Possible transformations example

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.670
Transformation C maps negative
2, 3 to 4, negative 1.

00:00:04.670 --> 00:00:08.875
So let me do negative
2 comma 3, and it

00:00:08.875 --> 00:00:11.555
maps that to 4, negative 1.

00:00:15.920 --> 00:00:28.390
And point negative 5 comma 5,
it maps that to 7, negative 3.

00:00:35.700 --> 00:00:37.900
And so let's think
about this a little bit.

00:00:37.900 --> 00:00:40.830
How could we get from
this point to this point,

00:00:40.830 --> 00:00:42.940
and that point to that point?

00:00:42.940 --> 00:00:45.830
Now it's tempting to view this
that maybe a translation is

00:00:45.830 --> 00:00:46.770
possible.

00:00:46.770 --> 00:00:52.590
Because if you imagined a
line like that, you could say,

00:00:52.590 --> 00:00:54.820
hey, let's just shift this
whole thing down and then

00:00:54.820 --> 00:00:55.620
to the right.

00:00:55.620 --> 00:00:58.980
These two things happen
to have the same slope.

00:00:58.980 --> 00:01:02.820
They both have a
slope of negative 2/3,

00:01:02.820 --> 00:01:04.680
and so this point would
map to this point,

00:01:04.680 --> 00:01:06.390
and that point would
map to that point.

00:01:06.390 --> 00:01:07.860
But that's not what we want.

00:01:07.860 --> 00:01:12.090
We don't want negative 2,
3 to map to 7, negative 3.

00:01:12.090 --> 00:01:15.330
We want negative 2, 3
to map to 4, negative 1.

00:01:15.330 --> 00:01:18.290
So you could get this
line over this line,

00:01:18.290 --> 00:01:21.330
but we won't map the
points that we want to map.

00:01:21.330 --> 00:01:25.600
So this can't be, at least
I can't think of a way,

00:01:25.600 --> 00:01:30.170
that this could actually
be a translation.

00:01:30.170 --> 00:01:33.800
Now let's think about
whether our transformation

00:01:33.800 --> 00:01:35.730
could be a reflection.

00:01:35.730 --> 00:01:39.610
Well, if we imagine a
line that has-- let's see,

00:01:39.610 --> 00:01:42.390
these both have a
slope of negative 3.

00:01:42.390 --> 00:01:45.770
These both have a
slope of negative 2/3.

00:01:45.770 --> 00:01:52.300
So if you imagined a line that
had a slope of positive 3/2

00:01:52.300 --> 00:01:55.115
that was equidistant from both--
and I don't know if this is.

00:01:55.115 --> 00:01:56.500
Let's see, is this equidistant?

00:01:56.500 --> 00:01:58.510
Is this equidistant
from both of them?

00:01:58.510 --> 00:02:02.980
It's either going to be that
line or this line right over--

00:02:02.980 --> 00:02:05.690
or that line, actually
that line looks better.

00:02:05.690 --> 00:02:06.720
So that one.

00:02:06.720 --> 00:02:10.720
And once again, I'm
just eyeballing it.

00:02:10.720 --> 00:02:14.660
So a line that has
slope of positive 3/2.

00:02:14.660 --> 00:02:17.685
So this one looks right
in between the two.

00:02:17.685 --> 00:02:19.560
Or actually it could be
someplace in between.

00:02:19.560 --> 00:02:22.101
But either way, we just have to
think about it qualitatively.

00:02:22.101 --> 00:02:24.450
If you had a line that
looked something like that,

00:02:24.450 --> 00:02:29.520
and if you were to
reflect over this line,

00:02:29.520 --> 00:02:32.430
then this point would
map to this point,

00:02:32.430 --> 00:02:33.740
which is what we want.

00:02:33.740 --> 00:02:37.120
And this purple point,
negative 5 comma 5,

00:02:37.120 --> 00:02:38.440
would map to that point.

00:02:38.440 --> 00:02:39.970
It would be reflected over.

00:02:39.970 --> 00:02:43.740
So it's pretty clear that
this could be a reflection.

00:02:43.740 --> 00:02:45.574
Now rotation actually
makes even more sense,

00:02:45.574 --> 00:02:47.656
or at least in my brain
makes a little more sense.

00:02:47.656 --> 00:02:50.190
If you were to rotate around
to this point right over here,

00:02:50.190 --> 00:02:52.260
this point would
map to that point,

00:02:52.260 --> 00:02:54.810
and that point would
map to that point.

00:02:54.810 --> 00:02:57.820
So a rotation also
seems like a possibility

00:02:57.820 --> 00:03:00.210
for transformation C.

00:03:00.210 --> 00:03:01.710
Now let's think
about transformation

00:03:01.710 --> 00:03:14.910
D. We are going from 4,
negative 1 to 7, negative 3.

00:03:17.900 --> 00:03:20.150
Actually maybe I'll put
that in magenta, as well.

00:03:20.150 --> 00:03:23.800
To 7, negative 3,
just like that.

00:03:23.800 --> 00:03:35.845
And we want to go from
negative 5, 5 to negative 2, 3.

00:03:38.810 --> 00:03:43.270
So I could definitely imagine
a translation right over here.

00:03:43.270 --> 00:03:46.870
This point went 3 to
the right and 2 down.

00:03:46.870 --> 00:03:51.050
This point went 3 to
the right and 2 down.

00:03:51.050 --> 00:03:54.720
So a translation
definitely makes sense.

00:03:54.720 --> 00:03:56.990
Now let's think
about a reflection.

00:03:56.990 --> 00:04:01.380
So it would be
tempting to-- let's

00:04:01.380 --> 00:04:04.170
see, if I were to get from
this point to this point,

00:04:04.170 --> 00:04:07.651
I could reflect around that, but
that won't help this one over

00:04:07.651 --> 00:04:08.150
here.

00:04:08.150 --> 00:04:09.858
And to get from that
point to that point,

00:04:09.858 --> 00:04:12.420
I could reflect around
that, but once again, that's

00:04:12.420 --> 00:04:14.086
not going to help
that point over there.

00:04:14.086 --> 00:04:16.990
So a reflection really
doesn't seem to do the trick.

00:04:16.990 --> 00:04:19.082
And what about a rotation?

00:04:19.082 --> 00:04:20.790
Well to go from this
point to this point,

00:04:20.790 --> 00:04:23.190
we could rotate
around this point.

00:04:23.190 --> 00:04:27.351
We could go there, but that
won't help this point right

00:04:27.351 --> 00:04:27.850
over here.

00:04:27.850 --> 00:04:29.770
While this is rotating
there, this point

00:04:29.770 --> 00:04:31.540
is going to rotate
around like that

00:04:31.540 --> 00:04:33.800
and it's going to end
up someplace out here.

00:04:33.800 --> 00:04:35.360
So that's not going to help.

00:04:35.360 --> 00:04:39.630
So it looks like this one
can only be a translation.


Rotating segment about origin example

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.980
We're asked to rotate the line
segment negative 90 degrees

00:00:03.980 --> 00:00:05.160
around the origin.

00:00:05.160 --> 00:00:06.440
So this is our line segment.

00:00:06.440 --> 00:00:08.080
We want to rotate it
around the origin.

00:00:08.080 --> 00:00:10.160
So let's use our Rotate tool.

00:00:10.160 --> 00:00:12.790
So let's define our
point of rotation.

00:00:12.790 --> 00:00:14.540
So we're going to
rotate around the origin

00:00:14.540 --> 00:00:16.140
so that's why I put it there.

00:00:16.140 --> 00:00:18.847
And I'm going to rotate
it negative 90 degrees.

00:00:18.847 --> 00:00:20.680
So I'm going to rotate
it in this direction.

00:00:20.680 --> 00:00:22.240
So let's see if I
can pull this off.

00:00:25.030 --> 00:00:27.330
So actually, let me
undo what I just did.

00:00:27.330 --> 00:00:29.750
Let me get my Rotate
tool back here.

00:00:29.750 --> 00:00:30.780
So negative 90 degrees.

