Shape properties after a sequence of transformations

WEBVTT
Kind: captions
Language: en

00:00:00.871 --> 00:00:02.619
- [Instructor] In past
videos, we've thought about

00:00:02.619 --> 00:00:05.141
whether segment lengths or angle measures

00:00:05.141 --> 00:00:07.638
are preserved with a transformation.

00:00:07.638 --> 00:00:09.798
What we're now gonna think
about is what is preserved

00:00:09.798 --> 00:00:12.358
with a sequence of transformations?

00:00:12.358 --> 00:00:15.643
And in particular, we're gonna
think about angle measure.

00:00:15.643 --> 00:00:18.287
Angle measure

00:00:18.287 --> 00:00:20.067
and segment lengths.

00:00:20.067 --> 00:00:22.568
So if you're transforming
some type of a shape.

00:00:22.568 --> 00:00:25.907
Segment, segment lengths.

00:00:25.907 --> 00:00:27.538
So let's look at this first example.

00:00:27.538 --> 00:00:30.419
They say a sequence of
transformations is described below.

00:00:30.419 --> 00:00:34.348
So we first do a translation,
then we do a reflection

00:00:34.348 --> 00:00:37.335
over a horizontal line, PQ,

00:00:37.335 --> 00:00:41.354
then we do vertical stretch about PQ.

00:00:41.354 --> 00:00:42.478
What is this going to do?

00:00:42.478 --> 00:00:44.718
Is this going to preserve angle measures

00:00:44.718 --> 00:00:48.222
and is this going to
preserve segment lengths?

00:00:48.222 --> 00:00:51.505
Well a translation is
a rigid transformation

00:00:51.505 --> 00:00:53.081
and so that will preserve both

00:00:53.081 --> 00:00:55.149
angle measures and segment lengths.

00:00:55.149 --> 00:00:57.502
So after that, angle
measures and segment lengths

00:00:57.502 --> 00:00:59.324
are still going to be the same.

00:00:59.324 --> 00:01:03.820
A reflection over a horizontal line PQ.

00:01:03.820 --> 00:01:07.107
Well a reflection is also
a rigid transformation

00:01:07.107 --> 00:01:09.158
and so we will continue to preserve

00:01:09.158 --> 00:01:11.536
angle measure and segment lengths.

00:01:11.536 --> 00:01:15.770
Then they say a vertical stretch about PQ.

00:01:15.770 --> 00:01:18.489
Well let's just think about
what a vertical stretch does.

00:01:18.489 --> 00:01:22.749
So if I have some
triangle right over here.

00:01:22.749 --> 00:01:25.595
If I have some triangle
that looks like this.

00:01:25.595 --> 00:01:28.656
Let's say it's triangle A, B, C.

00:01:28.656 --> 00:01:31.018
And if you were to do a vertical stretch,

00:01:31.018 --> 00:01:32.369
what's going to happen?

00:01:32.369 --> 00:01:35.401
Well let's just imagine
that we take these sides

00:01:35.401 --> 00:01:38.896
and we stretch them
out so that we now have

00:01:38.896 --> 00:01:42.307
A is over here or A prime
I should say is over there.

00:01:42.307 --> 00:01:45.974
Let's say that B prime is now over here.

00:01:47.346 --> 00:01:49.103
This isn't going to be exact.

00:01:49.103 --> 00:01:51.854
Well what just happened to my triangle?

00:01:51.854 --> 00:01:54.973
Well the measure of angle C is for sure

00:01:54.973 --> 00:01:56.891
going to be different now.

00:01:56.891 --> 00:01:59.153
And my segment lengths are for sure

00:01:59.153 --> 00:02:00.777
going to be different now.

00:02:00.777 --> 00:02:04.425
A prime C prime is going
to be different than AC

00:02:04.425 --> 00:02:06.126
in terms of segment length.

00:02:06.126 --> 00:02:08.592
So a vertical stretch, if
we're talking about a stretch

00:02:08.592 --> 00:02:11.711
in general, this is going
to preserve neither.

00:02:11.711 --> 00:02:14.771
So neither preserved,

00:02:14.771 --> 00:02:16.536
neither preserved.

00:02:16.536 --> 00:02:19.569
So in general, if you're
doing rigid transformation

00:02:19.569 --> 00:02:21.568
after rigid transformation,
you're gonna preserve

00:02:21.568 --> 00:02:23.569
both angles and segment lengths.

00:02:23.569 --> 00:02:27.113
But if you throw a stretch in
there, then all bets are off.

00:02:27.113 --> 00:02:29.466
You're not going to
preserve either of them.

00:02:29.466 --> 00:02:32.335
Let's do another example.

00:02:32.335 --> 00:02:34.794
A sequence of transformations
is described below.

00:02:34.794 --> 00:02:36.786
And so they give three transformations.

00:02:36.786 --> 00:02:40.060
So pause this video and think
about whether angle measures,

00:02:40.060 --> 00:02:43.181
segment lengths, or will either both

00:02:43.181 --> 00:02:47.176
or neither or only one
of them be preserved?

00:02:47.176 --> 00:02:50.492
Alright so first we have a
rotation about a point P.

00:02:50.492 --> 00:02:53.268
That's a rigid transformation,
it would preserve both

00:02:53.268 --> 00:02:56.189
segment lengths and angle measures.

00:02:56.189 --> 00:02:57.910
Then you have a translation which is

00:02:57.910 --> 00:03:00.038
also a rigid transformation and so

00:03:00.038 --> 00:03:02.358
that would preserve both again.

00:03:02.358 --> 00:03:06.364
Then we have a rotation about point P.

00:03:06.364 --> 00:03:09.019
So once again, another
rigid transformation.

00:03:09.019 --> 00:03:13.590
So in this situation, everything
is going to be preserved.

00:03:13.590 --> 00:03:17.241
So both angle measure,

00:03:17.241 --> 00:03:19.074
angle measure

00:03:20.379 --> 00:03:24.715
and segment length are
going to be preserved

00:03:24.715 --> 00:03:26.181
in this example.

00:03:26.181 --> 00:03:28.348
Let's do one more example.

00:03:29.852 --> 00:03:32.634
So here once again we have a
sequence of transformations.

00:03:32.634 --> 00:03:35.359
And so pause this video again
and see if you can figure out

00:03:35.359 --> 00:03:38.515
whether measures, segment
lengths, both or neither

00:03:38.515 --> 00:03:40.952
are going to be preserved.

00:03:40.952 --> 00:03:44.964
So the first transformation is a dilation.

00:03:44.964 --> 00:03:48.081
So a dilation is a
nonrigid transformation.

00:03:48.081 --> 00:03:50.631
So segment lengths not preserved.

00:03:50.631 --> 00:03:54.714
Segment lengths not preserved.

00:03:56.011 --> 00:03:58.591
And we've seen this in
multiple videos already.

00:03:58.591 --> 00:04:01.337
But in a dilation, angles are preserved.

00:04:01.337 --> 00:04:03.420
Angles preserved.

00:04:04.988 --> 00:04:06.642
So already we've lost our segment lengths

00:04:06.642 --> 00:04:08.168
but we still got our angles.

00:04:08.168 --> 00:04:10.743
Then we have a rotation
about another point Q.

00:04:10.743 --> 00:04:12.708
So this is a rigid transformation,

00:04:12.708 --> 00:04:14.460
it would preserve both but we've already

00:04:14.460 --> 00:04:15.907
lost our segment lengths.

00:04:15.907 --> 00:04:18.840
But angles are going to
continue to be preserved.

00:04:18.840 --> 00:04:21.636
And then finally a
reflection which is still

00:04:21.636 --> 00:04:24.065
a rigid transformation and
it would preserve both,

00:04:24.065 --> 00:04:26.002
but once again our
segment lengths got lost

00:04:26.002 --> 00:04:28.716
through the dilation but we will preserve,

00:04:28.716 --> 00:04:30.959
continue to preserve the angles.

00:04:30.959 --> 00:04:34.663
So in this series of after
these three transformations,

00:04:34.663 --> 00:04:36.722
the only thing that's
going to be preserved

00:04:36.722 --> 00:04:38.635
are going to be your angles.


Dilations and properties

WEBVTT
Kind: captions
Language: en

00:00:01.449 --> 00:00:04.792
- [Instructor] We are told
that quadrilateral ABCD

00:00:04.792 --> 00:00:07.254
is dilated about point P.

00:00:07.254 --> 00:00:08.879
So this is our quadrilateral

00:00:08.879 --> 00:00:11.178
that's going to be dilated around point P.

00:00:11.178 --> 00:00:12.710
And then they ask us some questions.

00:00:12.710 --> 00:00:15.264
Are the coordinates of
the vertices preserved?

00:00:15.264 --> 00:00:18.097
Are corresponding line
segments on the same line?

00:00:18.097 --> 00:00:20.768
Are angle measures preserved?

00:00:20.768 --> 00:00:23.368
Are side lengths, perimeter,
and area preserved?

00:00:23.368 --> 00:00:26.898
So pause this video and
see if you can answer it.

00:00:26.898 --> 00:00:28.152
So now let's work through it together.

00:00:28.152 --> 00:00:29.405
So one thing you might be thinking

00:00:29.405 --> 00:00:30.729
is they're just saying it's dilated.

00:00:30.729 --> 00:00:32.494
They didn't say what the scale factor is.

00:00:32.494 --> 00:00:34.189
We know it's dilated about point P,

00:00:34.189 --> 00:00:35.953
but what's the scale factor?

00:00:35.953 --> 00:00:37.416
Well since they didn't tell us,

00:00:37.416 --> 00:00:40.481
I guess these questions would
apply to any scale factor.

00:00:40.481 --> 00:00:42.455
But to help us visualize this,

00:00:42.455 --> 00:00:44.289
let's just supply a scale factor here.

00:00:44.289 --> 00:00:47.772
Let's just say that our scale factor,

00:00:47.772 --> 00:00:51.070
scale factor, so it fits on this page,

00:00:51.070 --> 00:00:53.601
is equal to 1/2.

00:00:53.601 --> 00:00:55.180
So what would result here?

00:00:55.180 --> 00:00:58.384
Well right now point A is one, two, three,

00:00:58.384 --> 00:01:01.008
four, five, six, seven, eight

00:01:01.008 --> 00:01:02.796
away from point P.

00:01:02.796 --> 00:01:04.932
So its image if we do
a scale factor of 1/2

00:01:04.932 --> 00:01:06.325
would be four away.

00:01:06.325 --> 00:01:07.997
One, two, three, four.

00:01:07.997 --> 00:01:12.827
So right over there, point
A' could be right over there.

00:01:12.827 --> 00:01:14.290
It would be along the same line,

00:01:14.290 --> 00:01:16.170
it would just be 1/2 as far.

00:01:16.170 --> 00:01:19.816
Point D is right now one, two,
three, four, five six away.

00:01:19.816 --> 00:01:21.983
So D' would be three away,

00:01:23.624 --> 00:01:25.598
half as far away.

00:01:25.598 --> 00:01:28.709
And so this would be the segment A'D'

00:01:28.709 --> 00:01:31.008
which corresponds to the segment AD.

00:01:31.008 --> 00:01:32.192
Now let's see,

00:01:32.192 --> 00:01:34.816
point B, if we have a scale factor of 1/2,

00:01:34.816 --> 00:01:37.440
it is one, two, three, four,

00:01:37.440 --> 00:01:39.739
five, six below A.

00:01:39.739 --> 00:01:41.898
But if we have a scale factor of 1/2,

00:01:41.898 --> 00:01:43.965
it's going to be three below A.

00:01:43.965 --> 00:01:45.965
So this would be our B'.

00:01:47.169 --> 00:01:50.304
And so you could also see
that if you connect a line

00:01:50.304 --> 00:01:55.041
going from point P to B that
B' is also on that line.

00:01:55.041 --> 00:01:57.734
It's just 1/2 as far, scale factor of 1/2.

00:01:57.734 --> 00:01:58.734
Now point C,

00:01:59.592 --> 00:02:01.240
there's a couple of ways
we could think about it.

00:02:01.240 --> 00:02:04.468
To go from point P to point C,

00:02:04.468 --> 00:02:07.254
we have to go four down
and two to the left.

00:02:07.254 --> 00:02:09.135
So if you apply a scale factor of 1/2,

00:02:09.135 --> 00:02:12.897
you go two down and one to the left.

00:02:12.897 --> 00:02:15.799
So this would be C' right over here.

00:02:15.799 --> 00:02:17.132
And so A'B'C'D',

00:02:20.861 --> 00:02:24.460
the image of quadrilateral
ABCD after the dilation

00:02:24.460 --> 00:02:25.714
around point P,

00:02:25.714 --> 00:02:27.549
and I picked an arbitrary
scale factor here

00:02:27.549 --> 00:02:29.104
that I felt that I could draw,

00:02:29.104 --> 00:02:30.172
you see it right over here.

00:02:30.172 --> 00:02:31.844
And the reason why I
did that is it makes it

00:02:31.844 --> 00:02:34.213
a little bit easier to
answer this question.

00:02:34.213 --> 00:02:37.255
Are the coordinates of
the vertices preserved?

00:02:37.255 --> 00:02:40.320
Well every vertex, all the vertices,

00:02:40.320 --> 00:02:42.015
are different coordinates here.

00:02:42.015 --> 00:02:45.637
So we would say no, none
of those are preserved.

