reflecting over 2 lines

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:05.550
Two of the points that define
a certain quadrilateral

00:00:05.550 --> 00:00:07.780
are negative 4 comma negative 2.

00:00:07.780 --> 00:00:08.720
So let's plot that.

00:00:08.720 --> 00:00:12.080
So that's negative
4 comma negative 2.

00:00:12.080 --> 00:00:13.760
And 0 comma 5.

00:00:17.320 --> 00:00:19.580
So that's 0 comma
5 right over there.

00:00:19.580 --> 00:00:22.990
The quadrilateral
is left unchanged

00:00:22.990 --> 00:00:28.180
by a reflection over the
line y is equal to x over 2.

00:00:28.180 --> 00:00:30.460
So what does that
line look like?

00:00:30.460 --> 00:00:32.270
y is equal to x over 2.

00:00:32.270 --> 00:00:34.760
I'll do that in the blue.

00:00:34.760 --> 00:00:35.850
y is equal to x over 2.

00:00:35.850 --> 00:00:39.530
So when x is equal to 0, y is 0.

00:00:39.530 --> 00:00:41.330
The y-intercept is 0 here.

00:00:41.330 --> 00:00:42.520
And the slope is 1/2.

00:00:42.520 --> 00:00:45.660
Every time x increases by
1, y will increase by 1/2.

00:00:45.660 --> 00:00:49.020
Or when x increases by
2, y will increase by 1.

00:00:49.020 --> 00:00:51.700
So x increases by
2, y increases by 1.

00:00:51.700 --> 00:00:56.500
X increases by 2,
y increases by 1.

00:00:56.500 --> 00:00:58.890
Or another way to think about
it, y is always 1/2 of x.

00:00:58.890 --> 00:01:00.720
So when x is 4, y is 2.

00:01:00.720 --> 00:01:04.120
When x is 6, y is 3.

00:01:04.120 --> 00:01:07.940
When x is 8, y is 4.

00:01:07.940 --> 00:01:09.090
So we can connect these.

00:01:09.090 --> 00:01:11.490
Let me try my best
attempt to draw

00:01:11.490 --> 00:01:13.670
these in a relatively
straight line.

00:01:13.670 --> 00:01:15.230
And then I can keep going.

00:01:15.230 --> 00:01:17.870
When x is negative
2, y is negative 1.

00:01:17.870 --> 00:01:20.280
When x is negative
4, y is negative 2.

00:01:20.280 --> 00:01:22.920
So it actually goes through
that point right there.

00:01:22.920 --> 00:01:26.525
And it just keeps going
with a slope of 1/2.

00:01:26.525 --> 00:01:28.900
So this line, and I can draw
it a little bit thicker now,

00:01:28.900 --> 00:01:30.670
now that I've dotted it out.

00:01:30.670 --> 00:01:37.426
This is the line y
is equal to x over 2.

00:01:37.426 --> 00:01:39.760
And they also say that
the quadrilateral is left

00:01:39.760 --> 00:01:43.440
unchanged by reflection
over the line

00:01:43.440 --> 00:01:46.180
y is equal to
negative 2x plus 5.

00:01:46.180 --> 00:01:47.710
So the y-intercept here is 5.

00:01:47.710 --> 00:01:49.405
When x is 0, y is 5.

00:01:49.405 --> 00:01:51.030
So it actually goes
through that point.

00:01:51.030 --> 00:01:52.720
And the slope is negative 2.

00:01:52.720 --> 00:01:56.310
Every time we increase by 1-- or
every time we increase x by 1,

00:01:56.310 --> 00:01:58.330
we decrease y by 2.

00:01:58.330 --> 00:02:00.380
So that would go there.

00:02:00.380 --> 00:02:01.620
We go there.

00:02:01.620 --> 00:02:04.930
And we keep going at
a slope of negative 2.

00:02:04.930 --> 00:02:06.680
So it's going to look
something like this.

00:02:09.992 --> 00:02:11.450
It actually goes
through that point

00:02:11.450 --> 00:02:13.040
and just keeps going on and on.

00:02:13.040 --> 00:02:18.250
So this is my best attempt
at drawing that line.

00:02:18.250 --> 00:02:23.639
So that is y is equal
to negative 2x plus 5.

00:02:23.639 --> 00:02:24.680
Now let's think about it.

00:02:24.680 --> 00:02:27.370
Let's see if we can
draw this quadrilateral.