00:00:30.780 --> 00:00:32.530
So these arrows right
over here, right now

00:00:32.530 --> 00:00:34.410
they're straddling the x-axis.

00:00:34.410 --> 00:00:37.700
If I rotate negative 90
degrees, these arrows,

00:00:37.700 --> 00:00:41.140
this double-headed, curvy
arrow should exactly

00:00:41.140 --> 00:00:43.430
straddle the negative y-axis.

00:00:43.430 --> 00:00:45.350
So let's make sure I do that.

00:00:45.350 --> 00:00:48.150
So this is the negative y-axis.

00:00:48.150 --> 00:00:49.250
There we go.

00:00:49.250 --> 00:00:52.390
So I have rotated this line
segment negative 90 degrees

00:00:52.390 --> 00:00:53.509
about the origin.

00:00:53.509 --> 00:00:55.300
And now we've got to
answer some questions.

00:00:55.300 --> 00:00:58.170
When a line segment is rotated,
the image created is a, well,

00:00:58.170 --> 00:00:59.600
it's, of course, a line segment.

00:00:59.600 --> 00:01:02.640
We know this is true because
the rotated image has two end

00:01:02.640 --> 00:01:03.390
points.

00:01:03.390 --> 00:01:09.387
The rotated image is the same
length as the original image.

00:01:09.387 --> 00:01:10.970
We haven't changed
the length of this,

00:01:10.970 --> 00:01:14.930
we just rotated it
around this dot.


Determining the line of reflection

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.020
A reflection acting on
the coordinate plane

00:00:03.020 --> 00:00:08.000
takes point G to
point H and point

00:00:08.000 --> 00:00:15.100
S to point T. What is
the line of reflection?

00:00:15.100 --> 00:00:16.970
So the line of
reflection is going

00:00:16.970 --> 00:00:21.040
to look something like that.

00:00:21.040 --> 00:00:25.150
And it's going to go through
the midpoints of H and G,

00:00:25.150 --> 00:00:27.250
which is going to be
someplace over there,

00:00:27.250 --> 00:00:28.960
and the midpoint
of S and T, which

00:00:28.960 --> 00:00:31.860
is going to be some
place over there.

00:00:31.860 --> 00:00:33.572
And I'm drawing it
very imprecisely.

00:00:33.572 --> 00:00:35.030
And the way we
think about it, when

00:00:35.030 --> 00:00:38.490
we take the reflection of
G over this line you would

00:00:38.490 --> 00:00:41.300
drop a perpendicular and
then go the same distance

00:00:41.300 --> 00:00:44.190
on the other side to get
the point H. Similarly--

00:00:44.190 --> 00:00:46.720
I can never say
that word-- if we

00:00:46.720 --> 00:00:49.220
were to take the reflection
of S around that line

00:00:49.220 --> 00:00:52.010
we would drop a perpendicular
to that midpoint

00:00:52.010 --> 00:00:54.970
and then keep going
the same distance to T.

00:00:54.970 --> 00:00:57.180
So let's think about
what these midpoints are,

00:00:57.180 --> 00:00:59.770
and actually I'll clear this
right now just so that I can

00:00:59.770 --> 00:01:01.930
draw it a little
bit more precisely.

00:01:01.930 --> 00:01:03.970
So let's clear that.

00:01:03.970 --> 00:01:11.030
And so H is at the
point 2, negative 2.

00:01:11.030 --> 00:01:14.320
G is at the point negative
2 comma negative 6.

00:01:14.320 --> 00:01:16.060
So what's the midpoint?

00:01:16.060 --> 00:01:17.844
Well, the midpoint
is going to be

00:01:17.844 --> 00:01:19.260
the midpoint of
the x-coordinates.

00:01:19.260 --> 00:01:23.570
2 plus negative 2
is 0, over 2 is 0.

00:01:23.570 --> 00:01:25.360
So the x-coordinate
is going to be 0.

00:01:25.360 --> 00:01:27.960
The y-coordinate, the average
of negative 2 and negative 6

00:01:27.960 --> 00:01:29.200
is negative 4.

00:01:29.200 --> 00:01:31.390
So our midpoint is going
to be right over here.

00:01:31.390 --> 00:01:34.630
It's the point 0, negative 4.

00:01:34.630 --> 00:01:37.080
Now let's think about the
midpoint of these two points.

00:01:37.080 --> 00:01:42.600
S is at the point 1.5 comma 5.5.

00:01:42.600 --> 00:01:49.480
And T is at the point negative
9.5 comma negative 5.5.

00:01:49.480 --> 00:01:57.180
So the midpoint is going to be
1.5 plus negative 9.5 over 2.

00:01:57.180 --> 00:01:58.180
That's the x-coordinate.

00:01:58.180 --> 00:02:00.810
I just took the average of
these two x-coordinates.

00:02:00.810 --> 00:02:07.850
And the y-coordinate is going
to be 5.5 plus negative 5.5

00:02:07.850 --> 00:02:10.720
over 2.

00:02:10.720 --> 00:02:11.645
So what is this?

00:02:11.645 --> 00:02:17.130
This is 1.5 plus negative
9.5 is negative 8 over 2.

00:02:17.130 --> 00:02:21.290
So that is negative 4 comma 0.

00:02:21.290 --> 00:02:25.300
So their midpoint is at the
point negative 4 comma 0.

00:02:25.300 --> 00:02:28.285
So that point right over there.

00:02:28.285 --> 00:02:29.300
Let me just write this.

00:02:29.300 --> 00:02:31.760
This is negative 4 comma 0.

00:02:31.760 --> 00:02:33.330
So the line of
reflection is going

00:02:33.330 --> 00:02:36.145
to be the line that goes
through both of these points.

00:02:41.580 --> 00:02:44.200
And we can write it in the
traditional slope intercept

00:02:44.200 --> 00:02:48.810
form, y is equal to mx plus b.

00:02:48.810 --> 00:02:50.600
So let's figure out its slope.

00:02:50.600 --> 00:02:52.900
Let's figure out the slope
between these two points,

00:02:52.900 --> 00:02:58.130
to go from negative 4
comma 0 to 0, negative 4.

00:02:58.130 --> 00:03:05.770
Well, to go between those
two points, our change in y--

00:03:05.770 --> 00:03:08.210
Actually, let me do this
in a different color

00:03:08.210 --> 00:03:10.310
just to ease the monotony.

00:03:10.310 --> 00:03:16.180
So our change in y over
here is negative 4.

00:03:16.180 --> 00:03:18.970
As we go from this point
to that point we go 4 down,

00:03:18.970 --> 00:03:25.510
and our change in x is
equal to positive 4.

00:03:25.510 --> 00:03:30.520
So m, which is our change
in y over change in x,

00:03:30.520 --> 00:03:35.270
we see that when we
increased our x by 4,

00:03:35.270 --> 00:03:42.600
our y went down by 4, so
the slope is negative 1.

00:03:42.600 --> 00:03:44.780
And now let's think about
what the y-intercept is.

00:03:44.780 --> 00:03:45.863
Well we already know that.

00:03:45.863 --> 00:03:49.850
When x is equal to
0, y is equal to b.

00:03:49.850 --> 00:03:51.850
But we see over here
that this point is there.

00:03:51.850 --> 00:03:54.150
So when x is equal to
0, y is negative 4.

00:03:54.150 --> 00:03:56.530
So this is negative 4.

00:03:56.530 --> 00:03:59.230
So the equation of
our line of reflection

00:03:59.230 --> 00:04:02.430
is y is equal to negative
1x, or I could just

00:04:02.430 --> 00:04:05.690
write negative x minus 4.

00:04:05.690 --> 00:04:09.310
That's our line of reflection.

00:04:09.310 --> 00:04:11.390
Now they ask us what
are the coordinates

00:04:11.390 --> 00:04:14.843
of the image of point
negative 2 comma negative 2

00:04:14.843 --> 00:04:16.720
under this reflection?

00:04:16.720 --> 00:04:22.530
So the point negative
2 comma negative 2,

00:04:22.530 --> 00:04:24.380
that's this point
right over here.

00:04:24.380 --> 00:04:26.860
That sits on the
line of reflection.