00:02:45.637 --> 00:02:47.727
Are the, and the whole
idea here is to appreciate

00:02:47.727 --> 00:02:48.841
what gets preserved

00:02:48.841 --> 00:02:52.347
or doesn't get preserved under dilations.

00:02:52.347 --> 00:02:56.248
Are corresponding line
segments on the same line?

00:02:56.248 --> 00:02:58.083
Well let's see, no,

00:02:58.083 --> 00:02:59.987
they're all parallel to each other,

00:02:59.987 --> 00:03:02.866
so AB is parallel to A'B',

00:03:02.866 --> 00:03:06.783
AD is parallel to A'D',
and so on and so forth.

00:03:08.021 --> 00:03:09.948
But they're not all, none of
them are on the same line,

00:03:09.948 --> 00:03:11.759
so I would say no.

00:03:11.759 --> 00:03:14.824
And then we say are
angle measures preserved?

00:03:14.824 --> 00:03:17.843
So this is something that
dilations do preserve.

00:03:17.843 --> 00:03:21.419
So this angle is still
the same as that angle.

00:03:21.419 --> 00:03:24.159
This was a right angle before,
it's still a right angle.

00:03:24.159 --> 00:03:27.340
This angle is still equal to that angle

00:03:27.340 --> 00:03:29.174
or the measure of those angles.

00:03:29.174 --> 00:03:30.637
Likewise the measure of that angle

00:03:30.637 --> 00:03:32.936
is still equal to the
measure of that angle.

00:03:32.936 --> 00:03:35.072
So angle measures are preserved.

00:03:35.072 --> 00:03:36.535
And that's a general thing
that you could take away

00:03:36.535 --> 00:03:37.868
about dilations.

00:03:38.857 --> 00:03:41.899
Are side lengths, perimeter,
and area preserved?

00:03:41.899 --> 00:03:45.684
Well dilations are a
non-rigid transformation

00:03:45.684 --> 00:03:48.540
which tells you that side
lengths are not preserved.

00:03:48.540 --> 00:03:51.187
The distance between corresponding
points are not preserved.

00:03:51.187 --> 00:03:52.812
You can see it very clearly here.

00:03:52.812 --> 00:03:54.322
And if side lengths aren't preserved,

00:03:54.322 --> 00:03:56.249
then perimeter is
definitely not preserved,

00:03:56.249 --> 00:03:58.385
and then area would not be preserved.

00:03:58.385 --> 00:04:00.986
So this would be no for all of these

00:04:00.986 --> 00:04:02.403
under a dilation.

00:04:04.074 --> 00:04:05.955
If we were talking about
a rigid transformation

00:04:05.955 --> 00:04:08.532
like a translation or a rotation,

00:04:08.532 --> 00:04:11.133
then these three would be preserved.

00:04:11.133 --> 00:04:13.216
Let's do another example.

00:04:15.034 --> 00:04:19.887
So we're told triangle ABC
is dilated about point C.

00:04:19.887 --> 00:04:21.930
Once again they don't tell
us what the scale factor,

00:04:21.930 --> 00:04:24.368
and then they ask us
the same four questions.

00:04:24.368 --> 00:04:26.295
So like before, pause this video,

00:04:26.295 --> 00:04:29.012
see if you can figure it out on your own.

00:04:29.012 --> 00:04:30.429
Now let's work through it together,

00:04:30.429 --> 00:04:32.333
and I'm just going to
apply a scale factor.

00:04:32.333 --> 00:04:35.514
In fact you know might as
well do a scale factor of 1/2.

00:04:35.514 --> 00:04:39.252
So scale factor, I'll
just do S.F. for short,

00:04:39.252 --> 00:04:40.808
let's say it's equal to 1/2.

00:04:40.808 --> 00:04:42.689
And I'm just picking one
so that we can visualize

00:04:42.689 --> 00:04:46.357
what its image would look
like after the dilation.

00:04:46.357 --> 00:04:47.867
So let's see,

00:04:47.867 --> 00:04:49.608
right now and remember this is interesting

00:04:49.608 --> 00:04:51.628
because we're dilating around a point

00:04:51.628 --> 00:04:55.483
that happens to be one of
the vertices of the triangle.

00:04:55.483 --> 00:04:58.014
So let's see, point A, to go from C to A,

00:04:58.014 --> 00:04:59.931
we go three to the left

00:05:01.404 --> 00:05:03.935
and we go two, four, six up.

00:05:03.935 --> 00:05:04.768
Six up.

00:05:06.930 --> 00:05:08.180
Just like that.

00:05:09.020 --> 00:05:11.992
So two to the left and then six up.

00:05:11.992 --> 00:05:13.548
So if you have a scale factor of 1/2,

00:05:13.548 --> 00:05:16.752
or sorry not two to the
left, three to the left,

00:05:16.752 --> 00:05:19.307
three to the left and six up.

00:05:19.307 --> 00:05:21.396
And so if you have a scale factor of 1/2,

00:05:21.396 --> 00:05:24.508
instead you're gonna go 1 1/2 to the left

00:05:24.508 --> 00:05:26.133
and then three up.

00:05:26.133 --> 00:05:28.734
And so you're gonna end
up right over there.

00:05:28.734 --> 00:05:29.984
Now this is A'.

00:05:31.381 --> 00:05:34.609
And so notice this is actually
sitting on the segment AC,

00:05:34.609 --> 00:05:36.420
and that happens to be the case

00:05:36.420 --> 00:05:37.674
because once again,

00:05:37.674 --> 00:05:39.554
our center of dilation
is one of the vertices

00:05:39.554 --> 00:05:42.155
of our actual triangle.

00:05:42.155 --> 00:05:45.359
Now let's look at think
about where B' would be.

00:05:45.359 --> 00:05:48.123
To go from C to B we go two to the right,

00:05:48.123 --> 00:05:51.745
and then we go four up, plus four.

00:05:51.745 --> 00:05:55.205
So if we are talking about
a scale factor of 1/2,

00:05:55.205 --> 00:05:57.271
we'll go 1/2 as far to the right

00:05:57.271 --> 00:05:59.082
and 1/2 as far up.

00:05:59.082 --> 00:06:00.267
So one to the right and two up,

00:06:00.267 --> 00:06:02.426
so this would be where B' is.

00:06:02.426 --> 00:06:04.678
And so there we have

00:06:04.678 --> 00:06:08.486
the image, and of
course, C would map to C'

00:06:08.486 --> 00:06:13.107
because it's actually the point
that we are dilating about.

00:06:13.107 --> 00:06:15.360
So this would be our
image right over here,

00:06:15.360 --> 00:06:16.610
A', B', and C'.

00:06:18.750 --> 00:06:21.908
So are the coordinates of
the vertices preserved?

00:06:21.908 --> 00:06:25.321
Well only, only C is preserved.

00:06:25.321 --> 00:06:26.900
The other ones are not preserved.

00:06:26.900 --> 00:06:28.781
And the only reason why C
was preserved is because

00:06:28.781 --> 00:06:30.661
that is the point we are dilating about.

00:06:30.661 --> 00:06:33.076
It's third time I said it, my apologies.

00:06:33.076 --> 00:06:36.049
Are corresponding line
segments on the same line?

00:06:36.049 --> 00:06:37.216
Well A'C' same

00:06:42.225 --> 00:06:44.802
on same line I guess you could say,

00:06:44.802 --> 00:06:45.802
on same line

00:06:48.727 --> 00:06:49.560
as AC.

00:06:51.258 --> 00:06:53.138
And similarly,

00:06:53.138 --> 00:06:55.471
C'B' or I can say B'C' same.

00:06:58.131 --> 00:06:59.803
I'm just gonna use a
little bit of shorthand.

00:06:59.803 --> 00:07:02.566
And once again that is
because we are dilating

00:07:02.566 --> 00:07:05.515
about a point that is one of the vertices.

00:07:05.515 --> 00:07:07.744
The other segment A'B' is not.

00:07:07.744 --> 00:07:08.577
Not A'B'.

00:07:11.436 --> 00:07:12.434
It's not on the same line.

00:07:12.434 --> 00:07:13.595
It is parallel.

00:07:13.595 --> 00:07:15.058
This is parallel to that,

00:07:15.058 --> 00:07:17.241
but it is not on the same line.

00:07:17.241 --> 00:07:18.866
Are angle measures preserved?

00:07:18.866 --> 00:07:20.004
Well we don't even have to look at this.

00:07:20.004 --> 00:07:21.653
This is going to be a yes.

00:07:21.653 --> 00:07:23.115
This is one of the things about dilation,

00:07:23.115 --> 00:07:25.019
angle measures are preserved.

00:07:25.019 --> 00:07:27.040
That angle's equal to that angle,

00:07:27.040 --> 00:07:29.571
the measure of this angle
is congruent to that angle,

00:07:29.571 --> 00:07:31.591
or this angle's congruent to that angle,

00:07:31.591 --> 00:07:34.934
and of course this is the
same angle in both cases.

00:07:34.934 --> 00:07:37.396
Are side lengths, perimeters,
and area preserved?

00:07:37.396 --> 00:07:38.626
Once again no.

00:07:38.626 --> 00:07:40.089
When we're doing a dilation side lengths,

00:07:40.089 --> 00:07:41.552
and if side lengths aren't preserved,

00:07:41.552 --> 00:07:42.643
perimeter's not gonna be preserved

00:07:42.643 --> 00:07:44.687
and area's not gonna be preserved.

00:07:44.687 --> 00:07:45.741
And we're done.


Example identifying the center of dilation

WEBVTT
Kind: captions
Language: en

00:00:01.170 --> 00:00:03.910
- [Instructor] We are
told that triangle N'

00:00:03.910 --> 00:00:06.232
is the image of triangle N

00:00:06.232 --> 00:00:08.159
under a dilation.

00:00:08.159 --> 00:00:10.853
So this is N' in this red color,

00:00:10.853 --> 00:00:13.918
and then N is the original
N is in this blue color.

00:00:13.918 --> 00:00:16.936
What is the center of dilation?

00:00:16.936 --> 00:00:18.864
And they give us some
choices here, choice A,

00:00:18.864 --> 00:00:21.580
B, C, or D as the center of dilation.

00:00:21.580 --> 00:00:23.833
So pause this video and see
if you can figure it out

00:00:23.833 --> 00:00:24.833
on your own.

00:00:26.340 --> 00:00:29.266
So there's a couple of
ways to think about it.

00:00:29.266 --> 00:00:31.147
One way I like to just first think about,

00:00:31.147 --> 00:00:33.794
well what is the scale factor here?

00:00:33.794 --> 00:00:36.090
So in our original N,
we have this side here,

00:00:36.090 --> 00:00:37.622
it has a length of two,

00:00:37.622 --> 00:00:40.107
and then once we dilated it by,

00:00:40.107 --> 00:00:41.895
and used that scale factor,

00:00:41.895 --> 00:00:45.262
the corresponding side
has a length of four.

00:00:45.262 --> 00:00:46.817
So we went from two to four.

00:00:46.817 --> 00:00:49.163
So we can figure out our scale factor,

00:00:49.163 --> 00:00:51.580
scale factor is equal to two.

00:00:53.040 --> 00:00:55.873
Two times two is equal to four.

00:00:55.873 --> 00:00:58.079
Now what about our center of dilation?

00:00:58.079 --> 00:01:00.331
So one way to think about it is,

00:01:00.331 --> 00:01:02.375
pick two corresponding points.

00:01:02.375 --> 00:01:04.999
So let's say we were to pick this point

00:01:04.999 --> 00:01:07.042
and this point.

00:01:07.042 --> 00:01:08.125
So the image,

00:01:09.434 --> 00:01:12.313
the corresponding point on N',

00:01:12.313 --> 00:01:14.751
is going to be the scale factor

00:01:14.751 --> 00:01:17.677
as far away from our center of dilation

00:01:17.677 --> 00:01:19.140
as the original point.

00:01:19.140 --> 00:01:21.508
So in this example we know
the scale factor is two,

00:01:21.508 --> 00:01:23.783
so this is going to be twice as far

00:01:23.783 --> 00:01:25.688
from our center of dilation

00:01:25.688 --> 00:01:28.520
as the corresponding point.

00:01:28.520 --> 00:01:29.890
Well you can immediately see,

00:01:29.890 --> 00:01:31.678
and it's going to be
in the same direction,

00:01:31.678 --> 00:01:35.184
so actually if you just draw
a line connecting these two,

00:01:35.184 --> 00:01:37.808
there's actually only one
choice that sits on that line,

00:01:37.808 --> 00:01:40.989
and that is choice D right over here

00:01:40.989 --> 00:01:43.428
as being the center of dilation.

00:01:43.428 --> 00:01:45.262
And you can also verify that notice,

00:01:45.262 --> 00:01:48.699
this first point on the original triangle,

00:01:48.699 --> 00:01:52.866
its change in x is two and
its change in y is three,

00:01:53.760 --> 00:01:57.220
two three, to go from
from point D to point

00:01:57.220 --> 00:01:58.497
to that point.

00:01:58.497 --> 00:02:01.005
And then if you wanna go
to point D to its image,

00:02:01.005 --> 00:02:03.002
well now you gotta go twice as far.

00:02:03.002 --> 00:02:05.603
Your change in x is four,

00:02:05.603 --> 00:02:08.389
and your change in y is six.

00:02:08.389 --> 00:02:09.643
You could use the Pythagorean Theorem

00:02:09.643 --> 00:02:11.222
to calculate this distance

00:02:11.222 --> 00:02:12.963
and then the longer distance,

00:02:12.963 --> 00:02:14.519
but what you see is,

00:02:14.519 --> 00:02:17.979
is that the corresponding
point is now twice as far

00:02:17.979 --> 00:02:21.044
from your center of dilation.

