WEBVTT

00:00:00.000 --> 00:00:05.788
[silence]

00:00:05.788 --> 00:00:06.928
So I heard you didn't really

00:00:06.928 --> 00:00:08.688
get the transformations,

00:00:08.688 --> 00:00:10.107
but I think I can help you

00:00:10.107 --> 00:00:11.736
out a little bit.

00:00:11.736 --> 00:00:13.865
So, transform just means

00:00:13.865 --> 00:00:14.794
change something.

00:00:14.794 --> 00:00:15.985
You already know that.

00:00:15.985 --> 00:00:17.383
And in geometry,

00:00:17.383 --> 00:00:19.301
there are some
types of transformations

00:00:19.301 --> 00:00:21.797
that are helpful to know about.

00:00:21.797 --> 00:00:22.962
So, the main thing is

00:00:22.962 --> 00:00:24.947
you have a first thing

00:00:24.947 --> 00:00:25.922
and then a second thing.

00:00:25.922 --> 00:00:27.587
So, one shape changes

00:00:27.587 --> 00:00:29.323
into another shape.

00:00:29.323 --> 00:00:31.030
And, the first one's called

00:00:31.030 --> 00:00:32.097
the pre-image,

00:00:32.097 --> 00:00:33.704
and the next one's
called the image.

00:00:33.704 --> 00:00:35.170
So, "pre" just means before.

00:00:35.170 --> 00:00:36.654
Just remember that,

00:00:36.654 --> 00:00:39.709
like pre-liminary and
pre-historic

00:00:39.714 --> 00:00:41.551
and then the next image.

00:00:41.551 --> 00:00:43.268
So, the first kind

00:00:43.268 --> 00:00:44.725
is a reflection.

00:00:44.725 --> 00:00:46.810
And again, you know what
this means already,

00:00:46.810 --> 00:00:48.480
but a reflection in geometry

00:00:48.480 --> 00:00:49.800
just means that

00:00:49.800 --> 00:00:51.167
you've flipped something

00:00:51.167 --> 00:00:52.664
over an axis.

00:00:52.664 --> 00:00:54.387
So, you could even look

00:00:54.387 --> 00:00:56.400
at your hands.

00:00:56.400 --> 00:00:58.112
This hand is a reflection
of the other hand.

00:00:58.112 --> 00:01:00.345
So, they've just flipped
onto each other,

00:01:00.358 --> 00:01:01.873
or you could think about a

00:01:01.873 --> 00:01:03.968
um, like a landscape scene.

00:01:03.968 --> 00:01:05.761
If you look at a lake

00:01:05.761 --> 00:01:07.134
that's reflecting,

00:01:07.134 --> 00:01:08.218
like the mountains,

00:01:08.218 --> 00:01:09.250
then it's just,

00:01:09.250 --> 00:01:11.822
the axis is that line

00:01:11.822 --> 00:01:14.374
of where the lake starts.

00:01:14.374 --> 00:01:16.853
So, that's a good way
to remember.

00:01:16.853 --> 00:01:19.380
Another kind
is called a rotation,

00:01:19.380 --> 00:01:20.865
and you know what that is too.

00:01:20.867 --> 00:01:22.292
But a rotation just means

00:01:22.292 --> 00:01:23.547
you're switching a shape

00:01:23.547 --> 00:01:26.484
onto a rotating axis.

00:01:26.484 --> 00:01:28.621
So, like this.

00:01:28.621 --> 00:01:30.369
And, another way to think

00:01:30.369 --> 00:01:31.833
about a rotation

00:01:31.833 --> 00:01:34.257
is two reflections.

00:01:34.257 --> 00:01:36.088
And, that doesn't need

00:01:36.088 --> 00:01:37.309
to be confusing.

00:01:37.309 --> 00:01:38.670
All you need to know is that

00:01:38.670 --> 00:01:40.691
if you have one thing

00:01:40.691 --> 00:01:42.217
reflected onto another

00:01:42.217 --> 00:01:43.863
and then it reflects again,

00:01:43.863 --> 00:01:45.750
it ends up being a rotation.

00:01:45.750 --> 00:01:47.447
So...

00:01:50.641 --> 00:01:53.633
That's a good way to
remember that too.

00:01:53.633 --> 00:01:55.762
And, one more type

00:01:55.762 --> 00:01:58.281
is called a translation.

00:01:58.281 --> 00:02:00.845
So, this is just like a slide.

00:02:00.845 --> 00:02:02.679
So, just like this

00:02:02.679 --> 00:02:03.778
moving over here.

00:02:03.778 --> 00:02:05.667
That's a translation.

00:02:05.667 --> 00:02:06.598
Another way to think

00:02:06.598 --> 00:02:08.859
about a translation is again,

00:02:08.859 --> 00:02:11.100
two reflections,

00:02:11.100 --> 00:02:12.549
but this time the reflections

00:02:12.549 --> 00:02:14.267
are in a straight line.

00:02:14.267 --> 00:02:17.795
So, if you reflect
this hand...here

00:02:17.795 --> 00:02:19.488
and you reflect it
one more time,

00:02:19.488 --> 00:02:20.311
then this hand

00:02:20.311 --> 00:02:22.034
has reflected over here.

00:02:22.034 --> 00:02:23.039
So, sorry

00:02:23.039 --> 00:02:26.134
this hand has translated
over here.

00:02:26.134 --> 00:02:29.767
So, these types, um,

00:02:29.767 --> 00:02:33.260
of transformations
are called isometric.

00:02:33.260 --> 00:02:36.421
A translation, a rotation,
and a reflection

00:02:36.421 --> 00:02:39.228
are all
isometric transformations.