00:02:27.370 --> 00:02:30.240
So let's first reflect
the quadrilateral,

00:02:30.240 --> 00:02:33.056
or let's reflect the points
we have over the line y

00:02:33.056 --> 00:02:33.930
is equal to x over 2.

00:02:33.930 --> 00:02:36.100
So this is the line y
is equal to x over 2.

00:02:36.100 --> 00:02:39.310
This magenta point,
the point negative 4, 2

00:02:39.310 --> 00:02:40.750
is already on that line.

00:02:40.750 --> 00:02:43.170
So it's its own reflection,
I guess you could say.

00:02:43.170 --> 00:02:46.580
It's on the mirror is one
way to think about it.

00:02:46.580 --> 00:02:49.310
But we can easily reflect
at this line over here.

00:02:49.310 --> 00:02:56.790
This line, if we were to drop
a perpendicular-- And actually,

00:02:56.790 --> 00:02:59.520
this line right over here,
y is equal to negative 2x

00:02:59.520 --> 00:03:02.395
plus 5 is perpendicular
to y is equal to x over 2.

00:03:02.395 --> 00:03:03.760
How do we know?

00:03:03.760 --> 00:03:07.694
Well if one line
has a slope of m,

00:03:07.694 --> 00:03:09.110
then the line
that's perpendicular

00:03:09.110 --> 00:03:10.818
would be the negative
reciprocal of this.

00:03:10.818 --> 00:03:13.040
It would be negative 1 over m.

00:03:13.040 --> 00:03:16.390
So this first line
has a slope of 1/2.

00:03:16.390 --> 00:03:18.710
Well what's the negative
reciprocal of 1/2?

00:03:18.710 --> 00:03:21.230
Well the reciprocal
of 1/2 is 2/1.

00:03:21.230 --> 00:03:22.450
and you make that negative.

00:03:22.450 --> 00:03:24.250
So it is equal to negative 2.

00:03:24.250 --> 00:03:27.730
So this slope is a negative
reciprocal of this slope.

00:03:27.730 --> 00:03:32.220
So these lines are
indeed-- I'm trying

00:03:32.220 --> 00:03:35.410
to erase that-- perpendicular.

00:03:35.410 --> 00:03:37.540
So we literally could
drop a perpendicular,

00:03:37.540 --> 00:03:39.910
literally go along this
line right over here,

00:03:39.910 --> 00:03:41.660
in our attempt to reflect.

00:03:41.660 --> 00:03:47.260
And we see that we're
going down 2, over 1 twice.

00:03:47.260 --> 00:03:51.224
So let's go down 2, over 1,
down 2, over 1, twice again.

00:03:51.224 --> 00:03:52.640
Let me do that in
that same color.

00:03:52.640 --> 00:03:57.326
The reflection of this point,
across y is equal to x/2

00:03:57.326 --> 00:03:59.589
is this point right over there.

00:03:59.589 --> 00:04:01.630
So now we have three points
of our quadrilateral.

00:04:01.630 --> 00:04:03.270
Let's see if we
can get a fourth.

00:04:03.270 --> 00:04:04.999
So let's go to
the magenta point.

00:04:04.999 --> 00:04:06.540
The magenta point
we've already seen.

00:04:06.540 --> 00:04:09.910
It's sitting on top
of y equals x/2,

00:04:09.910 --> 00:04:12.040
so trying to reflect it
doesn't help us much,

00:04:12.040 --> 00:04:13.700
but we could try to
reflect it across y

00:04:13.700 --> 00:04:15.663
is equal to negative 2x plus 5.

00:04:15.663 --> 00:04:18.079
So once again these lines are
perpendicular to each other.

00:04:18.079 --> 00:04:19.750
Actually let me mark that off.

00:04:19.750 --> 00:04:21.029
These lines are perpendicular.

00:04:21.029 --> 00:04:24.100
So we can drop a perpendicular
and try to find its reflection.

00:04:24.100 --> 00:04:26.320
So we're going to
the right 2 and up 1.

00:04:26.320 --> 00:04:29.790
We're doing that once, twice,
three times on the left side.

00:04:29.790 --> 00:04:34.020
So let's do that once, twice,
three times on the right side.

00:04:34.020 --> 00:04:36.150
So the reflection
is right there.

00:04:36.150 --> 00:04:38.000
We essentially want
to go to that line.

00:04:38.000 --> 00:04:40.400
And however far we
were to the left of it,

00:04:40.400 --> 00:04:42.970
we want to go that same, that
bottom left direction, which

00:04:42.970 --> 00:04:45.440
you want to go in the same
direction, to the top right,

00:04:45.440 --> 00:04:47.770
the same distance to
get the reflection.