00:02:21.044 --> 00:02:22.460
So there's a couple of
ways to think about it.

00:02:22.460 --> 00:02:25.432
One, if you connect corresponding points,

00:02:25.432 --> 00:02:27.778
your center of dilation
is going to be on a line

00:02:27.778 --> 00:02:30.285
that connects those two points.

00:02:30.285 --> 00:02:33.583
And that the image should
be the scale factor

00:02:33.583 --> 00:02:35.626
as far away from the center of dilation,

00:02:35.626 --> 00:02:37.344
in this case it should be twice as far

00:02:37.344 --> 00:02:39.388
from the center of dilation as the point

00:02:39.388 --> 00:02:41.388
that it is the image of.


Dilation scale factor examples

WEBVTT
Kind: captions
Language: en

00:00:01.124 --> 00:00:02.586
- [Instructor] We are told that pentagon

00:00:02.586 --> 00:00:03.503
A'B'C'D'E',

00:00:05.233 --> 00:00:06.650
which is in red right over here,

00:00:06.650 --> 00:00:09.761
is the image of pentagon ABCDE

00:00:09.761 --> 00:00:11.387
under a dilation.

00:00:11.387 --> 00:00:12.720
So that's ABCDE.

00:00:14.080 --> 00:00:17.517
What is the scale factor of the dilation?

00:00:17.517 --> 00:00:19.862
So they don't even tell us
the center of the dilation,

00:00:19.862 --> 00:00:21.905
but in order to figure
out the scale factor

00:00:21.905 --> 00:00:23.972
you just have to realize
when you do a dilation,

00:00:23.972 --> 00:00:26.433
the distance between corresponding points

00:00:26.433 --> 00:00:29.127
will change according to the scale factor.

00:00:29.127 --> 00:00:31.124
So for example we could
look at the distance

00:00:31.124 --> 00:00:35.164
between point A and
point B right over here.

00:00:35.164 --> 00:00:37.207
What is our change in y?

00:00:37.207 --> 00:00:39.924
Our change in, or even
what is our distance?

00:00:39.924 --> 00:00:41.387
Our change in y is our distance

00:00:41.387 --> 00:00:43.059
because we don't have a change in x.

00:00:43.059 --> 00:00:44.777
Well this is one, two, three,

00:00:44.777 --> 00:00:46.611
four, five, six.

00:00:46.611 --> 00:00:48.632
So this length right over here

00:00:48.632 --> 00:00:50.628
is equal to six.

00:00:50.628 --> 00:00:52.045
Now what about the corresponding side

00:00:52.045 --> 00:00:53.212
from A' to B'?

00:00:54.390 --> 00:00:57.385
Well this length right
over here is equal to two,

00:00:57.385 --> 00:00:59.104
and so you can see we went from having

00:00:59.104 --> 00:01:01.170
a length of six to a length of two,

00:01:01.170 --> 00:01:03.492
so you would have to multiply by 1/3.

00:01:03.492 --> 00:01:05.536
So our scale factor right over here

00:01:05.536 --> 00:01:07.834
is 1/3.

00:01:07.834 --> 00:01:09.669
Now you might be saying okay
that was pretty straightforward

00:01:09.669 --> 00:01:11.178
because we had a very clear,

00:01:11.178 --> 00:01:13.430
you could just see the
distance between A and B.

00:01:13.430 --> 00:01:15.311
How would you do it if
you didn't have a vertical

00:01:15.311 --> 00:01:17.424
or a horizontal line?

00:01:17.424 --> 00:01:19.468
Well one way to think about it is,

00:01:19.468 --> 00:01:22.092
the changes in y and the changes in x

00:01:22.092 --> 00:01:23.601
would scale accordingly.

00:01:23.601 --> 00:01:24.599
So if you looked at the distance

00:01:24.599 --> 00:01:26.480
between point A and point E,

00:01:26.480 --> 00:01:30.706
our change in y is negative
three right over here,

00:01:30.706 --> 00:01:33.887
and our change in x is
positive three right over here.

00:01:33.887 --> 00:01:36.302
And you can see over
here between A' and E',

00:01:36.302 --> 00:01:38.485
our change in y is negative one,

00:01:38.485 --> 00:01:40.760
which is 1/3 of negative three,

00:01:40.760 --> 00:01:42.432
and our change in x is one,

00:01:42.432 --> 00:01:44.476
which is 1/3 of three.

00:01:44.476 --> 00:01:46.101
So once again you see our scale factor

00:01:46.101 --> 00:01:48.655
being 1/3.

00:01:48.655 --> 00:01:50.738
Let's do another example.

00:01:53.020 --> 00:01:56.270
So we are told that pentagon A'B'C'D'E'

00:01:58.454 --> 00:01:59.801
is the image, and they don't,

00:01:59.801 --> 00:02:01.055
they haven't drawn that here,

00:02:01.055 --> 00:02:03.469
is the image of pentagon ABCDE

00:02:03.469 --> 00:02:07.045
under a dilation with
a scale factor of 5/2.

00:02:07.045 --> 00:02:08.764
So they're giving us our scale factor.

00:02:08.764 --> 00:02:11.681
What is the length of segment A'E'?

00:02:13.013 --> 00:02:15.056
So as I was mentioning while I read it,

00:02:15.056 --> 00:02:19.120
they didn't actually draw this one out.

00:02:19.120 --> 00:02:21.256
So how do we figure out
the length of a segment?

00:02:21.256 --> 00:02:23.253
Well I encourage you to pause the video

00:02:23.253 --> 00:02:25.598
and try to think about it.

00:02:25.598 --> 00:02:27.107
Well they give us the scale factor,

00:02:27.107 --> 00:02:28.686
and so what it tells us,

00:02:28.686 --> 00:02:30.614
the scale factor is 5/2.

00:02:30.614 --> 00:02:34.213
That means that the
corresponding lengths will change

00:02:34.213 --> 00:02:36.535
by a factor of 5/2.

00:02:36.535 --> 00:02:40.436
So to figure out the
length of segment A'E',

00:02:40.436 --> 00:02:41.991
this is going to be,

00:02:41.991 --> 00:02:44.035
you could think of it as the image

00:02:44.035 --> 00:02:45.202
of segment AE.

00:02:46.450 --> 00:02:48.702
And so you can see that

00:02:48.702 --> 00:02:50.035
the length of AE

00:02:51.024 --> 00:02:52.357
is equal to two.

00:02:54.089 --> 00:02:56.172
And so the length of A'E'

00:02:57.735 --> 00:03:00.312
is going to be equal to

00:03:00.312 --> 00:03:03.540
AE which is two times the scale factor,

00:03:03.540 --> 00:03:07.766
times 5/2, this is our scale
factor right over here.

00:03:07.766 --> 00:03:09.832
And of course what's two times 5/2?

00:03:09.832 --> 00:03:12.711
Well it is going to be equal to five,

00:03:12.711 --> 00:03:15.196
five of these units right over here.

00:03:15.196 --> 00:03:16.589
So in this case we
didn't even have to draw

00:03:16.589 --> 00:03:17.506
A'B'C'D'E'.

00:03:19.050 --> 00:03:20.885
In fact they haven't even
given us enough information.

00:03:20.885 --> 00:03:23.393
I could draw the scale of that,

00:03:23.393 --> 00:03:24.693
but I actually don't know where to put it

00:03:24.693 --> 00:03:26.086
because they didn't even give us

00:03:26.086 --> 00:03:27.874
our center of dilation.

00:03:27.874 --> 00:03:30.289
But we know that corresponding sides,

00:03:30.289 --> 00:03:32.402
or the lengths between
corresponding points,

00:03:32.402 --> 00:03:36.210
are going to be scaled
by the scale factor.

00:03:36.210 --> 00:03:37.232
Now with that in mind,

00:03:37.232 --> 00:03:39.315
let's do another example.

00:03:42.828 --> 00:03:45.939
So we are told that triangle A'B'C',

00:03:45.939 --> 00:03:47.657
which they depicted right over here,

00:03:47.657 --> 00:03:49.933
is the image of triangle ABC,

00:03:49.933 --> 00:03:51.187
which they did not depict,

00:03:51.187 --> 00:03:55.157
under a dilation with
a scale factor of two.

00:03:55.157 --> 00:03:57.897
What is the length of segment AB?

00:03:57.897 --> 00:03:59.639
Once again they haven't drawn AB here,

00:03:59.639 --> 00:04:00.730
how do we figure it out?

00:04:00.730 --> 00:04:03.331
Well it's gonna be a similar
way as the last example,

00:04:03.331 --> 00:04:05.142
but here they've given us the image

00:04:05.142 --> 00:04:06.489
and they didn't give us the original.

00:04:06.489 --> 00:04:08.323
So how do we do it?

00:04:08.323 --> 00:04:09.902
Well the key, and pause the video again

00:04:09.902 --> 00:04:12.433
and try to do it on your own.

00:04:12.433 --> 00:04:15.103
Well the key realization here is

00:04:15.103 --> 00:04:18.702
that if you take the length of segment AB

00:04:18.702 --> 00:04:21.001
and you were to multiply
by the scale factor,

00:04:21.001 --> 00:04:22.882
so you multiply it by two,

00:04:22.882 --> 00:04:24.066
then you're going to get the length

00:04:24.066 --> 00:04:25.399
of segment A'B'.

00:04:28.014 --> 00:04:32.031
The image's length is
equal to the scale factor

00:04:32.031 --> 00:04:36.373
times the corresponding length
on our original triangle.

00:04:36.373 --> 00:04:38.695
So what is the length of A'B'?

00:04:38.695 --> 00:04:40.390
Well this is straightforward
to figure out.

00:04:40.390 --> 00:04:42.851
It is one, two, three, four,

00:04:42.851 --> 00:04:45.429
five, six, seven, eight.

00:04:45.429 --> 00:04:47.054
So this right over here is eight,

00:04:47.054 --> 00:04:50.189
so we have two times
the length of segment AB

00:04:50.189 --> 00:04:51.814
is equal to eight.

00:04:51.814 --> 00:04:53.881
And then you get the length of segment AB,

00:04:53.881 --> 00:04:55.645
just divide both sides by two,

00:04:55.645 --> 00:04:57.224
is equal to four.

00:04:57.224 --> 00:04:58.474
And we're done.


Dilating points example

WEBVTT
Kind: captions
Language: en

00:00:01.095 --> 00:00:03.770
- [Instructor] We're asked
to plot the image of point A

00:00:03.770 --> 00:00:07.937
under a dilation about point P
with a scale factor of three.

00:00:10.481 --> 00:00:12.738
So what they're saying when
they say under a dilation,

00:00:12.738 --> 00:00:16.043
they're saying stretching
it or scaling it up or down

00:00:16.043 --> 00:00:18.385
around the point P.

00:00:18.385 --> 00:00:20.256
Now so what we're going
to do is just think about,

00:00:20.256 --> 00:00:24.541
well how far is point A and
then we want to dilate it

00:00:24.541 --> 00:00:26.463
with a scale factor of three.

00:00:26.463 --> 00:00:29.073
So however far A is from point P,

00:00:29.073 --> 00:00:32.765
it's going to be three times
further under the dilation.

00:00:32.765 --> 00:00:34.938
Three times further in the same direction.

00:00:34.938 --> 00:00:36.701
So how do we think about that?

00:00:36.701 --> 00:00:38.073
Well, one way to think about it

00:00:38.073 --> 00:00:41.133
is to go from P to A
you have to go one down

00:00:41.133 --> 00:00:45.701
and two to the left, so
minus one and minus two.

00:00:45.701 --> 00:00:48.443
And so if you dilate it
with a factor of three,

00:00:48.443 --> 00:00:51.614
then you're going to want to
go three times as far down.

00:00:51.614 --> 00:00:55.781
So minus three, and three
times as far to the left,

00:00:57.731 --> 00:00:59.430
so you'll go minus six.

00:00:59.430 --> 00:01:03.868
So one, let me do this, so
negative one, negative two,

00:01:03.868 --> 00:01:07.543
negative three, negative four,
negative five, negative six.

00:01:07.543 --> 00:01:10.798
So you will end up right over there.

00:01:10.798 --> 00:01:12.592
And you can even see
it, that this is indeed

00:01:12.592 --> 00:01:17.009
three times as far from
P in the same direction.

00:01:17.009 --> 00:01:18.517
So we could call the image of point A,

00:01:18.517 --> 00:01:21.502
maybe we call that A prime,
and so there you have it.

00:01:21.502 --> 00:01:24.803
It has been dilated with
a scale factor of three.

00:01:24.803 --> 00:01:25.927
And so you might be saying, wait,

00:01:25.927 --> 00:01:28.489
I'm used to dilating being
stretching or scaling.

00:01:28.489 --> 00:01:30.423
How have I stretched or scaled something?

00:01:30.423 --> 00:01:32.093
Well imagine a bunch of points here

00:01:32.093 --> 00:01:34.107
that represents some type of picture

00:01:34.107 --> 00:01:38.087
and if you push them all three
times further from point P,

00:01:38.087 --> 00:01:40.379
which you could do as
your center of dilation,

00:01:40.379 --> 00:01:42.532
then you would expand
the size of your picture

00:01:42.532 --> 00:01:44.888
by a scale factor of three.