00:04:26.860 --> 00:04:30.030
When x is equal to negative
2, negative negative 2

00:04:30.030 --> 00:04:33.680
is positive 2, minus
4 is negative 2.

00:04:33.680 --> 00:04:36.530
This sits on this
line of reflection.

00:04:36.530 --> 00:04:38.590
So the coordinates of
the image of this point

00:04:38.590 --> 00:04:40.090
under the reflection,
well it's just

00:04:40.090 --> 00:04:42.250
going to map back to itself.

00:04:42.250 --> 00:04:46.480
It's going to be negative
2 comma negative 2.

00:04:46.480 --> 00:04:50.570
Anything that's on the line of
reflection, when you reflect it

00:04:50.570 --> 00:04:53.160
across that line, since
it sits on that line,

00:04:53.160 --> 00:04:56.598
it's just going to
map back to itself.


Drawing line of reflection

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.750
- [Voiceover] We are asked to
draw the line of reflection

00:00:02.750 --> 00:00:06.070
that will map quadrilateral SORT,

00:00:06.070 --> 00:00:08.950
S-O-R-T

00:00:08.950 --> 00:00:11.760
onto the other quadrilateral below.

00:00:11.760 --> 00:00:14.490
So this is the other
quadrilateral right over here.

00:00:14.490 --> 00:00:17.190
So we wanna reflect SORT

00:00:17.190 --> 00:00:20.130
so it maps onto this other quadrilateral,

00:00:20.130 --> 00:00:22.580
and we wanna draw or define

00:00:22.580 --> 00:00:26.200
the line of reflection using
this tool right over here.

00:00:26.200 --> 00:00:28.720
And so see if I just kind of eyeball it,

00:00:28.720 --> 00:00:32.480
it looks like, it looks like something,

00:00:32.480 --> 00:00:37.030
something like, that is
starting to get close

00:00:37.030 --> 00:00:39.310
but not quite.

00:00:39.310 --> 00:00:40.860
And the way I could tackle this,

00:00:40.860 --> 00:00:42.730
I could look at corresponding points.

00:00:42.730 --> 00:00:45.500
I could say look, point S right over here

00:00:45.500 --> 00:00:48.030
in the image after the reflection

00:00:48.030 --> 00:00:50.360
the corresponding point
is that right over there.

00:00:50.360 --> 00:00:53.087
And so the line of reflection
should contain a point

00:00:53.087 --> 00:00:56.810
that is half way between these two points.

00:00:56.810 --> 00:00:59.000
And so what's a point
that's halfway between them?

00:00:59.000 --> 00:01:01.760
Well, let's see, where I have
it right now isn't halfway.

00:01:01.760 --> 00:01:04.000
If I put it there, that
actually looks halfway

00:01:04.000 --> 00:01:07.750
between this point and the
corresponding point in the image.

00:01:07.750 --> 00:01:09.960
Notice, to get from here to there,

00:01:09.960 --> 00:01:12.220
I took one to the left,

00:01:12.220 --> 00:01:14.530
I had to take one step to
the left, three steps up

00:01:14.530 --> 00:01:15.730
to go from here to here,

00:01:15.730 --> 00:01:18.460
I take one step to the
left, three steps up.

00:01:18.460 --> 00:01:20.730
So these are definitely the same distance

00:01:20.730 --> 00:01:23.830
away from this point and this point,

00:01:23.830 --> 00:01:27.760
are both the same distance
away from this line

00:01:28.520 --> 00:01:30.776
or this point that's on this line.

00:01:30.776 --> 00:01:32.930
So now let's see where
this point could go,

00:01:32.930 --> 00:01:35.490
and I could do something similar.

00:01:35.490 --> 00:01:37.610
I could try to find,

00:01:37.610 --> 00:01:39.750
let's see, I need to
find a different point.

00:01:39.750 --> 00:01:42.390
Let's see, maybe a point that's halfway

00:01:42.390 --> 00:01:44.978
between this point and T

00:01:44.978 --> 00:01:46.450
and the corresponding point of the image

00:01:46.450 --> 00:01:48.060
which is that over there.

00:01:48.060 --> 00:01:50.460
And let's see, I can probably put it,

00:01:50.460 --> 00:01:52.550
that actually looks pretty good

00:01:52.550 --> 00:01:55.590
because this is one step to the left,

00:01:55.590 --> 00:01:59.184
three steps up, and then
to get from here to there,

00:01:59.184 --> 00:02:01.580
one step to the left and three steps up.

00:02:01.580 --> 00:02:03.090
So this looks pretty good.

00:02:03.090 --> 00:02:05.620
This looks like I have drawn

00:02:05.620 --> 00:02:07.090
the line of reflection.

00:02:07.090 --> 00:02:09.100
This point that I'm
highlighting right now in orange

00:02:09.100 --> 00:02:12.060
is halfway between T and
the corresponding point

00:02:12.060 --> 00:02:13.200
to the image.

00:02:13.200 --> 00:02:14.760
This point right over here is halfway

00:02:14.760 --> 00:02:18.560
between S and the corresponding
point of the image.

00:02:18.560 --> 00:02:19.820
You need to just to have two points

00:02:19.820 --> 00:02:21.550
to define our line, this looks like

00:02:21.550 --> 00:02:23.230
my line of reflection.

00:02:23.230 --> 00:02:25.590
Let's see whether I'd got it right.

00:02:25.590 --> 00:02:26.750
And I did.


Reflecting segments over line

WEBVTT
Kind: captions
Language: en

00:00:00.468 --> 00:00:04.160
- Line segments IN, this
is segment IN over here,

00:00:04.160 --> 00:00:06.575
and TO, this is TO here,

00:00:06.575 --> 00:00:07.991
are reflected over the line

00:00:07.991 --> 00:00:10.661
Y is equal to negative X minus two.

00:00:10.661 --> 00:00:12.589
So this is the line that
they're reflected about

00:00:12.589 --> 00:00:14.214
this dashed, purple line.

00:00:14.214 --> 00:00:17.070
And it is indeed Y equals
negative X minus two.

00:00:17.070 --> 00:00:18.927
This right over here is
in slope intercept form.

00:00:18.927 --> 00:00:20.646
The slope should be negative one,

00:00:20.646 --> 00:00:22.248
and we see that the
slope of this purple line

00:00:22.248 --> 00:00:23.850
is indeed negative one.

00:00:23.850 --> 00:00:25.313
If X changes by a certain amount,

00:00:25.313 --> 00:00:26.984
Y changes by the negative of that.

00:00:26.984 --> 00:00:30.142
If X changes by one, Y
changes by negative one

00:00:30.142 --> 00:00:31.721
to get back to that line.

00:00:31.721 --> 00:00:33.021
If X changes by positive two,

00:00:33.021 --> 00:00:34.832
Y changes by negative two

00:00:34.832 --> 00:00:37.154
to get back to another point on that line.

00:00:37.154 --> 00:00:38.756
And the Y intercept?

00:00:38.756 --> 00:00:40.591
We see when X is equal to zero,

00:00:40.591 --> 00:00:41.891
Y should be negative two.

00:00:41.891 --> 00:00:43.331
When X is equal to zero,

00:00:43.331 --> 00:00:44.817
Y is indeed negative two.

00:00:44.817 --> 00:00:45.954
So we've validated that.

00:00:45.954 --> 00:00:48.160
Then they say, "Draw the
image of this reflection,

00:00:48.160 --> 00:00:50.203
using the interactive graph."

00:00:50.203 --> 00:00:53.059
Alright, so we can move
these lines around,

00:00:53.059 --> 00:00:54.731
and we want to reflect these.

00:00:54.731 --> 00:00:56.380
And I could try to eyeball it, you know,

00:00:56.380 --> 00:00:58.446
maybe it's something like this.

00:00:58.446 --> 00:01:00.327
I don't know, it doesn't exactly right.

00:01:00.327 --> 00:01:03.113
That looks close to the reflection of IN,

00:01:03.113 --> 00:01:06.689
and for TO I'd want to
move this down here.

