0:00:00.000,0:00:05.788
[silence]

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So I heard you didn't really

0:00:06.928,0:00:08.688
get the transformations,

0:00:08.688,0:00:10.107
but I think I can help you

0:00:10.107,0:00:11.736
out a little bit.

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So, transform just means

0:00:13.865,0:00:14.794
change something.

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You already know that.

0:00:15.985,0:00:17.383
And in geometry,

0:00:17.383,0:00:19.301
there are some[br]types of transformations

0:00:19.301,0:00:21.797
that are helpful to know about.

0:00:21.797,0:00:22.962
So, the main thing is

0:00:22.962,0:00:24.947
you have a first thing

0:00:24.947,0:00:25.922
and then a second thing.

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So, one shape changes

0:00:27.587,0:00:29.323
into another shape.

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And, the first one's called

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the pre-image,

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and the next one's[br]called the image.

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So, "pre" just means before.

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Just remember that,

0:00:36.654,0:00:39.709
like pre-liminary and[br]pre-historic

0:00:39.714,0:00:41.551
and then the next image.

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So, the first kind

0:00:43.268,0:00:44.725
is a reflection.

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And again, you know what[br]this means already,

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but a reflection in geometry

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just means that

0:00:49.800,0:00:51.167
you've flipped something

0:00:51.167,0:00:52.664
over an axis.

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So, you could even look

0:00:54.387,0:00:56.400
at your hands.

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This hand is a reflection[br]of the other hand.

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So, they've just flipped[br]onto each other,

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or you could think about a

0:01:01.873,0:01:03.968
um, like a landscape scene.

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If you look at a lake

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that's reflecting,

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like the mountains,

0:01:08.218,0:01:09.250
then it's just,

0:01:09.250,0:01:11.822
the axis is that line

0:01:11.822,0:01:14.374
of where the lake starts.

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So, that's a good way[br]to remember.

0:01:16.853,0:01:19.380
Another kind[br]is called a rotation,

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and you know what that is too.

0:01:20.867,0:01:22.292
But a rotation just means

0:01:22.292,0:01:23.547
you're switching a shape

0:01:23.547,0:01:26.484
onto a rotating axis.

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So, like this.

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And, another way to think

0:01:30.369,0:01:31.833
about a rotation

0:01:31.833,0:01:34.257
is two reflections.

0:01:34.257,0:01:36.088
And, that doesn't need

0:01:36.088,0:01:37.309
to be confusing.

0:01:37.309,0:01:38.670
All you need to know is that

0:01:38.670,0:01:40.691
if you have one thing

0:01:40.691,0:01:42.217
reflected onto another

0:01:42.217,0:01:43.863
and then it reflects again,

0:01:43.863,0:01:45.750
it ends up being a rotation.

0:01:45.750,0:01:47.447
So...

0:01:50.641,0:01:53.633
That's a good way to[br]remember that too.

0:01:53.633,0:01:55.762
And, one more type

0:01:55.762,0:01:58.281
is called a translation.

0:01:58.281,0:02:00.845
So, this is just like a slide.

0:02:00.845,0:02:02.679
So, just like this

0:02:02.679,0:02:03.778
moving over here.

0:02:03.778,0:02:05.667
That's a translation.

0:02:05.667,0:02:06.598
Another way to think

0:02:06.598,0:02:08.859
about a translation is again,

0:02:08.859,0:02:11.100
two reflections,

0:02:11.100,0:02:12.549
but this time the reflections

0:02:12.549,0:02:14.267
are in a straight line.

0:02:14.267,0:02:17.795
So, if you reflect[br]this hand...here

0:02:17.795,0:02:19.488
and you reflect it[br]one more time,

0:02:19.488,0:02:20.311
then this hand

0:02:20.311,0:02:22.034
has reflected over here.

0:02:22.034,0:02:23.039
So, sorry

0:02:23.039,0:02:26.134
this hand has translated[br]over here.

0:02:26.134,0:02:29.767
So, these types, um,

0:02:29.767,0:02:33.260
of transformations[br]are called isometric.

0:02:33.260,0:02:36.421
A translation, a rotation,[br]and a reflection

0:02:36.421,0:02:39.228
are all[br]isometric transformations.

0:02:39.228,0:02:40.916
That just means same.

0:02:40.916,0:02:42.974
So, when you start out

0:02:42.974,0:02:44.032
with a pre-image,

0:02:44.032,0:02:45.710
it will always be the same size

0:02:45.710,0:02:48.238
and shape as the image.

0:02:48.238,0:02:49.289
But, there's one more type

0:02:49.289,0:02:50.476
of transformation

0:02:50.476,0:02:52.233
that isn't isometric,

0:02:52.233,0:02:54.312
and that's called dilation.