00:01:44.888 --> 00:01:48.051
Let's do another example with a point.

00:01:48.051 --> 00:01:51.727
So, here we're told,
plot the image of point A

00:01:51.727 --> 00:01:55.894
under a dilation about the
origin with a scale factor of 1/3

00:01:58.872 --> 00:02:01.391
so first of all we don't
even see the point A here,

00:02:01.391 --> 00:02:03.355
so it's actually below the fold.

00:02:03.355 --> 00:02:05.814
So let's see, there we
go, that's our point A.

00:02:05.814 --> 00:02:07.930
We want it to be about the origin,

00:02:07.930 --> 00:02:10.457
so about the point zero zero.

00:02:10.457 --> 00:02:13.763
This is what we want to, the
dilation about the origin

00:02:13.763 --> 00:02:18.554
with a scale factor of 1/3, scale is 1/3.

00:02:18.554 --> 00:02:20.207
Scale factor, I should say.

00:02:20.207 --> 00:02:21.780
So how do we do this?

00:02:21.780 --> 00:02:26.063
Well here, however far
A is from the origin

00:02:26.063 --> 00:02:29.903
we now want to be in the same
direction, but 1/3 as far.

00:02:29.903 --> 00:02:32.599
So one way to think about it,
to go from the origin to A

00:02:32.599 --> 00:02:36.516
you have to go six down
and three to the right.

00:02:39.226 --> 00:02:44.118
So 1/3 of that would be two
down and one to the right.

00:02:44.118 --> 00:02:48.285
Two is 1/3 of six and one is 1/3 of three,

00:02:49.911 --> 00:02:52.210
so you will end up right over here.

00:02:52.210 --> 00:02:53.796
That would be our A prime.

00:02:53.796 --> 00:02:57.992
So notice, you are 1/3
away from the origin

00:02:57.992 --> 00:03:00.066
as we were before because once again,

00:03:00.066 --> 00:03:03.992
this is point A under a
dilation about the origin

00:03:03.992 --> 00:03:06.406
with a scale factor of 1/3.


Mapping shapes example

WEBVTT
Kind: captions
Language: en

00:00:01.059 --> 00:00:03.402
- [Instructor] So I'm here
on the Khan Academy Exercise

00:00:03.402 --> 00:00:04.635
for mapping shapes.

00:00:04.635 --> 00:00:08.081
And I'm asked to map the
movable quadrilateral

00:00:08.081 --> 00:00:10.744
onto quadrilateral ABCD

00:00:10.744 --> 00:00:12.962
using rigid transformations.

00:00:12.962 --> 00:00:16.207
So here in blue, I have
the movable quadrilateral.

00:00:16.207 --> 00:00:18.290
And I want to map it onto

00:00:19.216 --> 00:00:20.996
this quadrilateral in gray.

00:00:20.996 --> 00:00:24.009
And we have a series of tools here

00:00:24.009 --> 00:00:25.878
rigid transformations of translation,

00:00:25.878 --> 00:00:27.597
rotation and reflection

00:00:27.597 --> 00:00:29.112
on the Khan Academy tool.

00:00:29.112 --> 00:00:30.561
And of course, we can undo it.

00:00:30.561 --> 00:00:33.161
So the technique I'm
going to use to do this is

00:00:33.161 --> 00:00:34.973
I'm gonna first use translation

00:00:34.973 --> 00:00:37.480
to make one of the corresponding points

00:00:37.480 --> 00:00:40.734
overlap with the point
that it corresponds to.

00:00:40.734 --> 00:00:42.562
So, for example, it looks like

00:00:42.562 --> 00:00:45.178
this corresponds to
Point C right over here.

00:00:45.178 --> 00:00:47.575
And so I am going to translate.

00:00:47.575 --> 00:00:49.855
And so notice, once I click that I can

00:00:49.855 --> 00:00:50.862
translate this around.

00:00:50.862 --> 00:00:53.138
So I'm gonna translate
right over here to Point C.

00:00:53.138 --> 00:00:54.564
And now, let's see.

00:00:54.564 --> 00:00:55.652
To make

00:00:55.652 --> 00:00:57.178
these two overlap

00:00:57.178 --> 00:00:59.255
I really can't do anymore translation,

00:00:59.255 --> 00:01:00.776
I made one point overlap.

00:01:00.776 --> 00:01:03.436
Do I rotate, or do I reflect?

00:01:03.436 --> 00:01:06.016
Well, if I eyeball it right over here

00:01:06.016 --> 00:01:08.680
it looks like I am doing a rotation.

00:01:08.680 --> 00:01:11.041
Let me try to make user rotation to make

00:01:11.041 --> 00:01:14.519
this segment eight over here
overlap with segment CD.

00:01:14.519 --> 00:01:16.631
So let me do a rotation now.

00:01:16.631 --> 00:01:18.205
And so

00:01:18.205 --> 00:01:20.946
let's see, yep, this is looking good

00:01:20.946 --> 00:01:21.943
There you go.

00:01:21.943 --> 00:01:23.246
We did the rotation

00:01:23.246 --> 00:01:24.978
and we are done.

00:01:24.978 --> 00:01:27.873
Now, let's do another example.

00:01:27.873 --> 00:01:31.302
So, here, what do we need to do?

00:01:31.302 --> 00:01:33.356
All right, so I'm gonna
do the same technique.

00:01:33.356 --> 00:01:35.546
This seems to correspond to Point C

00:01:35.546 --> 00:01:37.634
so I'm gonna translate first.

00:01:37.634 --> 00:01:38.717
So, translate

00:01:40.310 --> 00:01:41.935
first.

00:01:41.935 --> 00:01:44.340
And then, there's something interesting

00:01:44.340 --> 00:01:46.033
going on right over here.

00:01:46.033 --> 00:01:48.172
Because I've actually been able to overlap

00:01:48.172 --> 00:01:50.672
points C and A by shifting it.

00:01:51.800 --> 00:01:53.776
By translating it, I should say.

00:01:53.776 --> 00:01:55.576
And so it's not clear
if I were to rotate it

00:01:55.576 --> 00:01:58.161
then I would lose the fact that A

00:01:58.161 --> 00:02:00.040
that the point that corresponds to A

00:02:00.040 --> 00:02:01.126
is now sitting on top of A.

00:02:01.126 --> 00:02:02.360
The point that corresponds to C

00:02:02.360 --> 00:02:03.599
is now sitting on top of C.

00:02:03.599 --> 00:02:05.299
It feels like a reflection.

00:02:05.299 --> 00:02:07.291
And it looks like a
line that would actually

00:02:07.291 --> 00:02:09.332
contain the points A and C.

00:02:09.332 --> 00:02:10.797
If we reflect over that line

00:02:10.797 --> 00:02:11.792
we will be in good shape.

00:02:11.792 --> 00:02:12.816
So let me see.

00:02:12.816 --> 00:02:14.349
Reflection.

00:02:14.349 --> 00:02:17.187
So let me see, move of the line.

00:02:17.187 --> 00:02:18.609
See how it looks.

00:02:18.609 --> 00:02:21.511
There, that's not what I wanted to do.

00:02:21.511 --> 00:02:22.529
So let's see.

00:02:22.529 --> 00:02:24.532
Let me move my line.

00:02:24.532 --> 00:02:27.304
So that is, I think, a
good line of reflection.

00:02:27.304 --> 00:02:29.060
And then let me actually

00:02:29.060 --> 00:02:30.303
try to reflect, and there you go.

00:02:30.303 --> 00:02:32.020
I was able to reflect over that line.

00:02:32.020 --> 00:02:34.375
And my clue that I had
to reflect over the line

00:02:34.375 --> 00:02:36.294
that contained A and C is that

00:02:36.294 --> 00:02:37.794
the points A and C

00:02:38.754 --> 00:02:40.393
and their images

00:02:40.393 --> 00:02:42.076
after the transformation

00:02:42.076 --> 00:02:43.387
were all sitting on top of each other.

00:02:43.387 --> 00:02:45.694
So, that was a good clue
that on a reflection

00:02:45.694 --> 00:02:48.493
if they're both sitting
on the line of reflection

00:02:48.493 --> 00:02:50.380
that they wouldn't move, so to speak.

00:02:50.380 --> 00:02:51.522
And there you have it.

00:02:51.522 --> 00:02:55.028
So this was a translation
and then a reflection.


Properties perserved after rigid transformations

WEBVTT
Kind: captions
Language: en

00:00:00.668 --> 00:00:02.334
- [Instructor] What we're
going to do in this video

00:00:02.334 --> 00:00:05.506
is think about what properties
of a shape are preserved

00:00:05.506 --> 00:00:07.177
or not preserved, as

00:00:07.177 --> 00:00:09.677
they undergo a transformation.

00:00:10.558 --> 00:00:12.997
In particular, we're gonna
think about rotations

00:00:12.997 --> 00:00:14.652
and reflections in this video.

00:00:14.652 --> 00:00:17.256
And both of those are
rigid transformations

00:00:17.256 --> 00:00:20.434
which means that the length
between corresponding

00:00:20.434 --> 00:00:22.112
points do not change.

00:00:22.112 --> 00:00:25.418
So for example, let's say
we take this circle A,

00:00:25.418 --> 00:00:27.040
it's centered at Point A.

00:00:27.040 --> 00:00:29.998
And we were to rotate it around Point P.

00:00:29.998 --> 00:00:32.424
Point P is the center of rotation.

00:00:32.424 --> 00:00:35.404
And just for the sake
of argument we rotate it

00:00:35.404 --> 00:00:37.450
clockwise a certain angle.

00:00:37.450 --> 00:00:40.088
So let's say we end up right over

00:00:40.088 --> 00:00:42.343
so we're gonna rotate that way.

00:00:42.343 --> 00:00:45.850
And let's say our center
ends up right over here.

00:00:45.850 --> 00:00:47.772
So our new circle, the image

00:00:47.772 --> 00:00:50.739
after the rotation might
look something like this.

00:00:50.739 --> 00:00:51.791
And I'm hand drawing it.

00:00:51.791 --> 00:00:53.299
So you got to forgive

00:00:53.299 --> 00:00:56.753
that it's not that well
hand drawn of a circle.

00:00:56.753 --> 00:01:00.468
But the circle might
look something like this.

00:01:00.468 --> 00:01:03.084
And so, the clear things
that are preserved

00:01:03.084 --> 00:01:04.704
or maybe it's not so clear, we're gonna

00:01:04.704 --> 00:01:07.105
hope we make them clear right now.

00:01:07.105 --> 00:01:09.305
Things that are preserved
under a rigid transformation

00:01:09.305 --> 00:01:11.353
like this rotation right over here.

00:01:11.353 --> 00:01:13.152
This is clearly a rotation.

00:01:13.152 --> 00:01:15.888
Things that are preserved

00:01:15.888 --> 00:01:19.075
well, you have things like the radius

00:01:19.075 --> 00:01:20.806
of the circle.

00:01:20.806 --> 00:01:23.766
The radius length, I could
say, to be more particular.

00:01:23.766 --> 00:01:25.683
The radius here is two.

00:01:26.518 --> 00:01:28.435
The radius here is also

00:01:29.708 --> 00:01:32.208
is also two, right over there.

00:01:33.100 --> 00:01:35.450
You have things like the perimeter.

00:01:35.450 --> 00:01:37.372
Well, if the radius is preserved

00:01:37.372 --> 00:01:38.852
the perimeter of a circle

00:01:38.852 --> 00:01:40.483
which we call a circumference

00:01:40.483 --> 00:01:42.274
well, that's just a
function of the radius.

00:01:42.274 --> 00:01:45.056
We're talking about two
times pi times the radius.

00:01:45.056 --> 00:01:48.453
So the perimeter, of course,
is going to be preserved.

00:01:48.453 --> 00:01:50.087
In fact, that follows from the fact that

00:01:50.087 --> 00:01:51.495
the length of the radius is preserved.

00:01:51.495 --> 00:01:53.521
And of course, if the
radius is preserved and then

00:01:53.521 --> 00:01:56.238
the area is also going to be preserved.

00:01:56.238 --> 00:01:58.532
The area is just pi
times the radius squared.

00:01:58.532 --> 00:01:59.784
So they have the same radius.

00:01:59.784 --> 00:02:01.078
They're gonna have all of these in common.

00:02:01.078 --> 00:02:03.497
And you could also that
feels intuitively right.

00:02:03.497 --> 00:02:05.613
So what is not preserved?

00:02:05.613 --> 00:02:06.780
Not preserved.

00:02:08.064 --> 00:02:09.752
And this is in general true of

00:02:09.752 --> 00:02:11.302
rigid transformations is that

00:02:11.302 --> 00:02:13.488
they will preserve the distance between

00:02:13.488 --> 00:02:16.306
corresponding points if
we're transforming a shape

00:02:16.306 --> 00:02:19.414
they'll preserve things
like perimeter and area.

00:02:19.414 --> 00:02:21.034
And this case, I can set a perimeter.

00:02:21.034 --> 00:02:22.719
I can say circumference.

00:02:22.719 --> 00:02:23.886
Circumference.

00:02:25.236 --> 00:02:26.820
So they'll preserve things like that.

00:02:26.820 --> 00:02:27.821
They'll preserve angles.

00:02:27.821 --> 00:02:29.545
We don't have clear
angles in this picture.

00:02:29.545 --> 00:02:31.418
But, they'll preserve things like angles.

00:02:31.418 --> 00:02:34.081
But what they won't
preserve is the coordinates.