00:01:06.689 --> 00:01:08.709
TO looks like it would be,

00:01:08.709 --> 00:01:10.125
I don't know, I'm eyeballing it.

00:01:10.125 --> 00:01:13.260
This is close, but it can't be close.

00:01:13.260 --> 00:01:14.653
I want to get exact.

00:01:14.653 --> 00:01:17.300
So let's, I've copied and
pasted the original problem

00:01:17.300 --> 00:01:19.547
on my scratch pad so we
can find the exact points

00:01:19.547 --> 00:01:22.171
and so I don't just have to estimate this.

00:01:22.171 --> 00:01:23.908
So let's go to the scratch pad.

00:01:23.908 --> 00:01:25.626
So exactly what we just saw.

00:01:25.626 --> 00:01:27.368
And the main realization is,

00:01:27.368 --> 00:01:29.411
is that if we want to
reflect a given point,

00:01:29.411 --> 00:01:31.245
if we want to reflect a given point,

00:01:31.245 --> 00:01:32.940
say point I right over here,

00:01:32.940 --> 00:01:33.985
what we want to do

00:01:33.985 --> 00:01:36.353
is we want to drop a perpendicular.

00:01:36.353 --> 00:01:38.234
We want to find a line
that's perpendicular,

00:01:38.234 --> 00:01:40.765
or a line that has the point I on it,

00:01:40.765 --> 00:01:43.505
and it's perpendicular to
this line right over here.

00:01:43.505 --> 00:01:45.270
Remember this is the line,

00:01:45.270 --> 00:01:47.522
let me do this is that purple color.

00:01:47.522 --> 00:01:51.910
This is the line, Y is equal
to negative X minus two.

00:01:51.910 --> 00:01:56.673
Its slope is equal to negative one.

00:01:56.673 --> 00:01:59.718
So I want a line that
goes through I, point I,

00:01:59.718 --> 00:02:04.222
that is perpendicular to this line,

00:02:04.222 --> 00:02:05.778
and I want to drop it,

00:02:05.778 --> 00:02:07.659
I want to drop it to,

00:02:07.659 --> 00:02:10.329
I want to drop it to the line
that I'm going to reflect on,

00:02:10.329 --> 00:02:11.792
and then I'm going to go the same distance

00:02:11.792 --> 00:02:13.510
onto the other side to find

00:02:13.510 --> 00:02:16.849
to find the corresponding
point in the image.

00:02:16.849 --> 00:02:18.329
So how do I do that?

00:02:18.329 --> 00:02:20.628
Well if this line, if this purple line

00:02:20.628 --> 00:02:21.998
has a slope of negative one,

00:02:21.998 --> 00:02:26.192
a line that is perpendicular to it

00:02:26.192 --> 00:02:27.940
a line that is perpendicular to it

00:02:27.940 --> 00:02:30.192
so this thing that I'm drawing
in purple right over here,

00:02:30.192 --> 00:02:33.164
its slope is going to be the
negative reciprocal of this.

00:02:33.164 --> 00:02:34.975
So the reciprocal of negative one

00:02:34.975 --> 00:02:36.206
is still just negative one.

00:02:36.206 --> 00:02:37.669
One over negative one
is still negative one.

00:02:37.669 --> 00:02:39.317
But we want the negative of that.

00:02:39.317 --> 00:02:41.035
So the slope here needs to be one.

00:02:41.035 --> 00:02:42.963
And luckily, that's how I drew it.

00:02:42.963 --> 00:02:45.586
The slope here needs to be equal to one

00:02:45.586 --> 00:02:47.931
which is however much I
change in the X direction

00:02:47.931 --> 00:02:48.999
I change in the Y direction.

00:02:48.999 --> 00:02:49.766
We see that.

00:02:49.766 --> 00:02:52.088
To go from this point to
this point right over here,

00:02:52.088 --> 00:02:55.324
we decrease Y by four

00:02:55.324 --> 00:02:58.384
and we decrease X by four.

00:02:58.384 --> 00:03:00.869
Now, if we want to stay on this line

00:03:00.869 --> 00:03:03.492
to find the reflection,
we just do the same thing.

00:03:03.492 --> 00:03:06.789
We could decrease X by four,

00:03:06.789 --> 00:03:09.042
so we'll go from negative
two to negative six,

00:03:09.042 --> 00:03:10.806
and decrease Y by four,

00:03:10.806 --> 00:03:13.709
and we end up at this
point right over here.

00:03:13.709 --> 00:03:15.241
So we end up at the point,

00:03:15.241 --> 00:03:18.561
this is X equals negative six,
Y is equal to negative four.

00:03:18.561 --> 00:03:21.185
So this is, this point corresponds

00:03:21.185 --> 00:03:23.368
to this point right over there.

00:03:23.368 --> 00:03:24.807
Now, let's do the same thing.

00:03:24.807 --> 00:03:27.570
Let's do the same thing for point N.

00:03:27.570 --> 00:03:29.916
For point N, we already know,

00:03:29.916 --> 00:03:32.237
if we drop a perpendicular,

00:03:32.237 --> 00:03:34.908
and this is perpendicular, it's
going to have a slope of one

00:03:34.908 --> 00:03:36.974
because this purple line
has a slope of negative one.

00:03:36.974 --> 00:03:40.109
The negative reciprocal of
negative one is positive one.

00:03:40.109 --> 00:03:42.407
And let's see, to go from this point

00:03:42.407 --> 00:03:44.497
to this point of intersection,

00:03:44.497 --> 00:03:47.725
we have to go down one and a half.

00:03:47.725 --> 00:03:49.954
We're going down one and a half,

00:03:49.954 --> 00:03:52.438
and we're going to the
left one and a half.

00:03:52.438 --> 00:03:54.040
So we want to do that on the other side.

00:03:54.040 --> 00:03:55.642
We want to stay on this
perpendicular line.

00:03:55.642 --> 00:03:57.918
So we want to go left one and a half,

00:03:57.918 --> 00:04:00.031
and down one and a half.

00:04:00.031 --> 00:04:03.026
And we get to this point right
over here, which is the point

00:04:03.026 --> 00:04:07.554
X equals three, Y is
equal to negative eight.

00:04:07.554 --> 00:04:09.550
So we are now equidistant.

00:04:09.550 --> 00:04:11.337
We're on this perpendicular line, still,

00:04:11.337 --> 00:04:14.402
but we're equidistant on the other side.

00:04:14.402 --> 00:04:17.235
So the image of IN is going to go through

00:04:17.235 --> 00:04:20.091
negative six, negative four,
and three, negative eight.

00:04:20.091 --> 00:04:22.339
So let me draw that.

00:04:22.339 --> 00:04:23.077
Let me see if I can remember,

00:04:23.077 --> 00:04:26.049
negative six, negative four,
and three, negative eight.

00:04:26.049 --> 00:04:27.883
So I have a bad memory.

00:04:27.883 --> 00:04:32.778
So negative six, negative four,
and three, negative eight.

00:04:32.778 --> 00:04:33.856
And I was close when I estimated,

00:04:33.856 --> 00:04:35.564
but I wasn't exactly right.

00:04:35.564 --> 00:04:38.570
So that's looking pretty good.

00:04:38.570 --> 00:04:40.427
And actually, we can
do the exact same thing

00:04:40.427 --> 00:04:42.587
with points T and point O.

00:04:42.587 --> 00:04:44.119
Let me do that.

00:04:44.119 --> 00:04:46.557
So point T, to point T,

00:04:46.557 --> 00:04:50.133
To get from point T to the
line in the shortest distance,

00:04:50.133 --> 00:04:51.758
once again, we drop a perpendicular.

00:04:51.758 --> 00:04:53.987
This line is going to have a slope of one,

00:04:53.987 --> 00:04:55.891
because it's perpendicular to the line

00:04:55.891 --> 00:04:57.423
that has a slope of negative one.

00:04:57.423 --> 00:05:01.117
And so to get there, we
have to decrease our X by

00:05:01.117 --> 00:05:02.323
we have to decrease our X.

00:05:02.323 --> 00:05:04.045
We're going from X
equals five to X equals,

00:05:04.045 --> 00:05:05.456
it looks like, half.