00:04:47.770 --> 00:04:48.690
So there you have it.

00:04:48.690 --> 00:04:52.770
There is our other point.

00:04:52.770 --> 00:04:57.720
So now we have the four
points of this quadrilateral.

00:04:57.720 --> 00:05:02.486
Four points of this
quadrilateral are-- or the four

00:05:02.486 --> 00:05:04.610
sides, let me actually just
draw the quadrilateral.

00:05:04.610 --> 00:05:06.060
We have our four points.

00:05:06.060 --> 00:05:09.290
So this is one side
right over here.

00:05:09.290 --> 00:05:13.060
This is one side,
right over here.

00:05:13.060 --> 00:05:16.050
This is another side,
right over here.

00:05:16.050 --> 00:05:18.130
And you can verify that
these are parallel.

00:05:18.130 --> 00:05:19.921
How would you verify
that they're parallel?

00:05:19.921 --> 00:05:21.240
Well they have the same slope.

00:05:21.240 --> 00:05:24.130
To get from this point to that
point, you have to go over,

00:05:24.130 --> 00:05:26.240
so your run has to
be 4, and you have

00:05:26.240 --> 00:05:30.570
to rise 1, 2, 3, 4, 5, 6, 7.

00:05:30.570 --> 00:05:32.363
So the slope here is 7/4.

00:05:36.010 --> 00:05:39.275
Rise over run, or change in
y over change in x, is 7/4.

00:05:39.275 --> 00:05:42.960
And over here,
you go 1, 2, 3, 4.

00:05:42.960 --> 00:05:47.940
So you run 4 and then you
rise 1, 2, 3, 4, 5, 6, 7.

00:05:47.940 --> 00:05:51.020
So the slope here is also 7/4.

00:05:51.020 --> 00:05:58.220
So these two lines are
going to be parallel.

00:05:58.220 --> 00:06:02.270
And then we could draw
these lines over here.

00:06:02.270 --> 00:06:04.850
So this one at the top.

00:06:04.850 --> 00:06:06.990
The one at the top
right over there.

00:06:06.990 --> 00:06:07.960
And what's its slope?

00:06:07.960 --> 00:06:08.626
Well, let's see.

00:06:08.626 --> 00:06:10.790
We go from x equals
0 to x equals 8.

00:06:10.790 --> 00:06:14.790
So we go down, our change in
y is negative 1 every time

00:06:14.790 --> 00:06:17.040
we increase x by 8.

00:06:17.040 --> 00:06:21.779
So this is the slope of--
slope is equal to negative 1/8.

00:06:21.779 --> 00:06:24.320
And that's the exact same slope
that we have right over here.

00:06:24.320 --> 00:06:25.880
Negative 1/8.

00:06:25.880 --> 00:06:28.130
So these two lines
are parallel as well.

00:06:28.130 --> 00:06:31.842
This line is parallel
to this line as well.

00:06:31.842 --> 00:06:33.925
So at minimum, we're dealing
with a parallelogram.

00:06:33.925 --> 00:06:35.940
But let's see if we can
get even more specific.

00:06:35.940 --> 00:06:38.120
Because this kind of
looks like a rhombus.

00:06:38.120 --> 00:06:39.840
It looks like a
parallelogram, where

00:06:39.840 --> 00:06:43.211
all four sides have
the same length.

00:06:43.211 --> 00:06:44.710
So there's a couple
of ways that you

00:06:44.710 --> 00:06:47.034
could verify that this
parallelogram is a rhombus.

00:06:47.034 --> 00:06:48.450
One way is you
could actually find

00:06:48.450 --> 00:06:50.205
the distance between the points.

00:06:50.205 --> 00:06:52.830
We know the coordinates, so you
could use the distance formula,

00:06:52.830 --> 00:06:55.090
which really comes straight
out of the Pythagorean theorem.

00:06:55.090 --> 00:06:57.070
Or even better, you could
look at the diagonals

00:06:57.070 --> 00:06:59.114
of this rhombus-- we could
look at the diagonals

00:06:59.114 --> 00:07:00.030
of this parallelogram.

00:07:00.030 --> 00:07:02.110
We are trying to figure
out if it's a rhombus.

00:07:02.110 --> 00:07:03.840
And if the diagonals
are perpendicular,

00:07:03.840 --> 00:07:04.860
then you're dealing
with a rhombus.