00:02:34.081 --> 00:02:34.998
Coordinates

00:02:35.869 --> 00:02:37.524
of

00:02:37.524 --> 00:02:38.853
corresponding points.

00:02:38.853 --> 00:02:40.691
They might sometimes, but not always.

00:02:40.691 --> 00:02:43.519
So for example, the
coordinate of the center here

00:02:43.519 --> 00:02:45.151
is for sure, going to change.

00:02:45.151 --> 00:02:48.081
We go from the coordinate
negative three comma zero.

00:02:48.081 --> 00:02:50.142
To here we went to the coordinate

00:02:50.142 --> 00:02:52.396
we went to the coordinate
negative one comma two.

00:02:52.396 --> 00:02:54.144
So the coordinates are not preserved.

00:02:54.144 --> 00:02:56.264
Coordinates of the center.

00:02:56.264 --> 00:02:58.770
Let's do another example with a

00:02:58.770 --> 00:03:00.099
non-circular shape.

00:03:00.099 --> 00:03:02.582
And we'll do a different
type of transformation.

00:03:02.582 --> 00:03:04.832
In this situation let us do

00:03:06.469 --> 00:03:08.331
a reflection.

00:03:08.331 --> 00:03:10.950
So, we have a quadrilateral here.

00:03:10.950 --> 00:03:12.533
Quadrilateral ABCD.

00:03:14.035 --> 00:03:16.352
And we want to think about
what is preserved, or not

00:03:16.352 --> 00:03:19.177
preserved as we do a reflection

00:03:19.177 --> 00:03:20.677
across the line L.

00:03:21.865 --> 00:03:23.270
So let me write that down.

00:03:23.270 --> 00:03:27.424
We're gonna have a
reflection in this situation.

00:03:27.424 --> 00:03:29.267
And we can even think about this

00:03:29.267 --> 00:03:30.918
without even doing the
reflection ourselves.

00:03:30.918 --> 00:03:33.831
But let's just do the
reflection really fast.

00:03:33.831 --> 00:03:37.998
So we're reflecting across
the line XYZ equal to X.

00:03:39.006 --> 00:03:41.008
So what it essentially
does to the coordinates is

00:03:41.008 --> 00:03:43.136
it swaps the X and Y coordinates.

00:03:43.136 --> 00:03:44.505
But you don't have to know that

00:03:44.505 --> 00:03:46.069
for the sake of this video.

00:03:46.069 --> 00:03:47.069
So,

00:03:47.069 --> 00:03:49.738
B prime would be right over here.

00:03:49.738 --> 00:03:52.667
A prime would be right over there.

00:03:52.667 --> 00:03:54.000
D prime would be

00:03:55.519 --> 00:03:56.957
right over here.

00:03:56.957 --> 00:03:59.345
And since C is right on the line now

00:03:59.345 --> 00:04:02.184
its image, C prime, won't change.

00:04:02.184 --> 00:04:03.435
And so our new

00:04:03.435 --> 00:04:05.102
when we reflect over

00:04:06.299 --> 00:04:07.829
the line L.

00:04:07.829 --> 00:04:10.624
And you don't have to
know for the sake of this

00:04:10.624 --> 00:04:13.359
video, exactly how I
did that fairly quickly.

00:04:13.359 --> 00:04:14.682
I really just want you to see

00:04:14.682 --> 00:04:16.195
what the reflection looks like.

00:04:16.195 --> 00:04:18.327
The real appreciation here is think about,

00:04:18.327 --> 00:04:21.123
well, what happens with
rigid transformations.

00:04:21.123 --> 00:04:24.651
So, it's gonna look something like this.

00:04:24.651 --> 00:04:25.901
The reflection.

00:04:26.811 --> 00:04:29.726
The reflection looks something like this.

00:04:29.726 --> 00:04:32.054
So what's preserved?

00:04:32.054 --> 00:04:33.943
And in general, this is good to know

00:04:33.943 --> 00:04:35.364
for any rigid transformation
what's preserved.

00:04:35.364 --> 00:04:36.599
Well, side lengths.

00:04:36.599 --> 00:04:38.587
That's actually one way that we even

00:04:38.587 --> 00:04:40.605
use to define what a
rigid transformation is.

00:04:40.605 --> 00:04:43.456
A transformation that
preserves the lengths

00:04:43.456 --> 00:04:45.073
between corresponding points.

00:04:45.073 --> 00:04:46.945
Angle measures.

00:04:46.945 --> 00:04:48.195
Angle measures.

00:04:49.345 --> 00:04:51.748
So, for example, this angle here,

00:04:51.748 --> 00:04:53.900
the angle A, is gonna be the same

00:04:53.900 --> 00:04:55.601
as the angle A prime over here.

00:04:55.601 --> 00:04:57.816
Side lengths, the distance between A and B

00:04:57.816 --> 00:04:59.645
is going to be the same
as the distance between

00:04:59.645 --> 00:05:01.417
A prime and B prime.

00:05:01.417 --> 00:05:02.250
Perimeter.

00:05:03.647 --> 00:05:06.197
If you have the same side
lengths and the same angles,

00:05:06.197 --> 00:05:07.608
the perimeter and area

00:05:07.608 --> 00:05:09.025
are also going to be preserved.

00:05:09.025 --> 00:05:10.751
Just like we saw with
the rotation example.

00:05:10.751 --> 00:05:12.252
These are rigid transformations.

00:05:12.252 --> 00:05:14.911
These are the types of
things that are preserved.

00:05:14.911 --> 00:05:16.930
Well, what is not preserved?

00:05:16.930 --> 00:05:17.928
Not preserved.

00:05:17.928 --> 00:05:19.618
And this just goes back to the example

00:05:19.618 --> 00:05:20.613
we just looked at.

00:05:20.613 --> 00:05:22.499
Well, coordinates are not preserved.

00:05:22.499 --> 00:05:24.820
So as we see, the image of A

00:05:24.820 --> 00:05:27.980
A prime has different coordinates than A.

00:05:27.980 --> 00:05:30.060
B prime has different coordinates than B.

00:05:30.060 --> 00:05:32.369
C prime, in this case, happens to have

00:05:32.369 --> 00:05:34.052
the same coordinates as C because C

00:05:34.052 --> 00:05:36.213
happens to sit on our, the line

00:05:36.213 --> 00:05:37.781
that we're reflecting over.

00:05:37.781 --> 00:05:39.540
But D prime definitely does not have

00:05:39.540 --> 00:05:41.211
the same coordinates as D.

00:05:41.211 --> 00:05:42.044
So, most

00:05:43.834 --> 00:05:44.876
of

00:05:44.876 --> 00:05:47.645
or let me say, coordinates of AB

00:05:47.645 --> 00:05:49.285
ABD.

00:05:49.285 --> 00:05:50.202
Coordinates

00:05:51.983 --> 00:05:52.816
of

00:05:53.701 --> 00:05:54.700
A

00:05:54.700 --> 00:05:55.715
B

00:05:55.715 --> 00:05:57.048
D not preserved.

00:05:58.254 --> 00:06:00.653
After transformation, or their images

00:06:00.653 --> 00:06:02.513
don't have the same coordinates.

00:06:02.513 --> 00:06:03.346
After

00:06:04.225 --> 00:06:05.475
transformation.

00:06:08.285 --> 00:06:10.607
The one coordinate that
happened to be preserved here

00:06:10.607 --> 00:06:12.214
is Cs

00:06:12.214 --> 00:06:13.238
coordinates.

00:06:13.238 --> 00:06:16.345
Because it was right on
the line of reflection.

00:06:16.345 --> 00:06:18.253
And you can also look at other properties

00:06:18.253 --> 00:06:19.562
so how it might relate

00:06:19.562 --> 00:06:21.594
how different segments might relate to

00:06:21.594 --> 00:06:23.359
lines that would not be

00:06:23.359 --> 00:06:25.329
that were not being transformed.

00:06:25.329 --> 00:06:27.647
So, for example, right over here

00:06:27.647 --> 00:06:30.730
before transformation, CD is parallel

00:06:31.869 --> 00:06:33.405
to the Y axis.

00:06:33.405 --> 00:06:35.191
You see this right over here.

00:06:35.191 --> 00:06:37.473
But after the transformation

00:06:37.473 --> 00:06:39.189
C prime D prime.

00:06:39.189 --> 00:06:41.695
So this could be C prime D prime

00:06:41.695 --> 00:06:44.887
is no longer parallel to the Y axis.

00:06:44.887 --> 00:06:47.992
In fact, now it is parallel to the X axis.

00:06:47.992 --> 00:06:49.941
So, when you have the
relationship to things

00:06:49.941 --> 00:06:52.518
outside of the things we're transformed

00:06:52.518 --> 00:06:54.130
that relationship might not.

00:06:54.130 --> 00:06:57.461
Those relationships may no longer be true

00:06:57.461 --> 00:06:59.544
after the transformation.


Finding measures using rigid transformations

WEBVTT
Kind: captions
Language: en

00:00:00.343 --> 00:00:03.122
- [Instructor] We are
told that triangle ABC,

00:00:03.122 --> 00:00:07.407
which is right over here,
is reflected across line l,

00:00:07.407 --> 00:00:10.417
so it's reflected across
line l right over here,

00:00:10.417 --> 00:00:13.933
to get to triangle A
prime B prime C prime.

00:00:13.933 --> 00:00:14.995
Fair enough.

00:00:14.995 --> 00:00:18.062
So based on that, they're
going to ask us some questions.

00:00:18.062 --> 00:00:19.435
And I encourage you to pause this video

00:00:19.435 --> 00:00:21.448
and see if you can figure out
the answers to these questions

00:00:21.448 --> 00:00:24.132
on your own before I work through them.

00:00:24.132 --> 00:00:25.368
So the first question they say

00:00:25.368 --> 00:00:27.658
is well, what's A prime C prime?

00:00:27.658 --> 00:00:31.284
This is really what's the length
of segment A prime C prime?

00:00:31.284 --> 00:00:35.092
So they want the length
of this right over here.

00:00:35.092 --> 00:00:37.174
How do we figure that out?

00:00:37.174 --> 00:00:40.741
Well, the key realization
here is a reflection

00:00:40.741 --> 00:00:42.908
is a rigid transformation.

00:00:43.966 --> 00:00:48.049
Rigid transformation,
which is a very fancy word.

00:00:48.931 --> 00:00:51.227
But it's really just saying
that it's a transformation

00:00:51.227 --> 00:00:56.155
where the length between
corresponding points don't change.

00:00:56.155 --> 00:00:58.923
If we're talking about
a shape like a triangle,

00:00:58.923 --> 00:01:01.442
the angle measures won't change,

00:01:01.442 --> 00:01:05.015
the perimeter won't change,
and the area won't change.

00:01:05.015 --> 00:01:07.702
So we're gonna use the
fact that the length

00:01:07.702 --> 00:01:10.140
between corresponding points won't change.

00:01:10.140 --> 00:01:12.293
So the length between A prime and C prime

00:01:12.293 --> 00:01:16.274
is gonna be the same as
the length between A and C.

00:01:16.274 --> 00:01:20.527
So A prime C prime is
going to be equal to AC,

00:01:20.527 --> 00:01:23.030
which is equal to they
tell us right over there.

00:01:23.030 --> 00:01:25.443
That's this corresponding
side of the triangle.

00:01:25.443 --> 00:01:27.237
That has a length of three.

00:01:27.237 --> 00:01:28.696
So we answered the first question.

00:01:28.696 --> 00:01:30.027
And maybe that gave you a good clue.

00:01:30.027 --> 00:01:32.582
And so I encourage you
to keep pausing the video

00:01:32.582 --> 00:01:34.919
when you feel like you
can have a go at it.

00:01:34.919 --> 00:01:36.793
Alright, the next question is

00:01:36.793 --> 00:01:39.404
what is the measure of angle B prime?

00:01:39.404 --> 00:01:41.471
So that's this angle right over here.

00:01:41.471 --> 00:01:44.280
And we're gonna use the
exact same property.

00:01:44.280 --> 00:01:47.349
Angle B prime corresponds to angle B.

00:01:47.349 --> 00:01:51.152
It underwent a rigid
transformation of a reflection.

00:01:51.152 --> 00:01:53.812
This would also be true
if we had a translation,

00:01:53.812 --> 00:01:56.420
or if we had a rotation.

00:01:56.420 --> 00:01:59.231
And so right over here, the
measure of angle B prime

00:01:59.231 --> 00:02:01.785
would be the same as
the measure of angle B.

00:02:01.785 --> 00:02:03.901
But what is that going to be equal to?

00:02:03.901 --> 00:02:06.584
Well, we can use the fact
that if we call that measure,

00:02:06.584 --> 00:02:08.479
let's just call that X.

00:02:08.479 --> 00:02:12.893
X plus 53 degrees, we'll
do it all in degrees,

00:02:12.893 --> 00:02:16.266
plus 90 degrees, this right angle here.

00:02:16.266 --> 00:02:18.212
Well, the sum of the
interior angles of a triangle

00:02:18.212 --> 00:02:20.674
add up to 180 degrees.

00:02:20.674 --> 00:02:21.689
And so what do we have?

00:02:21.689 --> 00:02:25.430
We could subtract, let's
see, 53 plus 90 is X,

00:02:25.430 --> 00:02:28.847
plus 143 degrees is equal to 180 degrees.

00:02:30.713 --> 00:02:34.026
And so subtract 143
degrees from both sides.