00:05:05.456 --> 00:05:09.891
So X went down by four and
a half in the X direction

00:05:09.891 --> 00:05:12.492
and Y also needs to go
down by four and a half.

00:05:12.492 --> 00:05:13.954
So if we want to stay on that line,

00:05:13.954 --> 00:05:16.253
let's decrease our X by four and a half.

00:05:16.253 --> 00:05:19.388
So that's half, one, two, three, four.

00:05:19.388 --> 00:05:21.617
And Y needs to go down by four and a half.

00:05:21.617 --> 00:05:25.216
So that's half, one, two, three, four,

00:05:25.216 --> 00:05:26.980
and we get to this point right over here,

00:05:26.980 --> 00:05:30.486
which is the point X equals negative four,

00:05:30.486 --> 00:05:34.201
Y is equal to negative seven.

00:05:34.201 --> 00:05:36.123
And it's negative four, negative seven.

00:05:36.123 --> 00:05:38.190
Let me, so this should be at,

00:05:38.190 --> 00:05:41.951
X equals negative four,
Y equals negative seven.

00:05:41.951 --> 00:05:43.879
And there's a couple of
things you could do here.

00:05:43.879 --> 00:05:45.481
You could just say, hey, this is two long,

00:05:45.481 --> 00:05:48.843
or two units long, not like,
too long in length, somehow.

00:05:48.843 --> 00:05:51.211
This is two units long, so
maybe this is two units long.

00:05:51.211 --> 00:05:52.604
So this is feeling pretty good.

00:05:52.604 --> 00:05:55.483
But let's just go through the
exercise for point O as well.

00:05:55.483 --> 00:05:58.177
So point O, once again,
this is going to have,

00:05:58.177 --> 00:06:00.173
if we drop a perpendicular,

00:06:00.173 --> 00:06:02.054
it's going to have a slope of one.

00:06:02.054 --> 00:06:04.794
So whatever our change
in X between this point

00:06:04.794 --> 00:06:05.718
and this point, we're going to have

00:06:05.718 --> 00:06:07.757
the same change in Y.

00:06:07.757 --> 00:06:12.026
And our change in X,
to go from seven to...

00:06:12.026 --> 00:06:14.719
Seven to one and a half,

00:06:14.719 --> 00:06:17.343
our change in X is five and a half.

00:06:17.343 --> 00:06:19.642
So then we do it this way.

00:06:19.642 --> 00:06:21.453
So the change in X here,

00:06:21.453 --> 00:06:26.166
so the change in X is equal
to negative five and a half,

00:06:26.166 --> 00:06:27.815
five point five.

00:06:27.815 --> 00:06:29.301
If you subtract five
point five from seven,

00:06:29.301 --> 00:06:30.624
you get to one point five.

00:06:30.624 --> 00:06:32.366
And our change in Y.

00:06:32.366 --> 00:06:35.268
Our change in Y is also
negative five point five.

00:06:35.268 --> 00:06:37.125
Change is Y is negative,

00:06:37.125 --> 00:06:37.831
you're not going to see that

00:06:37.831 --> 00:06:38.620
so I'm going to do it differently,

00:06:38.620 --> 00:06:41.198
negative five point five.

00:06:41.198 --> 00:06:43.171
And so we need to stay on this line.

00:06:43.171 --> 00:06:45.772
So we need to change by those same amounts

00:06:45.772 --> 00:06:47.444
onto the other side of that line.

00:06:47.444 --> 00:06:50.532
So if we decrease our
X by five and a half.

00:06:50.532 --> 00:06:54.502
So half, one, two, three,
four, five, we get there,

00:06:54.502 --> 00:06:56.383
and Y by five and a half,

00:06:56.383 --> 00:06:59.410
half, one, two,

00:06:59.410 --> 00:07:02.126
three, four, five.

00:07:02.126 --> 00:07:04.889
We get to the point, X
equals negative four,

00:07:04.889 --> 00:07:08.349
Y is equal to negative nine.

00:07:08.349 --> 00:07:11.646
So we get to the point
X equals negative four,

00:07:11.646 --> 00:07:13.736
Y is equal to negative
nine, and we're done!

00:07:13.736 --> 00:07:17.863
We have figured out the
reflection of these two segments.


Points on line of reflection

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:02.650
- [Voiceover] We're asked to
use the "Reflection" tool to

00:00:02.650 --> 00:00:05.850
define a reflection that
will map line segment ME,

00:00:05.850 --> 00:00:10.540
line segment ME, onto the
other line segment below.

00:00:10.890 --> 00:00:13.420
So we want to map ME to
this segment over here

00:00:13.420 --> 00:00:15.350
and we want to use a Reflection.

00:00:15.350 --> 00:00:16.920
Let's see what they expect from us

00:00:16.920 --> 00:00:18.050
if we want to add a Reflection.

00:00:18.050 --> 00:00:19.520
So if I click on this it says

00:00:19.520 --> 00:00:21.550
Reflection over the line from,

00:00:21.550 --> 00:00:23.450
and then we have two coordinate pairs.

00:00:23.450 --> 00:00:25.420
So they want us to define
the line that we're going

00:00:25.420 --> 00:00:29.150
to reflect over with
two points on that line.

00:00:29.150 --> 00:00:30.320
So let's see if we can do that.

00:00:30.320 --> 00:00:31.982
To do that I think I need
to write something down

00:00:31.982 --> 00:00:34.840
so let me get my scratch pad out

00:00:34.840 --> 00:00:37.340
and I copied and pasted the same diagram.

00:00:38.010 --> 00:00:40.390
And the line of reflection,
one way to think about it,

00:00:40.390 --> 00:00:43.450
we want to map point E,
we want to map point E,

00:00:43.450 --> 00:00:45.640
to this point right over here.

00:00:46.420 --> 00:00:47.960
We want to map point M

00:00:49.040 --> 00:00:50.700
to this point over here.

00:00:51.500 --> 00:00:54.950
And so between any point
and its corresponding point

00:00:54.950 --> 00:00:57.820
on the image after the reflection,

00:00:57.820 --> 00:01:00.550
these should be equidistant
from the line of reflection.

00:01:00.550 --> 00:01:02.910
This and this should
be equidistant from the

00:01:02.910 --> 00:01:04.180
line of reflection.

00:01:04.180 --> 00:01:06.490
This and this should be E,
and this point should be

00:01:06.490 --> 00:01:08.650
equidistant from the line of reflection.

00:01:08.650 --> 00:01:11.210
Or another way of thinking about
it, that line of reflection

00:01:11.210 --> 00:01:14.450
should contain the midpoint
between these two magenta points

00:01:14.450 --> 00:01:17.400
and it should contain the
midpoint between these two

00:01:17.400 --> 00:01:19.620
deep navy blue points.

00:01:19.620 --> 00:01:21.750
So let's just calculate the midpoints.

00:01:21.750 --> 00:01:24.720
So we could do that with a
little bit of mathematics.

00:01:24.720 --> 00:01:28.120
The coordinates for E
right over here, that is,

00:01:28.120 --> 00:01:29.230
let's see that is

00:01:29.670 --> 00:01:31.150
x equals negative four,

00:01:31.520 --> 00:01:34.720
y is equal to negative four,

00:01:35.600 --> 00:01:38.450
and the coordinates for the
corresponding point to E

00:01:38.450 --> 00:01:39.520
in the image.

00:01:39.520 --> 00:01:43.790
This is x is equal to
two, x is equal to two,

00:01:43.790 --> 00:01:45.750
and y is equal to negative six.

00:01:46.570 --> 00:01:47.710
So what's the midpoint between

00:01:47.710 --> 00:01:48.950
negative four, negative four,

00:01:48.950 --> 00:01:50.720
and two, comma, negative six?

00:01:50.720 --> 00:01:52.750
Well you just have to take
the average of the x's

00:01:52.750 --> 00:01:54.230
and take the average of the y's.

00:01:54.230 --> 00:01:56.150
Let me do that, actually
I'll do it over here.