00:07:04.860 --> 00:07:06.735
And we've already shown
that these diagonals,

00:07:06.735 --> 00:07:10.500
that this diagonal, this
diagonal, and this diagonal

00:07:10.500 --> 00:07:13.140
are perpendicular.

00:07:13.140 --> 00:07:15.330
They intersect at right angles.

00:07:15.330 --> 00:07:18.589
And so this must be a rhombus.


Rotating polygons 180 degrees about their center

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.764
We have two copies of
six different figures

00:00:02.764 --> 00:00:03.430
right over here.

00:00:03.430 --> 00:00:05.590
And I want to think about
which of these figures

00:00:05.590 --> 00:00:10.450
are going to be unchanged if I
were to rotate it 180 degrees?

00:00:10.450 --> 00:00:12.280
So let's do two
examples of that.

00:00:12.280 --> 00:00:13.780
So I have two copies
of this square.

00:00:13.780 --> 00:00:15.890
If I were to take
one of these copies

00:00:15.890 --> 00:00:17.570
and rotate it 180 degrees.

00:00:17.570 --> 00:00:19.236
So let me show you
what that looks like.

00:00:19.236 --> 00:00:21.620
And we're going to rotate
around its center 180 degrees.

00:00:21.620 --> 00:00:23.970
So we're going to rotate
around the center.

00:00:23.970 --> 00:00:25.300
So this is it.

00:00:25.300 --> 00:00:26.410
So we're rotating it.

00:00:26.410 --> 00:00:28.730
That's rotated 90 degrees.

00:00:28.730 --> 00:00:31.120
And then we've
rotated 180 degrees.

00:00:31.120 --> 00:00:34.110
And notice the figure
looks exactly the same.

00:00:34.110 --> 00:00:36.380
This one, the
square is unchanged

00:00:36.380 --> 00:00:39.704
by a 180-degree rotation.

00:00:39.704 --> 00:00:41.620
Now what about this
trapezoid right over here?

00:00:41.620 --> 00:00:43.078
Let's think about
what happens when

00:00:43.078 --> 00:00:45.250
it's rotated by 180 degrees.

00:00:45.250 --> 00:00:51.280
So that is 90 degrees
and 180 degrees.

00:00:51.280 --> 00:00:53.460
So this has now been changed.

00:00:53.460 --> 00:00:56.120
Now I have the short side.

00:00:56.120 --> 00:00:58.850
I have my base is short
and my top is long.

00:00:58.850 --> 00:01:01.640
Before my base was long
and my top was short.

00:01:01.640 --> 00:01:03.320
So when I rotate
it 180 degrees I

00:01:03.320 --> 00:01:05.379
didn't get to the
exact same figure.

00:01:05.379 --> 00:01:08.322
I have essentially an
upside down version of it.

00:01:08.322 --> 00:01:10.280
So what I want you to do
for the rest of these,

00:01:10.280 --> 00:01:12.530
is pause the video and think
about which of these will

00:01:12.530 --> 00:01:14.250
be unchanged and
which of them will

00:01:14.250 --> 00:01:17.730
be changed when you
rotate by 180 degrees.

00:01:17.730 --> 00:01:19.200
So let's look at
this star thing.

00:01:19.200 --> 00:01:21.130
And one way that my
brain visualizes it,

00:01:21.130 --> 00:01:22.160
is imagine the center.

00:01:22.160 --> 00:01:23.940
That's what we're
rotating around.

00:01:23.940 --> 00:01:26.150
And then if you
rotate 180 degrees.

00:01:26.150 --> 00:01:26.960
Imagine any point.

00:01:26.960 --> 00:01:29.200
Say this point,
relative to the center.

00:01:29.200 --> 00:01:31.670
If you were to
rotate it 90 degrees,

00:01:31.670 --> 00:01:32.680
you would get over here.

00:01:32.680 --> 00:01:34.846
And then if rotate it 180
degrees, you go over here.

00:01:34.846 --> 00:01:37.260
You go the opposite side of
the center from where it is.

00:01:37.260 --> 00:01:39.050
So from that point,
to the center,

00:01:39.050 --> 00:01:40.530
you keep going
that same distance.

00:01:40.530 --> 00:01:42.080
You'll end up right over there.

00:01:42.080 --> 00:01:43.910
So this one looks like
it won't be changed.

00:01:43.910 --> 00:01:45.440
But let's verify it.

00:01:45.440 --> 00:01:48.690
So we're going to
rotate 90 degrees.

00:01:48.690 --> 00:01:51.995
And then we have 180 degrees.

00:01:51.995 --> 00:01:54.880
It is unchanged.