00:02:39.228 --> 00:02:40.916
That just means same.

00:02:40.916 --> 00:02:42.974
So, when you start out

00:02:42.974 --> 00:02:44.032
with a pre-image,

00:02:44.032 --> 00:02:45.710
it will always be the same size

00:02:45.710 --> 00:02:48.238
and shape as the image.

00:02:48.238 --> 00:02:49.289
But, there's one more type

00:02:49.289 --> 00:02:50.476
of transformation

00:02:50.476 --> 00:02:52.233
that isn't isometric,

00:02:52.233 --> 00:02:54.312
and that's called dilation.

00:02:54.312 --> 00:02:55.669
And, if you ever go

00:02:55.669 --> 00:02:56.964
to the eye doctor,

00:02:56.964 --> 00:02:58.709
and they dilate your eyes,

00:02:58.709 --> 00:02:59.895
it means they made them

00:02:59.905 --> 00:03:01.211
really big so that they could

00:03:01.211 --> 00:03:02.629
see inside them.

00:03:02.629 --> 00:03:05.915
So, when you dilate a shape,

00:03:05.915 --> 00:03:06.789
you make it either

00:03:06.789 --> 00:03:08.553
smaller or bigger,

00:03:08.553 --> 00:03:10.308
one of those two.

00:03:11.118 --> 00:03:13.250
So, one more term

00:03:13.250 --> 00:03:15.632
is a glide reflection.

00:03:15.632 --> 00:03:16.829
And a glide reflection

00:03:16.829 --> 00:03:18.546
is exactly what it sounds like.

00:03:18.546 --> 00:03:20.061
It's just one reflection

00:03:20.061 --> 00:03:21.746
plus one translation,

00:03:21.746 --> 00:03:23.499
and these can be
in either order.

00:03:23.499 --> 00:03:25.921
You could either
translate and then reflect

00:03:25.921 --> 00:03:28.545
or you could 
reflect and then translate.

00:03:28.545 --> 00:03:30.337
Either way.

00:03:30.337 --> 00:03:31.807
[pause]

00:03:31.807 --> 00:03:34.593
So, transformations are helpful,

00:03:34.593 --> 00:03:36.339
because we can know about

00:03:36.339 --> 00:03:38.706
something called a tessellation.

00:03:38.706 --> 00:03:40.222
And a tessellation is

00:03:40.222 --> 00:03:42.278
just a repeating shape,

00:03:42.278 --> 00:03:44.338
but only certain types
of shapes can repeat

00:03:44.338 --> 00:03:47.078
in patterns that don't overlap.

00:03:47.078 --> 00:03:48.684
So, if we had like

00:03:48.684 --> 00:03:50.390
a crazy star or something,

00:03:50.390 --> 00:03:51.359
it wouldn't tessellate,

00:03:51.359 --> 00:03:53.168
because it's a weird shape.

00:03:53.168 --> 00:03:56.238
But, if we had just like
a square or a rectangle,

00:03:56.238 --> 00:03:57.810
it'll tessellate perfectly,

00:03:57.810 --> 00:03:59.192
because they can overlap

00:03:59.192 --> 00:04:00.524
in any sort of pattern.

00:04:00.524 --> 00:04:02.066
So like, it's why we can have

00:04:02.066 --> 00:04:03.704
a brick wall or something,

00:04:03.704 --> 00:04:04.502
you know.

00:04:04.502 --> 00:04:06.140
They can...

00:04:06.140 --> 00:04:07.712
they perfectly fit
next to each other

00:04:07.712 --> 00:04:09.996
and on top of each other.

00:04:09.996 --> 00:04:11.175
But, if you need to know

00:04:11.175 --> 00:04:13.434
what types of shapes
can tessellate,

00:04:13.434 --> 00:04:16.093
we can remember
by angle measure.

00:04:16.093 --> 00:04:20.387
So, if all the angles of a shape

00:04:20.387 --> 00:04:22.783
equal 360 degrees,

00:04:22.783 --> 00:04:24.468
then they'll be able
to tessellate.

00:04:24.468 --> 00:04:25.640
So, sometimes you can look

00:04:25.640 --> 00:04:27.082
at a shape and figure it out.

00:04:27.082 --> 00:04:29.066
Other times, maybe you can't.

00:04:29.066 --> 00:04:30.452
So, if you need to remember

00:04:30.452 --> 00:04:32.418
what types of shapes can,

00:04:32.418 --> 00:04:34.187
if all the angles that meet up

00:04:34.187 --> 00:04:36.186
equal 360,

00:04:36.186 --> 00:04:37.552
then you're in business.

00:04:37.552 --> 00:04:38.838
You can tessellate.

00:04:38.838 --> 00:04:40.664
So, you know rectangles can,

00:04:40.664 --> 00:04:42.260
like in this picture.

00:04:42.260 --> 00:04:43.768
But then in the next one,

00:04:43.768 --> 00:04:45.136
we have two different shapes.

00:04:45.136 --> 00:04:48.662
We have squares and triangles,

00:04:48.662 --> 00:04:50.708
and you can see that

00:04:50.708 --> 00:04:52.655
all these angles together,

00:04:52.655 --> 00:04:56.832
the 90 plus 90 plus
60 plus 60 plus 60

00:04:56.832 --> 00:04:59.651
equals 360 degrees.

00:04:59.651 --> 00:05:01.179
And, if you just add up

00:05:01.179 --> 00:05:02.627
all the angles that touch

00:05:02.627 --> 00:05:04.341
and they equal 360,

00:05:04.341 --> 00:05:06.355
then you can see,

00:05:06.355 --> 00:05:07.151
like right here,

00:05:07.151 --> 00:05:09.520
that they tessellate perfectly.