00:01:56.150 --> 00:01:58.260
So if I take the average
of the x's it's going to be

00:01:58.260 --> 00:02:00.920
negative four, negative four,

00:02:00.920 --> 00:02:03.850
plus two, plus two,

00:02:03.850 --> 00:02:06.620
over two, that's the average of the x's.

00:02:06.620 --> 00:02:08.790
And then the average of
the y's, it's going to be

00:02:08.790 --> 00:02:11.950
negative four plus negative six over two.

00:02:12.350 --> 00:02:14.870
Negative four plus negative six,

00:02:15.280 --> 00:02:19.050
over two and then close the parentheses.

00:02:19.050 --> 00:02:22.590
Let's see, negative four
plus two is negative two,

00:02:22.890 --> 00:02:24.450
divided by two is negative one.

00:02:24.450 --> 00:02:26.920
So it's going to be negative one, comma.

00:02:26.920 --> 00:02:28.620
Negative four plus negative six,

00:02:28.620 --> 00:02:30.250
that's the same thing as
negative four minus six

00:02:30.250 --> 00:02:31.990
which is going to be negative 10.

00:02:31.990 --> 00:02:34.490
Divided by two is negative five.

00:02:35.730 --> 00:02:37.090
Let me do that in a blue color

00:02:37.090 --> 00:02:38.620
so you see where it came from.

00:02:38.620 --> 00:02:41.380
Is going to be negative five.

00:02:41.650 --> 00:02:42.550
So there you have it.

00:02:42.550 --> 00:02:45.250
That's going to be the midpoint between E

00:02:45.250 --> 00:02:47.020
and the corresponding point on its image.

00:02:47.020 --> 00:02:49.980
So let's see if I can plot that.

00:02:49.980 --> 00:02:51.490
So this is going to be,

00:02:51.490 --> 00:02:54.050
this point right over here is going to be

00:02:54.050 --> 00:02:56.520
negative one, comma, negative five.

00:02:56.520 --> 00:02:57.890
So x is negative one,

00:02:57.890 --> 00:02:59.420
y is negative five.

00:02:59.420 --> 00:03:00.820
So it's this point right over here

00:03:00.820 --> 00:03:02.650
and it does indeed look like the midpoint.

00:03:02.650 --> 00:03:04.820
It looks like it's
equidistant between and E

00:03:04.820 --> 00:03:06.490
and this point right over here.

00:03:06.490 --> 00:03:09.620
And so this should sit on
the line of reflection.

00:03:09.620 --> 00:03:11.820
So now let's find the midpoint
between M and this point

00:03:11.820 --> 00:03:12.950
right over here.

00:03:12.950 --> 00:03:16.470
The coordinates of M
are x is negative five,

00:03:17.190 --> 00:03:19.340
and y is equal to three.

00:03:19.980 --> 00:03:24.120
The coordinates here
are x is equal to seven

00:03:24.120 --> 00:03:27.110
and y is equal to negative one.

00:03:27.620 --> 00:03:29.910
So the midpoint, the x
coordinate of the midpoint,

00:03:29.910 --> 00:03:32.010
is going to be the
average of the x's here.

00:03:32.010 --> 00:03:34.790
So let's see it's going
to be negative five

00:03:34.790 --> 00:03:36.750
plus seven over two.

00:03:36.750 --> 00:03:38.720
And the y coordinate of the
midpoint is going to be the

00:03:38.720 --> 00:03:40.120
average of the y coordinates.

00:03:40.120 --> 00:03:44.290
So three plus negative one over two.

00:03:44.290 --> 00:03:47.550
Let's see, negative five of plus seven is

00:03:47.880 --> 00:03:50.500
positive two, over two is one.

00:03:51.010 --> 00:03:53.650
Three minus one, three plus negative one,

00:03:53.650 --> 00:03:55.720
that's positive two over two is one.

00:03:56.280 --> 00:03:59.050
So the point one, comma, one
is a midpoint between these two

00:03:59.050 --> 00:04:02.450
so one, comma, one, just like that.

00:04:02.450 --> 00:04:04.290
So the line of reflection
is going to contain

00:04:04.290 --> 00:04:04.950
these two points.

00:04:04.950 --> 00:04:06.850
And two points define a line.

00:04:07.800 --> 00:04:09.190
Let me draw the line of reflection,

00:04:09.190 --> 00:04:11.450
just 'cause we did all of this work,

00:04:11.450 --> 00:04:13.820
the line of reflection is
going to look something like,

00:04:17.050 --> 00:04:19.410
I want to draw this a little
bit straighter than that,

00:04:19.410 --> 00:04:24.220
it's going to look something like this.

00:04:24.220 --> 00:04:26.320
And this makes sense that
this is a line of reflection.

00:04:27.400 --> 00:04:29.430
I missed that magenta point a little bit,

00:04:30.050 --> 00:04:32.310
so let me go through the magenta point.

00:04:32.310 --> 00:04:33.320
Okay, there you go.

00:04:33.320 --> 00:04:35.450
This makes sense that this
is a line of reflection

00:04:35.450 --> 00:04:38.290
'cause you see that you
pick an arbitrary point

00:04:38.290 --> 00:04:39.420
on segment ME,

00:04:39.420 --> 00:04:41.220
say that point,

00:04:41.220 --> 00:04:43.080
and if you reflected over this line.

00:04:43.080 --> 00:04:46.390
This is the shortest
distance from the line.

00:04:46.390 --> 00:04:49.120
You just go onto the other
side of the line equal distant

00:04:49.120 --> 00:04:51.190
and you get to its corresponding point

00:04:51.190 --> 00:04:52.210
on the image.

00:04:52.210 --> 00:04:54.620
So it makes a lot of sense
that these are mirror images

00:04:54.620 --> 00:04:57.150
if this is kind of the mirror here.

00:04:57.150 --> 00:05:00.150
And you can image that this
is kind of the surface of

00:05:00.150 --> 00:05:02.100
the water, if you're
looking at it at an angle.

00:05:02.100 --> 00:05:03.350
I don't know if that helps you or not.

00:05:03.350 --> 00:05:05.710
But anyway we found two points.

00:05:05.710 --> 00:05:08.680
We found two points that
define that line of reflection

00:05:08.680 --> 00:05:10.870
so now let's use the tool to type them in.

00:05:10.870 --> 00:05:12.690
One is negative one, negative five.

00:05:12.690 --> 00:05:14.050
The other is one, comma, one

00:05:14.050 --> 00:05:15.490
so let me see if I can remember that.

00:05:15.490 --> 00:05:16.760
I have a bad memory.

00:05:17.760 --> 00:05:22.200
So one is negative one,
comma, negative five.

00:05:22.720 --> 00:05:25.090
And then the other one is one, comma, one.

00:05:25.490 --> 00:05:26.810
And we see it worked.

00:05:27.120 --> 00:05:28.120
We see it worked.

00:05:28.120 --> 00:05:30.840
When I did that, it actually
made the reflection happen

00:05:30.840 --> 00:05:33.050
and notice it completely
went from this point

00:05:33.050 --> 00:05:36.750
and now our blue is over the
image that we wanted to get to.

00:05:37.130 --> 00:05:39.340
So we are done.


Rotating about arbitrary point

WEBVTT
Kind: captions
Language: en

00:00:00.378 --> 00:00:03.040
- [Voiceover] Triangle SAM, S-A-M,

00:00:03.040 --> 00:00:05.680
and this is one over here, S-A-M,

00:00:05.680 --> 00:00:07.880
is rotated 270 degrees,

00:00:07.880 --> 00:00:10.277
about the point four comma negative two.

00:00:10.277 --> 00:00:12.538
So, this is four comma
negative two right over here.

00:00:12.538 --> 00:00:15.579
Draw the image of this rotation
using the interactive graph.

00:00:15.579 --> 00:00:17.179
We have this little interactive graph tool

00:00:17.179 --> 00:00:18.200
where we can draw points

00:00:18.200 --> 00:00:20.320
or if we wanna put them in the trash

00:00:20.320 --> 00:00:21.800
we can put them there.