0:02:54.312,0:02:55.669
And, if you ever go

0:02:55.669,0:02:56.964
to the eye doctor,

0:02:56.964,0:02:58.709
and they dilate your eyes,

0:02:58.709,0:02:59.895
it means they made them

0:02:59.905,0:03:01.211
really big so that they could

0:03:01.211,0:03:02.629
see inside them.

0:03:02.629,0:03:05.915
So, when you dilate a shape,

0:03:05.915,0:03:06.789
you make it either

0:03:06.789,0:03:08.553
smaller or bigger,

0:03:08.553,0:03:10.308
one of those two.

0:03:11.118,0:03:13.250
So, one more term

0:03:13.250,0:03:15.632
is a glide reflection.

0:03:15.632,0:03:16.829
And a glide reflection

0:03:16.829,0:03:18.546
is exactly what it sounds like.

0:03:18.546,0:03:20.061
It's just one reflection

0:03:20.061,0:03:21.746
plus one translation,

0:03:21.746,0:03:23.499
and these can be[br]in either order.

0:03:23.499,0:03:25.921
You could either[br]translate and then reflect

0:03:25.921,0:03:28.545
or you could [br]reflect and then translate.

0:03:28.545,0:03:30.337
Either way.

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[pause]

0:03:31.807,0:03:34.593
So, transformations are helpful,

0:03:34.593,0:03:36.339
because we can know about

0:03:36.339,0:03:38.706
something called a tessellation.

0:03:38.706,0:03:40.222
And a tessellation is

0:03:40.222,0:03:42.278
just a repeating shape,

0:03:42.278,0:03:44.338
but only certain types[br]of shapes can repeat

0:03:44.338,0:03:47.078
in patterns that don't overlap.

0:03:47.078,0:03:48.684
So, if we had like

0:03:48.684,0:03:50.390
a crazy star or something,

0:03:50.390,0:03:51.359
it wouldn't tessellate,

0:03:51.359,0:03:53.168
because it's a weird shape.

0:03:53.168,0:03:56.238
But, if we had just like[br]a square or a rectangle,

0:03:56.238,0:03:57.810
it'll tessellate perfectly,

0:03:57.810,0:03:59.192
because they can overlap

0:03:59.192,0:04:00.524
in any sort of pattern.

0:04:00.524,0:04:02.066
So like, it's why we can have

0:04:02.066,0:04:03.704
a brick wall or something,

0:04:03.704,0:04:04.502
you know.

0:04:04.502,0:04:06.140
They can...

0:04:06.140,0:04:07.712
they perfectly fit[br]next to each other

0:04:07.712,0:04:09.996
and on top of each other.

0:04:09.996,0:04:11.175
But, if you need to know

0:04:11.175,0:04:13.434
what types of shapes[br]can tessellate,

0:04:13.434,0:04:16.093
we can remember[br]by angle measure.

0:04:16.093,0:04:20.387
So, if all the angles of a shape

0:04:20.387,0:04:22.783
equal 360 degrees,

0:04:22.783,0:04:24.468
then they'll be able[br]to tessellate.

0:04:24.468,0:04:25.640
So, sometimes you can look

0:04:25.640,0:04:27.082
at a shape and figure it out.

0:04:27.082,0:04:29.066
Other times, maybe you can't.

0:04:29.066,0:04:30.452
So, if you need to remember

0:04:30.452,0:04:32.418
what types of shapes can,

0:04:32.418,0:04:34.187
if all the angles that meet up

0:04:34.187,0:04:36.186
equal 360,

0:04:36.186,0:04:37.552
then you're in business.

0:04:37.552,0:04:38.838
You can tessellate.

0:04:38.838,0:04:40.664
So, you know rectangles can,

0:04:40.664,0:04:42.260
like in this picture.

0:04:42.260,0:04:43.768
But then in the next one,

0:04:43.768,0:04:45.136
we have two different shapes.

0:04:45.136,0:04:48.662
We have squares and triangles,

0:04:48.662,0:04:50.708
and you can see that

0:04:50.708,0:04:52.655
all these angles together,

0:04:52.655,0:04:56.832
the 90 plus 90 plus[br]60 plus 60 plus 60

0:04:56.832,0:04:59.651
equals 360 degrees.

0:04:59.651,0:05:01.179
And, if you just add up

0:05:01.179,0:05:02.627
all the angles that touch

0:05:02.627,0:05:04.341
and they equal 360,

0:05:04.341,0:05:06.355
then you can see,

0:05:06.355,0:05:07.151
like right here,

0:05:07.151,0:05:09.520
that they tessellate perfectly.