00:02:34.026 --> 00:02:36.939
You will get X is equal to, let's see,

00:02:36.939 --> 00:02:39.022
180 minus 40 would be 40.

00:02:40.841 --> 00:02:43.820
80 minus 43 would be 37 degrees.

00:02:43.820 --> 00:02:47.643
X is equal to 37 degrees,
so that is 37 degrees.

00:02:47.643 --> 00:02:49.172
If that's 37 degrees,

00:02:49.172 --> 00:02:52.167
then this is also going to be 37 degrees.

00:02:52.167 --> 00:02:56.334
Next, they ask us what is
the area of triangle ABC?

00:02:57.277 --> 00:02:58.110
ABC.

00:02:58.960 --> 00:03:00.338
Well, it's gonna have the same area

00:03:00.338 --> 00:03:02.449
as A prime B prime C prime.

00:03:02.449 --> 00:03:04.855
And so a couple of ways
we could think about it.

00:03:04.855 --> 00:03:07.866
We could try to find the area
of A prime B prime C prime

00:03:07.866 --> 00:03:09.084
based on the fact that we already know

00:03:09.084 --> 00:03:12.238
that this length is three
and this is a right triangle.

00:03:12.238 --> 00:03:14.507
Or we can use the fact that

00:03:14.507 --> 00:03:18.110
this length right over here,
four, from A prime to B prime

00:03:18.110 --> 00:03:20.468
is gonna be the same thing as
this length right over here,

00:03:20.468 --> 00:03:23.464
from A prime to B prime, which is four.

00:03:23.464 --> 00:03:24.559
And so the area of this triangle,

00:03:24.559 --> 00:03:25.680
especially this is a right triangle,

00:03:25.680 --> 00:03:26.715
it's quite straightforward,

00:03:26.715 --> 00:03:29.843
it's the base times the height times 1/2.

00:03:29.843 --> 00:03:34.771
So this area is gonna be
1/2 times the base, four,

00:03:34.771 --> 00:03:39.164
times the height, three,
which is equal to 1/2 of 12,

00:03:39.164 --> 00:03:42.376
which is equal to six square units.

00:03:42.376 --> 00:03:43.628
And then last but not least,

00:03:43.628 --> 00:03:48.276
what's the perimeter of triangle
A prime B prime C prime?

00:03:48.276 --> 00:03:50.200
Well, here we just used
the Pythagorean Theorem

00:03:50.200 --> 00:03:54.103
to figure out the length
of this hypotenuse.

00:03:54.103 --> 00:03:56.441
And we know that this is a length of three

00:03:56.441 --> 00:03:58.679
based on the whole rigid transformation

00:03:58.679 --> 00:04:00.249
and lengths are preserved.

00:04:00.249 --> 00:04:02.270
And so you might immediately recognize

00:04:02.270 --> 00:04:03.117
that if you have a right triangle

00:04:03.117 --> 00:04:05.456
where one side is three
and the other side is four,

00:04:05.456 --> 00:04:07.405
that the hypotenuse is five.

00:04:07.405 --> 00:04:09.014
Three four five triangles.

00:04:09.014 --> 00:04:10.593
Or you could just the Pythagorean Theorem.

00:04:10.593 --> 00:04:13.926
You say three squared plus four squared,

00:04:15.673 --> 00:04:18.020
four squared is equal to let's just say

00:04:18.020 --> 00:04:19.937
the hypotenuse squared.

00:04:21.384 --> 00:04:24.927
Well, three squared plus four
squared, that's nine plus 16.

00:04:24.927 --> 00:04:28.094
25 is equal to the hypotenuse squared.

00:04:28.945 --> 00:04:33.172
And so the hypotenuse right
over here will be equal to five.

00:04:33.172 --> 00:04:35.304
And so they're not asking us
the length of the hypotenuse.

00:04:35.304 --> 00:04:36.453
They wanna know the perimeter.

00:04:36.453 --> 00:04:39.659
So it's gonna be four
plus three plus five,

00:04:39.659 --> 00:04:41.904
which is equal to 12.

00:04:41.904 --> 00:04:43.911
The perimeter of either
of those triangles,

00:04:43.911 --> 00:04:45.814
because it's just one's
the image of the other

00:04:45.814 --> 00:04:47.263
under a rigid transformation.

00:04:47.263 --> 00:04:49.456
They're gonna have the same
perimeter, the same area.

00:04:49.456 --> 00:04:51.472
The perimeter of either
of the triangles is 12.

00:04:51.472 --> 00:04:54.343
The area of either of
the triangles is six.

00:04:54.343 --> 00:04:55.593
And we're done.


Example reflecting quadrilateral over x axis

WEBVTT
Kind: captions
Language: en

00:00:00.648 --> 00:00:02.732
- [Instructor] We're
asked to plot the image

00:00:02.732 --> 00:00:04.936
of quadrilateral ABCD

00:00:04.936 --> 00:00:07.300
so that's this blue quadrilateral here.

00:00:07.300 --> 00:00:10.115
Under a reflection across the X axis.

00:00:10.115 --> 00:00:11.503
So that's the X axis.

00:00:11.503 --> 00:00:13.489
And we have our little
tool here on Khan Academy

00:00:13.489 --> 00:00:15.427
where we can construct a quadrilateral.

00:00:15.427 --> 00:00:18.376
And we need to construct a reflection

00:00:18.376 --> 00:00:19.421
of triangle A,

00:00:19.421 --> 00:00:20.465
B, C,

00:00:20.465 --> 00:00:21.467
D.

00:00:21.467 --> 00:00:22.827
And so what we can do
is, let me scroll down

00:00:22.827 --> 00:00:23.913
a little bit.

00:00:23.913 --> 00:00:26.067
So we can see the entire coordinate axis.

00:00:26.067 --> 00:00:28.717
We want to find the
reflection across the X axis.

00:00:28.717 --> 00:00:30.554
So I'm gonna reflect point by point.

00:00:30.554 --> 00:00:32.627
And actually, let me just
move this whole thing

00:00:32.627 --> 00:00:34.611
down here so that we can

00:00:34.611 --> 00:00:36.983
so that we can see what is going on

00:00:36.983 --> 00:00:38.319
a little bit clearer.

00:00:38.319 --> 00:00:41.424
So let's just first reflect point

00:00:41.424 --> 00:00:44.090
let me move this a little
bit out of the way.

00:00:44.090 --> 00:00:46.940
So let's first reflect Point A.

00:00:46.940 --> 00:00:49.367
So we're gonna reflect across the X axis.

00:00:49.367 --> 00:00:52.174
A is four units above the X axis.

00:00:52.174 --> 00:00:54.109
One, two, three, four.

00:00:54.109 --> 00:00:57.119
So, its image, A prime we could say,

00:00:57.119 --> 00:00:59.471
would be four units below the X axis.

00:00:59.471 --> 00:01:00.992
So, one, two,

00:01:00.992 --> 00:01:02.355
three, four.

00:01:02.355 --> 00:01:04.547
So let's make this right over here

00:01:04.547 --> 00:01:05.464
A, A prime.

00:01:07.449 --> 00:01:09.391
I'm having trouble putting the

00:01:09.391 --> 00:01:10.593
let's see if I move these other

00:01:10.593 --> 00:01:11.611
characters around.

00:01:11.611 --> 00:01:12.610
Okay, there you go.

00:01:12.610 --> 00:01:13.830
So this is gonna be my A prime.

00:01:13.830 --> 00:01:14.939
Now let me try B.

00:01:14.939 --> 00:01:18.112
B is two units above the X axis.

00:01:18.112 --> 00:01:21.157
So B prime is gonna have
the same X coordinate

00:01:21.157 --> 00:01:23.988
but it's gonna be two
units below the X axis.

00:01:23.988 --> 00:01:25.391
So let's make this our B.

00:01:25.391 --> 00:01:27.139
So this is our B right over here.

00:01:27.139 --> 00:01:29.315
Now let's make this our C.

00:01:29.315 --> 00:01:30.482
C, right here,

00:01:31.511 --> 00:01:33.654
has the X coordinate of negative five.

00:01:33.654 --> 00:01:36.237
A d a Y coordinate of negative four.

00:01:36.237 --> 00:01:39.215
Now C prime would have
the same X coordinate

00:01:39.215 --> 00:01:41.585
but instead of being four
units below the X axis,

00:01:41.585 --> 00:01:43.652
it will be four units above the X axis.

00:01:43.652 --> 00:01:45.702
So it would have the
coordinates negative five

00:01:45.702 --> 00:01:47.086
comma positive four.

00:01:47.086 --> 00:01:48.884
So this is going to be our C here.

00:01:48.884 --> 00:01:51.038
So this goes to negative five,

00:01:51.038 --> 00:01:54.419
one, two, three, positive four.

00:01:54.419 --> 00:01:56.737
And then last but not least, D.

00:01:56.737 --> 00:01:58.706
And so let's see, D right now

00:01:58.706 --> 00:02:00.739
is at negative two comma negative one.

00:02:00.739 --> 00:02:03.004
If we reflect across the X axis

00:02:03.004 --> 00:02:05.030
instead of being one
unit below the X axis,

00:02:05.030 --> 00:02:07.336
we'll be one unit above the X axis.

00:02:07.336 --> 00:02:09.502
And we'll keep our X
coordinate of negative two.

00:02:09.502 --> 00:02:11.470
And so, there you have it.

00:02:11.470 --> 00:02:13.585
We have constructed the reflection

00:02:13.585 --> 00:02:16.447
of ABCD across the X axis.

00:02:16.447 --> 00:02:18.413
And what's interesting about this example

00:02:18.413 --> 00:02:20.375
is that, the original quadrilateral

00:02:20.375 --> 00:02:22.506
is on top of the X axis.

00:02:22.506 --> 00:02:24.877
So you an kind of see this top part

00:02:24.877 --> 00:02:26.770
of the quadrilateral
gets reflected below it.

00:02:26.770 --> 00:02:28.634
And this bottom part of the quadrilateral

00:02:28.634 --> 00:02:29.907
gets reflected above it.

00:02:29.907 --> 00:02:31.283
And then you can see that indeed

00:02:31.283 --> 00:02:34.171
do they indeed do look like reflections

00:02:34.171 --> 00:02:36.254
flipped over the X axis.


Line of reflection example

WEBVTT
Kind: captions
Language: en

00:00:00.597 --> 00:00:03.041
- [Instructor] We're asked to
draw the line of reflection

00:00:03.041 --> 00:00:06.329
that reflects triangle ABC,
so that's this blue triangle,

00:00:06.329 --> 00:00:09.294
onto triangle A prime B prime C prime,

00:00:09.294 --> 00:00:11.260
which is this red
triangle right over here.

00:00:11.260 --> 00:00:14.151
And they give us a
little line drawing tool

00:00:14.151 --> 00:00:16.558
in order to draw the line of reflection.

00:00:16.558 --> 00:00:18.605
So the way I'm gonna think about it is

00:00:18.605 --> 00:00:20.835
well, when I just eyeball it,

00:00:20.835 --> 00:00:22.700
it looks like I'm just flipped over

00:00:22.700 --> 00:00:25.133
some type of a horizontal line here.

00:00:25.133 --> 00:00:26.289
But let's see if we can actually construct

00:00:26.289 --> 00:00:27.984
a horizontal line where
it does actually look

00:00:27.984 --> 00:00:29.833
like the line of reflection.

00:00:29.833 --> 00:00:31.468
So let's see, C and C prime,

00:00:31.468 --> 00:00:33.327
how far apart are they from each other?

00:00:33.327 --> 00:00:37.494
So if we go one, two,
three, four, five, six down.

00:00:38.807 --> 00:00:40.062
So they are six apart.

00:00:40.062 --> 00:00:43.954
So let's see if we just put
this three above C prime

00:00:43.954 --> 00:00:47.862
and three below C, let's see
if this horizontal line works

00:00:47.862 --> 00:00:49.581
as a line of reflection.

00:00:49.581 --> 00:00:53.086
So C, or C prime is
definitely the reflection

00:00:53.086 --> 00:00:55.633
of C across this line.

00:00:55.633 --> 00:00:57.943
C is exactly three units above it,

00:00:57.943 --> 00:01:00.645
and C prime is exactly
three units below it.

00:01:00.645 --> 00:01:02.518
Let's see if it works for A and A prime.

00:01:02.518 --> 00:01:06.815
A is one, two, three,
four, five units above it.

00:01:06.815 --> 00:01:11.787
A prime is one, two, three,
four, five units below it.

00:01:11.787 --> 00:01:13.056
So that's looking good.

00:01:13.056 --> 00:01:14.697
Now let's just check out B.

00:01:14.697 --> 00:01:18.166
So B, we can see it's at the
y-coordinate here is seven.

00:01:18.166 --> 00:01:20.836
This line right over here
is y is equal to one.

00:01:20.836 --> 00:01:23.510
And so what we would
have here is, let's see,

00:01:23.510 --> 00:01:27.458
this looks like it's six
units above this line,

00:01:27.458 --> 00:01:30.105
and B prime is six units below the line.

00:01:30.105 --> 00:01:31.209
So this indeed works.

00:01:31.209 --> 00:01:33.775
We've just constructed
the line of reflection

00:01:33.775 --> 00:01:36.370
that reflects the blue
triangle, triangle ABC,

00:01:36.370 --> 00:01:39.537
onto triangle A prime B prime C prime.