00:01:54.880 --> 00:01:58.100
Now let's look at this
parallelogram right over here.

00:01:58.100 --> 00:02:00.310
So its center, if we
think about its center

00:02:00.310 --> 00:02:03.402
where my cursor is right
now-- Think about this point.

00:02:03.402 --> 00:02:05.360
The distance between that
point and the center,

00:02:05.360 --> 00:02:08.050
if we were to keep going
that same distance again,

00:02:08.050 --> 00:02:09.345
you would get to that point.

00:02:09.345 --> 00:02:11.720
Likewise, the distance between
this point and the center,

00:02:11.720 --> 00:02:12.680
if we were to go that
same distance again,

00:02:12.680 --> 00:02:14.160
you would to get to that point.

00:02:14.160 --> 00:02:16.450
So it seems like that
point would end up there.

00:02:16.450 --> 00:02:17.970
That point would end up there.

00:02:17.970 --> 00:02:18.950
And vice versa.

00:02:18.950 --> 00:02:21.810
So I think this one will
be unchanged by rotation.

00:02:21.810 --> 00:02:23.500
So let's verify it.

00:02:23.500 --> 00:02:25.930
So you go 90 degrees.

00:02:25.930 --> 00:02:27.559
And then you go 180 degrees.

00:02:27.559 --> 00:02:29.600
Or I should say, it will
be unchanged by rotation

00:02:29.600 --> 00:02:32.360
of 180 degrees
around its center.

00:02:32.360 --> 00:02:34.660
We got the same figure.

00:02:34.660 --> 00:02:37.302
Now let's think
about this triangle.

00:02:37.302 --> 00:02:39.260
So if you think about
the center of the figure.

00:02:39.260 --> 00:02:43.250
Let's say the center of the
figure is right around here.

00:02:43.250 --> 00:02:46.200
If you take this point, go
to the center of the figure,

00:02:46.200 --> 00:02:47.965
and then go that
distance again, you

00:02:47.965 --> 00:02:50.090
end up in a place where
there's no point right now.

00:02:50.090 --> 00:02:52.380
So that point is
going to end up there.

00:02:52.380 --> 00:02:54.240
This point is going
to end up there.

00:02:54.240 --> 00:02:55.750
This point is going
to end up here.

00:02:55.750 --> 00:02:58.300
So you're not going to have
the same figure anymore.

00:02:58.300 --> 00:03:01.140
And so we can
rotate it to verify.

00:03:01.140 --> 00:03:04.510
So that's rotated 90 degrees.

00:03:04.510 --> 00:03:08.712
And then that's
rotated 180 degrees.

00:03:08.712 --> 00:03:10.670
So we've kind of turned
this thing on its side.

00:03:10.670 --> 00:03:12.900
It is not the same thing.

00:03:12.900 --> 00:03:15.852
Now let's think about this
figure right over here.

00:03:15.852 --> 00:03:17.810
Well this figure, if you
rotate it 180 degrees,

00:03:17.810 --> 00:03:19.897
this point is now
going to be down here.

00:03:19.897 --> 00:03:21.480
And this point is
going to be up here.

00:03:21.480 --> 00:03:22.980
So you're going to
make, essentially

00:03:22.980 --> 00:03:25.446
it's going to be an upside
down version of the same kite.

00:03:25.446 --> 00:03:26.320
And we can view that.

00:03:26.320 --> 00:03:27.461
We can visualize that now.

00:03:27.461 --> 00:03:28.710
So it's going to be different.

00:03:28.710 --> 00:03:31.910
But let's just show it.

00:03:31.910 --> 00:03:33.500
This is 90 degrees.

00:03:33.500 --> 00:03:38.160
And now this is 180 degrees.

00:03:38.160 --> 00:03:46.800
If it was actually symmetric
about the horizontal axis,

00:03:46.800 --> 00:03:48.607
then we would have a
different scenario.

00:03:48.607 --> 00:03:50.690
We would have a different
scenario with this thing

00:03:50.690 --> 00:03:51.356
right over here.

00:03:51.356 --> 00:03:54.240
If it was some type of a
parallelogram, or a rhombus,

00:03:54.240 --> 00:03:55.770
or something like
that, then this

00:03:55.770 --> 00:03:57.530
could have been
more interesting.

00:03:57.530 --> 00:04:01.890
But in this situation, if it was
just a more symmetrical diamond

00:04:01.890 --> 00:04:05.480
shape, then this rotation
would not have affected it.

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But this one clearly did.