00:00:21.800 --> 00:00:22.899
The direction of rotation by a

00:00:22.899 --> 00:00:25.739
positive angle is counter-clockwise.

00:00:25.739 --> 00:00:27.631
So, this would be 270 degrees in

00:00:27.631 --> 00:00:29.511
the counter-clockwise direction.

00:00:29.511 --> 00:00:31.252
So, to help us think about that I've

00:00:31.252 --> 00:00:33.512
copied and pasted this on my scratch pad

00:00:33.512 --> 00:00:35.674
and we can draw through it

00:00:35.674 --> 00:00:36.825
and the first thing that we might wanna

00:00:36.825 --> 00:00:39.663
think about is if you rotate,

00:00:39.663 --> 00:00:41.558
I've talked about this
in the previous video

00:00:41.558 --> 00:00:43.521
when we were rotating around the origin,

00:00:43.521 --> 00:00:45.379
if you rotate something by,

00:00:45.379 --> 00:00:48.161
last time we talked about a
negative 270 degree rotation,

00:00:48.161 --> 00:00:50.960
but if we're talking about
a 270 degree rotation.

00:00:50.960 --> 00:00:52.740
If you imagine a point right over here

00:00:52.740 --> 00:00:55.198
this would be 90 degrees, 180,

00:00:55.198 --> 00:00:57.459
and then that is 270 degrees.

00:00:57.459 --> 00:00:59.732
That is a 200 and 70 degree rotation.

00:00:59.732 --> 00:01:01.881
We're going in a
counter-clockwise direction.

00:01:01.881 --> 00:01:03.961
You see that that is equivalent,

00:01:03.961 --> 00:01:07.281
that is equivalent to a 90 degrees,

00:01:07.281 --> 00:01:10.059
to a 90 degrees clockwise rotation,

00:01:10.059 --> 00:01:13.139
or a negative 90 degree rotation.

00:01:13.139 --> 00:01:14.579
And 90 degree rotations are

00:01:14.579 --> 00:01:16.018
a little bit easier to think about.

00:01:16.018 --> 00:01:17.517
So, let's just, instead of thinking

00:01:17.517 --> 00:01:20.299
of this in terms of rotating 270 degrees

00:01:20.299 --> 00:01:21.411
in the positive direction,

00:01:21.411 --> 00:01:23.281
in the counter-clockwise direction,

00:01:23.281 --> 00:01:25.120
let's think about, let's think about this,

00:01:25.120 --> 00:01:28.378
rotating this 90 degrees
in the clockwise direction.

00:01:28.378 --> 00:01:30.037
They are going to be equivalent.

00:01:30.037 --> 00:01:32.117
But how do we do that?

00:01:32.117 --> 00:01:34.117
Well, like we've done in previous videos

00:01:34.117 --> 00:01:36.749
what I'm gonna do is I'm
gonna take each point,

00:01:36.749 --> 00:01:39.720
let's say point, using
a color you can see,

00:01:39.720 --> 00:01:42.579
point M, I'm having
trouble switching color,

00:01:42.579 --> 00:01:44.088
point M right over here,

00:01:44.088 --> 00:01:46.289
and I'm gonna construct a right triangle

00:01:46.289 --> 00:01:49.148
between point M and our center of rotation

00:01:49.148 --> 00:01:51.580
where the hypotenuse
of the right triangles

00:01:51.580 --> 00:01:53.358
is gonna be the line
that connects these two.

00:01:53.358 --> 00:01:55.621
So, let me draw that.

00:01:55.621 --> 00:01:57.198
So, that's the hypotenuse.

00:01:57.198 --> 00:01:59.899
Now, let me draw a right
triangle like this.

00:01:59.899 --> 00:02:03.417
So, it's gonna go, it's gonna be, like,

00:02:03.417 --> 00:02:05.154
it's gonna look like that,

00:02:05.154 --> 00:02:07.936
and then it's gonna look like, whoops,

00:02:07.936 --> 00:02:12.117
that wasn't to press my,
the command button before.

00:02:12.117 --> 00:02:13.136
Okay, there you go.

00:02:13.136 --> 00:02:14.718
All right, so I have this right triangle.

00:02:14.718 --> 00:02:18.594
So, you imagine, imagine
rotating this right triangle,

00:02:18.594 --> 00:02:21.176
this magenta one that I just constructed,

00:02:21.176 --> 00:02:24.535
by negative 90 degrees,
by negative 90 degrees,

00:02:24.535 --> 00:02:26.957
or 90 degrees in the clockwise direction.

00:02:26.957 --> 00:02:29.737
Well, what's gonna happen
to each of these sides?

00:02:29.737 --> 00:02:32.248
Well, this side right over here,

00:02:32.248 --> 00:02:34.927
if I rotate it by negative 90 degrees

00:02:34.927 --> 00:02:38.407
it is going to end up,
it is going to end up,

00:02:38.407 --> 00:02:40.667
it is going to end up pointing up

00:02:40.667 --> 00:02:42.746
and how long is it going to be?

00:02:42.746 --> 00:02:45.978
Well, let's see, this
one is one, two, three,

00:02:45.978 --> 00:02:50.427
four, five, six, seven units long,

00:02:50.427 --> 00:02:52.029
so it's still gonna be seven units long.

00:02:52.029 --> 00:02:56.387
One, two, three, four, five, six, seven,

00:02:56.387 --> 00:02:59.887
it's gonna put us right over here.

00:02:59.887 --> 00:03:03.591
It is going to put us right over there.

00:03:03.591 --> 00:03:07.007
And then this side,
let me make this clear,

00:03:07.007 --> 00:03:09.666
this side that I'm highlighting in green

00:03:09.666 --> 00:03:11.448
which we see are four units long.

00:03:11.448 --> 00:03:13.329
It forms a right angle.

00:03:13.329 --> 00:03:15.849
It forms a right angle.

00:03:15.849 --> 00:03:17.249
So, if you were to rotate this,

00:03:17.249 --> 00:03:21.165
the entire triangle 90 degrees
in the clockwise direction

00:03:21.165 --> 00:03:23.386
well, then this is gonna go four

00:03:23.386 --> 00:03:26.521
in this direction just like that.

00:03:26.521 --> 00:03:27.724
Just like that.

00:03:27.724 --> 00:03:30.663
And then your new triangle, the image of,

00:03:30.663 --> 00:03:31.924
I guess, what we just,

00:03:31.924 --> 00:03:33.585
this right triangle I've constructed

00:03:33.585 --> 00:03:35.581
if I rotate it by negative 90 degrees,

00:03:35.581 --> 00:03:37.581
90 degrees in the clockwise direction,

00:03:37.581 --> 00:03:40.382
is going to look like this.

00:03:40.382 --> 00:03:42.261
So, after rotation, so what did I just do?

00:03:42.261 --> 00:03:43.942
I just took this, I just took

00:03:43.942 --> 00:03:46.082
this triangle that I just constructed,

00:03:46.082 --> 00:03:48.941
and I rotated it negative 90 degrees

00:03:48.941 --> 00:03:51.342
to get to this triangle right over here.

00:03:51.342 --> 00:03:53.921
And so, the point M,

00:03:53.921 --> 00:03:55.850
its image after the 90 degree

00:03:55.850 --> 00:03:57.930
rotation would be right over here

00:03:57.930 --> 00:04:01.449
and I could call that,
I'll call that M prime.

00:04:01.449 --> 00:04:02.271
What I'm essentially going to do,

00:04:02.271 --> 00:04:03.449
I'm going to do that for point M,

00:04:03.449 --> 00:04:04.890
I'm going to do that for point A,

00:04:04.890 --> 00:04:06.428
and I'm gonna do that for point S,

00:04:06.428 --> 00:04:08.050
and I'm gonna connect the three,

00:04:08.050 --> 00:04:10.469
the M prime, the A prime, and the S prime,

00:04:10.469 --> 00:04:11.450
and then that will give me the image,

00:04:11.450 --> 00:04:14.391
that'll give me the
vertices of my new triangle.

00:04:14.391 --> 00:04:17.570
The image of triangle SAM after rotation.