Reflecting points across horizontal and vertical lines

WEBVTT
Kind: captions
Language: en

00:00:00.341 --> 00:00:03.013
- [Instructor] We're asked
to plot the image of point A

00:00:03.013 --> 00:00:06.096
under a reflection across the line l.

00:00:07.388 --> 00:00:09.419
So we have our line l here,

00:00:09.419 --> 00:00:12.296
and so we wanna plot the image of here,

00:00:12.296 --> 00:00:13.700
we wanna plot the image of point A

00:00:13.700 --> 00:00:17.121
under a reflection across line l.

00:00:17.121 --> 00:00:18.868
Well, one way to think about it is point A

00:00:18.868 --> 00:00:23.035
is exactly one, two, three,
four units to the right of l.

00:00:24.788 --> 00:00:27.277
And so its reflection is going to be

00:00:27.277 --> 00:00:29.118
four units to the left of l.

00:00:29.118 --> 00:00:31.951
So if we go one, two, three, four,

00:00:33.033 --> 00:00:35.557
that would be the image of point A.

00:00:35.557 --> 00:00:38.013
We could maybe denote that as A prime.

00:00:38.013 --> 00:00:39.859
So if you're doing this on
the Khan Academy exercise,

00:00:39.859 --> 00:00:42.116
you would actually just click
on a point right over there,

00:00:42.116 --> 00:00:43.180
and it would show up.

00:00:43.180 --> 00:00:44.554
But this would be the reflection

00:00:44.554 --> 00:00:46.971
of point A across the line l.

00:00:48.496 --> 00:00:51.226
Let's do another example.

00:00:51.226 --> 00:00:54.218
So here we're asked plot
the image of point B

00:00:54.218 --> 00:00:57.566
under a reflection across the x-axis.

00:00:57.566 --> 00:00:59.280
Alright, so this is point B,

00:00:59.280 --> 00:01:00.282
and we're going to reflect it

00:01:00.282 --> 00:01:03.115
across the x-axis right over here.

00:01:04.586 --> 00:01:06.770
So to go from B to the x-axis,

00:01:06.770 --> 00:01:09.918
it's exactly five units below the x-axis.

00:01:09.918 --> 00:01:12.251
One, two, three, four, five.

00:01:13.123 --> 00:01:16.387
So if we were to reflect
across the x-axis,

00:01:16.387 --> 00:01:18.062
essentially create its mirror image,

00:01:18.062 --> 00:01:20.440
it's going to be five
units above the x-axis.

00:01:20.440 --> 00:01:22.773
One, two, three, four, five.

00:01:23.905 --> 00:01:25.669
So that's where the image would be.

00:01:25.669 --> 00:01:28.612
Maybe we could denote that with a B prime.

00:01:28.612 --> 00:01:31.612
We are reflecting across the x-axis.

00:01:32.722 --> 00:01:35.256
Let's do another example.

00:01:35.256 --> 00:01:39.423
So here they tell us point
C prime is the image of C,

00:01:40.577 --> 00:01:42.424
which is at the coordinates negative four

00:01:42.424 --> 00:01:47.357
comma negative two, under a
reflection across the y-axis.

00:01:47.357 --> 00:01:50.276
What are the coordinates of C prime?

00:01:50.276 --> 00:01:51.495
So pause this video

00:01:51.495 --> 00:01:54.706
and see if you can figure
it out on your own.

00:01:54.706 --> 00:01:56.825
So there's several ways to approach it.

00:01:56.825 --> 00:01:59.425
It doesn't hurt to do
a quick visual diagram.

00:01:59.425 --> 00:02:02.426
So that could be my x-axis.

00:02:02.426 --> 00:02:04.426
This would be my y-axis.

00:02:06.161 --> 00:02:08.447
And it's the point negative
four comma negative two,

00:02:08.447 --> 00:02:09.446
so that might look like this.

00:02:09.446 --> 00:02:12.417
Negative four, negative two.

00:02:12.417 --> 00:02:15.517
So this is the point C right over here.

00:02:15.517 --> 00:02:18.806
And we wanna reflect across the y-axis.

00:02:18.806 --> 00:02:21.049
So we wanna reflect across the y-axis,

00:02:21.049 --> 00:02:24.425
which I am coloring it
in red right over here.

00:02:24.425 --> 00:02:25.592
So let's see.

00:02:25.592 --> 00:02:29.425
The point C is four to
the left of the y-axis.

00:02:30.269 --> 00:02:31.581
So its reflection is going to be

00:02:31.581 --> 00:02:34.342
four to the right of the y-axis.

00:02:34.342 --> 00:02:36.349
So let me do it like this.

00:02:36.349 --> 00:02:39.094
So instead of being four to the left,

00:02:39.094 --> 00:02:43.162
we wanna go four to the
right, so plus four.

00:02:43.162 --> 00:02:46.786
So where would that put our C prime?

00:02:46.786 --> 00:02:49.328
So our C prime would be right over there.

00:02:49.328 --> 00:02:51.055
And what would its coordinates be?

00:02:51.055 --> 00:02:53.280
Well, it would have the same y-coordinate,

00:02:53.280 --> 00:02:56.561
so C prime would have a
y-coordinate of negative two.

00:02:56.561 --> 00:02:58.535
But what would its x-coordinate be?

00:02:58.535 --> 00:03:00.318
Well, instead of it being negative four,

00:03:00.318 --> 00:03:02.984
it gets flipped over the y-axis,

00:03:02.984 --> 00:03:07.029
so now it's gonna have a
x-coordinate of positive four.

00:03:07.029 --> 00:03:10.108
So the coordinates here would
be four comma negative two.

00:03:10.108 --> 00:03:11.816
Four comma negative two.

00:03:11.816 --> 00:03:13.813
You might've been able
to do this in your head.

00:03:13.813 --> 00:03:16.096
Although, for me, even if
I try to do it in my head,

00:03:16.096 --> 00:03:19.356
I would still have this
visualization going on in my head.

00:03:19.356 --> 00:03:20.848
Negative four comma negative two.

00:03:20.848 --> 00:03:22.980
I'm sitting there in the third quadrant.

00:03:22.980 --> 00:03:26.136
If I'm flipping over the y-axis,

00:03:26.136 --> 00:03:28.302
my y-coordinate wouldn't change,

00:03:28.302 --> 00:03:30.719
but my x-coordinate
would become the opposite

00:03:30.719 --> 00:03:32.316
and I would end up in the fourth quadrant,

00:03:32.316 --> 00:03:33.664
and that's exactly what happened.

00:03:33.664 --> 00:03:36.989
Y-coordinate did not change,
but then my x-coordinate,

00:03:36.989 --> 00:03:38.955
since I'm flipping over the y-axis,

00:03:38.955 --> 00:03:40.523
it became the negative of this,

00:03:40.523 --> 00:03:44.672
so the opposite of negative
four which is positive four.


Determining angle of rotation

WEBVTT
Kind: captions
Language: en

00:00:00.445 --> 00:00:02.311
- [Instructor] We're told that triangle

00:00:02.311 --> 00:00:04.501
A-prime, B-prime, C-prime,

00:00:04.501 --> 00:00:06.598
so that's this red triangle over here,

00:00:06.598 --> 00:00:09.479
is the image of triangle ABC,

00:00:09.479 --> 00:00:11.383
so that's this blue triangle here,

00:00:11.383 --> 00:00:13.837
under rotation about the origin,

00:00:13.837 --> 00:00:16.200
so we're rotating about the origin here.

00:00:16.200 --> 00:00:18.766
Determine the angle of rotation.

00:00:18.766 --> 00:00:20.130
So like always, pause this video,

00:00:20.130 --> 00:00:22.242
see if you can figure it out.

00:00:22.242 --> 00:00:24.098
So I'm just gonna think about

00:00:24.098 --> 00:00:26.536
how did each of these
points have to be rotated

00:00:26.536 --> 00:00:29.175
to go from A to A-prime, or B to B-prime,

00:00:29.175 --> 00:00:31.044
or from C to C-prime?

00:00:31.044 --> 00:00:33.410
So let's just start with A.

00:00:33.410 --> 00:00:34.822
So this is where A starts.

00:00:34.822 --> 00:00:36.525
Remember we're rotating about the origin.

00:00:36.525 --> 00:00:39.045
That's why I'm drawing this
line from the origin to A.

00:00:39.045 --> 00:00:40.530
And where does it get rotated to?

00:00:40.530 --> 00:00:43.452
Well, it gets rotated to right over here.

00:00:43.452 --> 00:00:47.676
So the rotation is going in
the counterclockwise direction,

00:00:47.676 --> 00:00:49.295
so it's going to have a positive angle.

00:00:49.295 --> 00:00:51.761
So we can rule out these
two right over here.

00:00:51.761 --> 00:00:56.100
And the key question is, is
this 30 degrees or 60 degrees?

00:00:56.100 --> 00:00:58.261
And there's a bunch of ways
that you could think about it.

00:00:58.261 --> 00:01:01.316
One, 60 degrees would
be 2/3 of a right angle,

00:01:01.316 --> 00:01:03.828
while 30 degrees would
be 1/3 of a right angle.

00:01:03.828 --> 00:01:06.337
A right angle would look
something like this.

00:01:06.337 --> 00:01:09.039
So this looks much more
like 2/3 of a right angle,

00:01:09.039 --> 00:01:10.924
so I'll go with 60 degrees.

00:01:10.924 --> 00:01:13.483
Another way to think
about is that 60 degrees

00:01:13.483 --> 00:01:16.380
is 1/3 of 180 degrees,

00:01:16.380 --> 00:01:19.401
which this also looks
like right over here.

00:01:19.401 --> 00:01:20.942
And if you do that with any of the points,

00:01:20.942 --> 00:01:22.627
you would see a similar thing.

00:01:22.627 --> 00:01:25.332
So just looking at A to
A-prime makes me feel good

00:01:25.332 --> 00:01:28.338
that this was a 60-degree rotation.

00:01:28.338 --> 00:01:30.421
Let's do another example.

00:01:32.911 --> 00:01:34.590
So we are told quadrilateral

00:01:34.590 --> 00:01:37.506
A-prime, B-prime, C-prime,
D-prime, in red here,

00:01:37.506 --> 00:01:41.394
is the image of quadrilateral
ABCD, in blue here,

00:01:41.394 --> 00:01:44.438
under rotation about point Q.

00:01:44.438 --> 00:01:47.457
Determine the angle of rotation.

00:01:47.457 --> 00:01:48.530
So once again, pause this video,

00:01:48.530 --> 00:01:50.341
and see if you can figure it out.

00:01:50.341 --> 00:01:51.848
Well, I'm gonna tackle this the same way.

00:01:51.848 --> 00:01:53.356
I don't have a coordinate plane here,

00:01:53.356 --> 00:01:54.215
but it's the same notion.

00:01:54.215 --> 00:01:58.110
I can take some initial point
and then look at its image

00:01:58.110 --> 00:02:00.446
and think about, well, how
much did I have to rotate it?

00:02:00.446 --> 00:02:01.896
I could do B to B-prime,

00:02:01.896 --> 00:02:03.534
although this might be
a little bit too close.

00:02:03.534 --> 00:02:06.360
So I'm going from B to,
let me do a new color here,

00:02:06.360 --> 00:02:10.322
just 'cause this color is
too close to, I'll use black,

00:02:10.322 --> 00:02:14.104
so we're going from B to
B-prime right over here.

00:02:14.104 --> 00:02:16.587
We are going clockwise,

00:02:16.587 --> 00:02:18.426
so it's going to be a negative rotation.

00:02:18.426 --> 00:02:20.618
So we can rule that and that out.

00:02:20.618 --> 00:02:23.222
And it looks like a right angle.

00:02:23.222 --> 00:02:25.201
This looks like a right angle,

00:02:25.201 --> 00:02:27.978
so I feel good about
picking negative 90 degrees.

00:02:27.978 --> 00:02:30.122
We could try another
point and feel good that

00:02:30.122 --> 00:02:32.728
that also meets that negative 90 degrees.

00:02:32.728 --> 00:02:34.421
Let's say D to D-prime.

00:02:34.421 --> 00:02:37.187
So this is where D is initially.

00:02:37.187 --> 00:02:38.894
This is where D is,

00:02:38.894 --> 00:02:41.782
and this is where D-prime is.

00:02:41.782 --> 00:02:44.726
And once again, we are moving clockwise,

00:02:44.726 --> 00:02:46.893
so it's a negative rotation.

00:02:46.893 --> 00:02:48.956
And this looks like a right angle,

00:02:48.956 --> 00:02:51.764
definitely more like a right
angle than a 60-degree angle.

00:02:51.764 --> 00:02:54.232
And so this would be negative 90 degrees,

00:02:54.232 --> 00:02:56.468
definitely feel good about that.


Positive and negative rotaion of points example

WEBVTT
Kind: captions
Language: en

00:00:00.485 --> 00:00:03.608
- [Instructor] We're told
that point P was rotated

00:00:03.608 --> 00:00:07.196
about the origin (0,0) by 60 degrees.

00:00:07.196 --> 00:00:10.400
Which point is the image of P?

00:00:10.400 --> 00:00:13.152
Pause this video and see
if you can figure that out.

00:00:13.152 --> 00:00:14.371
All right, now let's think about it.

00:00:14.371 --> 00:00:16.054
This is point P.

00:00:16.054 --> 00:00:18.875
It's being rotated around the origin

00:00:18.875 --> 00:00:21.940
(0,0) by 60 degrees.

00:00:21.940 --> 00:00:25.785
So if originally point
P is right over here

00:00:25.785 --> 00:00:28.250
and we're rotating by positive 60 degrees,

00:00:28.250 --> 00:00:33.237
so that means we go counter
clockwise by 60 degrees.

00:00:33.237 --> 00:00:37.080
So this looks like about
60 degrees right over here.

00:00:37.080 --> 00:00:39.123
One way to think about 60 degrees,

00:00:39.123 --> 00:00:42.676
is that that's 1/3 of 180 degrees.

00:00:42.676 --> 00:00:45.358
So does this look like 1/3 of 180 degrees?

00:00:45.358 --> 00:00:48.388
Remember, 180 degrees would
be almost a full line.

00:00:48.388 --> 00:00:52.881
So that indeed does look
like 1/3 of 180 degrees,

00:00:52.881 --> 00:00:55.714
60 degrees, it gets us to point C.

00:00:56.933 --> 00:00:58.326
And it looks like it's the same distance

00:00:58.326 --> 00:00:59.510
from the origin.

00:00:59.510 --> 00:01:02.447
We have just rotated by 60 degrees.

00:01:02.447 --> 00:01:06.743
Point D looks like it's more
than 60 degree rotation,

00:01:06.743 --> 00:01:09.239
so I won't go with that one.

00:01:09.239 --> 00:01:12.406
All right, let's do one more of these.

00:01:14.185 --> 00:01:18.659
So we're told point P was
rotated by negative 90 degrees.

00:01:18.659 --> 00:01:20.292
The center of rotation is indicated.

00:01:20.292 --> 00:01:22.022
Which point is the image of P?

00:01:22.022 --> 00:01:23.346
So once again, pause this video

00:01:23.346 --> 00:01:24.684
and try to think about it.

00:01:24.684 --> 00:01:26.609
All right, so we have
our center of rotation,

00:01:26.609 --> 00:01:27.943
this is our point P,

00:01:27.943 --> 00:01:30.973
and we're rotating by negative 90 degrees.

00:01:30.973 --> 00:01:33.202
So this means we are going clockwise.

00:01:33.202 --> 00:01:34.688
So we're going in that direction.

00:01:34.688 --> 00:01:36.017
And 90 degrees is easy to spot.

00:01:36.017 --> 00:01:37.558
It's a right angle.

00:01:37.558 --> 00:01:39.599
And so it would look like
that and it looks like

00:01:39.599 --> 00:01:42.432
it is getting us right to point A.

00:01:43.473 --> 00:01:46.798
So this is a negative 90 degree
rotation right over here.

00:01:46.798 --> 00:01:48.465
Gets us to point A.


Message to LearnStormers from Paralympic ski racer Josh Sundquist

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.058
- LearnStormers, my
name is Josh Sundquist.

00:00:02.058 --> 00:00:03.961
I am a YouTuber, best-selling author,

00:00:03.961 --> 00:00:05.988
and a paralympic ski racer.

00:00:05.988 --> 00:00:08.994
I first started ski racing
when I was a teenager.

00:00:08.994 --> 00:00:13.247
I went to my first race
thinking I was the best skier

00:00:13.247 --> 00:00:15.853
of all time, and it was
gonna be amazing, right?

00:00:15.853 --> 00:00:18.500
I was super excited about
it, and I'm sure that's kinda

00:00:18.500 --> 00:00:20.431
how you feel when you start
anything in life, right?

00:00:20.431 --> 00:00:23.264
Maybe even when you started this class.

00:00:23.264 --> 00:00:26.198
I went through the course,
and I actually fell

00:00:26.198 --> 00:00:29.049
on the third turn in the course.

00:00:29.049 --> 00:00:31.238
That was really disappointing.

00:00:31.238 --> 00:00:34.831
I expected I was gonna ski
really well, and I fell.

00:00:34.831 --> 00:00:37.040
I think that is an
inevitable part of anything,

00:00:37.040 --> 00:00:38.203
including this course, right?

00:00:38.203 --> 00:00:40.625
You're really excited at first,
and then something happens,

00:00:40.625 --> 00:00:43.863
a problem, you fall, you get
discouraged, and it's in that

00:00:43.863 --> 00:00:47.372
moment I think you have choice
about do you want to quit.

00:00:47.372 --> 00:00:49.642
Do you just want to be
like, "eh, it's not for me

00:00:49.642 --> 00:00:52.616
because I fell" or do
you want to say, "no,

00:00:52.616 --> 00:00:54.003
falling is just part of the process."

00:00:54.003 --> 00:00:56.423
It's just something that
happens along the way.

00:00:56.423 --> 00:00:59.740
If you wanna cross the finish
line at the end race course,

00:00:59.740 --> 00:01:03.770
as if you wanna finish the
class, you have to get back up.

00:01:03.770 --> 00:01:06.470
I got back up, and I fell down again,

00:01:06.470 --> 00:01:07.491
and got back up and fell down.

00:01:07.491 --> 00:01:10.662
I actually fell down five
times in this race course.

00:01:10.662 --> 00:01:13.377
It was not very good, as it turns out.

00:01:13.377 --> 00:01:16.422
But I kept getting back
up, and for that reason,

00:01:16.422 --> 00:01:18.763
I eventually crossed the finish line.

00:01:18.763 --> 00:01:21.455
What I want to say to you
is that when you fall,

00:01:21.455 --> 00:01:23.830
when something doesn't work
out quite the way that you

00:01:23.830 --> 00:01:25.490
wanted, when you get discouraged,

00:01:25.490 --> 00:01:27.269
and that's not a reason to quit,

00:01:27.269 --> 00:01:30.768
that's just part of the process.

00:01:30.768 --> 00:01:32.444
It's in those moments that
you have to look at the finish

00:01:32.444 --> 00:01:35.022
line that you're trying to reach and say,

00:01:35.022 --> 00:01:36.673
"all right, I'm gonna get back up

00:01:36.673 --> 00:01:39.099
and I'm gonna keep getting
back up as many times

00:01:39.099 --> 00:01:42.766
as it takes until I
cross that finish line."

00:01:44.083 --> 00:01:46.750
So, keep up the good work,
keep getting back up,

00:01:46.750 --> 00:01:48.038
and good luck in the rest of your class.


Example translating points

WEBVTT
Kind: captions
Language: en

00:00:00.581 --> 00:00:02.324
- [Instructor] What we're going
to do in this video is look

00:00:02.324 --> 00:00:06.349
at all of the ways of describing
how to translate a point

00:00:06.349 --> 00:00:09.254
and then to actually translate that point

00:00:09.254 --> 00:00:11.163
on our coordinate plane.

00:00:11.163 --> 00:00:15.172
So, for example, they say
plot the image of point P

00:00:15.172 --> 00:00:17.676
under a translation by

00:00:17.676 --> 00:00:19.509
five units to the left

00:00:20.767 --> 00:00:22.221
and

00:00:22.221 --> 00:00:23.471
three units up.

00:00:24.710 --> 00:00:25.948
So let's just do that at first,

00:00:25.948 --> 00:00:27.743
and then we're gonna
think about other ways

00:00:27.743 --> 00:00:29.408
of describing this.

00:00:29.408 --> 00:00:32.340
So we want to go five units to the left.

00:00:32.340 --> 00:00:33.536
So we start right over here.

00:00:33.536 --> 00:00:36.786
We're gonna go one, two, three,

00:00:36.786 --> 00:00:39.652
four, five units to the left,

00:00:39.652 --> 00:00:42.267
and then we're gonna go three units up.

00:00:42.267 --> 00:00:46.253
So that's going to be one, two, three.

00:00:46.253 --> 00:00:48.860
And so the image of point P,

00:00:48.860 --> 00:00:50.352
I guess, would show up right over here,

00:00:50.352 --> 00:00:53.740
after this translation described this way.

00:00:53.740 --> 00:00:54.823
Now, there are other ways

00:00:54.823 --> 00:00:56.619
that you could describe this translation.

00:00:56.619 --> 00:00:58.515
Here, we described it
just in plain English,

00:00:58.515 --> 00:01:00.991
by five units to the
left and three units up.

00:01:00.991 --> 00:01:02.926
But you could, and this will look fancy,

00:01:02.926 --> 00:01:05.646
but, as we'll see, it's
hopefully a pretty intuitive way

00:01:05.646 --> 00:01:06.961
to describe a translation.

00:01:06.961 --> 00:01:09.357
You could say, look, I'm
gonna take some point

00:01:09.357 --> 00:01:11.715
with the coordinates x comma y.

00:01:11.715 --> 00:01:14.224
And the x coordinate tells
me what's my coordinate

00:01:14.224 --> 00:01:17.069
in the horizontal direction
to the left or the right.

00:01:17.069 --> 00:01:19.955
And so I want that to be five less.

00:01:19.955 --> 00:01:22.455
So I would say x minus five

00:01:24.234 --> 00:01:25.067
comma

00:01:26.551 --> 00:01:27.541
y.

00:01:27.541 --> 00:01:28.722
And what do we do to the y coordinate?

00:01:28.722 --> 00:01:30.433
Well, we're going to increase it by three.

00:01:30.433 --> 00:01:33.319
We're going to translate three units up,

00:01:33.319 --> 00:01:34.430
so y

00:01:34.430 --> 00:01:35.500
plus three.

00:01:35.500 --> 00:01:37.072
So all this is saying is whatever x

00:01:37.072 --> 00:01:39.157
and y coordinates you have,

00:01:39.157 --> 00:01:42.835
this translation will make
you take five from the x.

00:01:42.835 --> 00:01:45.529
That's what, meaning this is,

00:01:45.529 --> 00:01:48.446
this right over here, is five units

00:01:50.100 --> 00:01:51.100
to the left.

00:01:52.464 --> 00:01:54.909
And then this right over here,

00:01:54.909 --> 00:01:56.159
is saying three

00:01:57.280 --> 00:01:58.113
units up.

00:01:59.902 --> 00:02:01.400
Increase your y coordinate by three.

00:02:01.400 --> 00:02:03.745
Decrease your x coordinate by five.

00:02:03.745 --> 00:02:05.455
And so let's just test this out

00:02:05.455 --> 00:02:08.282
with this particular coordinate,
with this particular point.

00:02:08.282 --> 00:02:09.683
So at this point right over here,

00:02:09.683 --> 00:02:13.111
P has the coordinates,
its x coordinate is three,

00:02:13.111 --> 00:02:16.061
and its y coordinate is negative four.

00:02:16.061 --> 00:02:17.271
So let's see how that works.

00:02:17.271 --> 00:02:19.740
If I have three comma negative four,

00:02:19.740 --> 00:02:22.136
and I want to apply this translation,

00:02:22.136 --> 00:02:23.516
what happens?

00:02:23.516 --> 00:02:26.713
Well, let me just do my coordinates.

00:02:26.713 --> 00:02:30.173
And so I started off with
three and negative four,

00:02:30.173 --> 00:02:34.199
and I'm going to subtract
five from the three.

00:02:34.199 --> 00:02:38.063
So subtract five here, we
see that right over there,

00:02:38.063 --> 00:02:41.604
and we're going to add three to the y.

00:02:41.604 --> 00:02:43.902
So notice, well, instead of
an x, now I have a three.

00:02:43.902 --> 00:02:45.204
Instead of an x, now I have a three.

00:02:45.204 --> 00:02:47.304
Instead of a y, now I
have a negative four.

00:02:47.304 --> 00:02:49.368
Instead of a y, now I
have a negative four.

00:02:49.368 --> 00:02:51.675
And so another way of writing this,

00:02:51.675 --> 00:02:54.844
we're going from three comma negative four

00:02:54.844 --> 00:02:58.212
to three minus five is negative two,

00:02:58.212 --> 00:03:01.889
and negative four plus
three is negative one.

00:03:01.889 --> 00:03:03.817
So what are the coordinates
right over here?

00:03:03.817 --> 00:03:06.009
Well, the coordinate
of this point is indeed

00:03:06.009 --> 00:03:09.060
negative two comma negative one.

00:03:09.060 --> 00:03:10.643
So notice how this,

00:03:11.631 --> 00:03:13.327
I guess you could say this formula,

00:03:13.327 --> 00:03:17.473
the algebraic formula that shows
how we map our coordinates,

00:03:17.473 --> 00:03:19.515
how it's able to draw the connection

00:03:19.515 --> 00:03:21.284
between the coordinates.

00:03:21.284 --> 00:03:23.376
And so you'll see questions
where they'll tell you,

00:03:23.376 --> 00:03:25.339
hey, plot the image, and
they'll describe it like this.

00:03:25.339 --> 00:03:27.214
Translate x units to the left or the right

00:03:27.214 --> 00:03:28.658
or three units up or down.

00:03:28.658 --> 00:03:30.410
You'll sometimes see it like this,

00:03:30.410 --> 00:03:31.921
but just recognize this is just saying

00:03:31.921 --> 00:03:34.199
just take your x and
subtract five from it,

00:03:34.199 --> 00:03:36.234
which means move five to the left.

00:03:36.234 --> 00:03:37.801
And this just means take your y coordinate

00:03:37.801 --> 00:03:40.932
and add three to it,
which means move three up.

00:03:40.932 --> 00:03:42.054
And sometimes they'll ask you,

00:03:42.054 --> 00:03:43.507
hey, what's the new coordinate?

00:03:43.507 --> 00:03:46.003
Or sometimes they'll ask you
to plot something like that,

00:03:46.003 --> 00:03:50.129
but just realize that it's
all the same underlying idea.