00:04:17.570 --> 00:04:18.631
So, let's keep going.

00:04:18.631 --> 00:04:22.489
Let's do this, let's do
this now with point A.

00:04:22.489 --> 00:04:27.048
With point A then I can
construct another right triangle

00:04:27.048 --> 00:04:29.837
and I could do it, I could do it several,

00:04:29.837 --> 00:04:31.952
I could do it several ways.

00:04:31.952 --> 00:04:33.088
Actually, let me just do it this way.

00:04:33.088 --> 00:04:35.731
Let me, let me draw it.

00:04:35.731 --> 00:04:37.507
I could, I could draw this.

00:04:37.507 --> 00:04:41.369
The hypotenuse we know
is gonna go from here to,

00:04:41.369 --> 00:04:43.889
from point A to our center of origin.

00:04:43.889 --> 00:04:47.910
And I could draw a right
triangle that's up here like this

00:04:47.910 --> 00:04:50.448
or I could draw a right triangle

00:04:50.448 --> 00:04:54.052
that is down here like this.

00:04:54.052 --> 00:04:55.707
Maybe I'll focus on this one.

00:04:55.707 --> 00:04:57.748
Let me draw this right triangle.

00:04:57.748 --> 00:05:01.608
So, this right triangle right like this.

00:05:01.608 --> 00:05:04.329
So, if I draw this right
triangle like this.

00:05:04.329 --> 00:05:07.728
So, this side, let me do this in a color

00:05:07.728 --> 00:05:09.008
that I can see more clearly

00:05:09.008 --> 00:05:12.908
or that is mostly this side.

00:05:12.908 --> 00:05:14.208
No, you won't be able to see that.

00:05:14.208 --> 00:05:15.748
All right, I'mma stick with the purple.

00:05:15.748 --> 00:05:17.507
So, this side right over here

00:05:17.507 --> 00:05:19.569
that I am drawing in purple,

00:05:19.569 --> 00:05:22.309
if you rotate it clockwise
in the 90 degree direction

00:05:22.309 --> 00:05:25.927
or negative 90 rotation,
well, it's going to,

00:05:25.927 --> 00:05:27.408
instead of being straight horizontal,

00:05:27.408 --> 00:05:28.748
it's gonna be straight vertical.

00:05:28.748 --> 00:05:30.909
So, it's going to be, it's going to be,

00:05:30.909 --> 00:05:32.986
it's gonna overlap with
what we had already drawn

00:05:32.986 --> 00:05:33.989
and how long is it?

00:05:33.989 --> 00:05:35.787
Well, right now it goes from

00:05:35.787 --> 00:05:37.826
X equals negative five to X equals four.

00:05:37.826 --> 00:05:40.089
So, it's nine long and you can count that.

00:05:40.089 --> 00:05:42.907
One, two, three, four, five,

00:05:42.907 --> 00:05:46.128
six, seven, eight, nine,

00:05:46.128 --> 00:05:47.648
or you could say it's two longer

00:05:47.648 --> 00:05:48.928
than what we had already drawn.

00:05:48.928 --> 00:05:49.874
So, let's see, if you,

00:05:49.874 --> 00:05:52.687
if you start Y equals
negative two and you add nine

00:05:52.687 --> 00:05:54.573
you're gonna get to Y equals seven.

00:05:54.573 --> 00:05:58.544
So, this is gonna go all the way like this

00:05:58.544 --> 00:06:00.814
and then, and then let's see,

00:06:00.814 --> 00:06:05.814
this side, this side forms
a right angle with it.

00:06:05.995 --> 00:06:09.934
So, if you, just gonna go like this,

00:06:09.934 --> 00:06:11.255
it's gonna be a right angle

00:06:11.255 --> 00:06:12.153
and how long is this?

00:06:12.153 --> 00:06:16.273
It's one, two, three,
four, five units long.

00:06:16.273 --> 00:06:17.365
So, now we're gonna go one,

00:06:17.365 --> 00:06:21.043
two, three, four, five
units in that direction

00:06:21.043 --> 00:06:24.883
and just like that we get to our A prime.

00:06:24.883 --> 00:06:28.414
We get, we get to,

00:06:28.414 --> 00:06:31.884
we get to our A prime

00:06:31.884 --> 00:06:34.814
and we can draw the
hypotenuse of this triangle.

00:06:34.814 --> 00:06:37.062
So, let me see if I can draw it.

00:06:37.062 --> 00:06:39.644
Press, I can only draw the straight line.

00:06:39.644 --> 00:06:43.174
All right, it's gonna look
like, gonna look like that.

00:06:43.174 --> 00:06:44.575
So, we know where the image

00:06:44.575 --> 00:06:46.432
of point M is after the rotation.

00:06:46.432 --> 00:06:48.912
We know where the image of
point A is after rotation

00:06:48.912 --> 00:06:51.413
and we see how these
triangles we constructed.

00:06:51.413 --> 00:06:52.993
We've just rotated them.

00:06:52.993 --> 00:06:54.663
We've just rotated them 90

00:06:54.663 --> 00:06:56.682
degrees in the clockwise direction.

00:06:56.682 --> 00:06:58.943
Now, we just have to do this for point S

00:06:58.943 --> 00:07:02.033
and then we have the three
points for our image.

00:07:02.033 --> 00:07:06.895
So, point S, once again we can
construct a right triangle.

00:07:06.895 --> 00:07:08.093
Let's do that.

00:07:08.093 --> 00:07:11.153
So, hypotenuse, we'll connect point S

00:07:11.153 --> 00:07:13.493
to our center of rotation,

00:07:13.493 --> 00:07:17.131
and I can do this several ways.

00:07:17.131 --> 00:07:20.341
Let me think about the best way to do it.

00:07:20.341 --> 00:07:22.490
Well, let me construct this right triangle

00:07:22.490 --> 00:07:24.973
so it looks like this.

00:07:24.973 --> 00:07:28.473
And then looks like this.

00:07:28.473 --> 00:07:33.473
So, this top side, this
top side right over here.

00:07:34.000 --> 00:07:35.478
If I were to rotate it,

00:07:35.478 --> 00:07:38.999
if I were to rotate it
negative 90 degrees,

00:07:38.999 --> 00:07:40.559
90 degrees in the clockwise direction,

00:07:40.559 --> 00:07:41.839
where does that put us?

00:07:41.839 --> 00:07:44.817
Well, once again it's gonna
put us straight up and down

00:07:44.817 --> 00:07:47.137
and this length is it goes from

00:07:47.137 --> 00:07:49.098
X equals negative three to X equals four.

00:07:49.098 --> 00:07:50.697
So, it's seven long.

00:07:50.697 --> 00:07:54.999
So, it gets us to this
point right over here.

00:07:54.999 --> 00:07:59.718
Now, this side, let me be
careful when I just say this,

00:07:59.718 --> 00:08:02.197
and let me do it in a different color.

00:08:02.197 --> 00:08:05.036
This side over here forms a right angle

00:08:05.036 --> 00:08:07.297
and so it's, and if you
see when you go from here

00:08:07.297 --> 00:08:09.857
you go down so here you're gonna go left

00:08:09.857 --> 00:08:11.559
and you see that it's three long.

00:08:11.559 --> 00:08:14.617
So, it's three, it's three long.

00:08:14.617 --> 00:08:16.418
This is a right angle right over here

00:08:16.418 --> 00:08:18.756
and that gets us to our S prime.

00:08:18.756 --> 00:08:21.077
That gets us to our S prime.

00:08:21.077 --> 00:08:24.490
We can now draw the hypotenuse.

00:08:24.490 --> 00:08:27.742
So, notice this whole, like,

00:08:27.742 --> 00:08:30.617
kind of crazy complex that we have drawn

00:08:30.617 --> 00:08:31.478
of these different triangles,

00:08:31.478 --> 00:08:34.158
I've just rotated the whole
thing by negative 90 degrees.

00:08:34.158 --> 00:08:37.216
But this triangle that
we have, that we were,

00:08:37.216 --> 00:08:40.500
we had just made when you rotated it

00:08:40.500 --> 00:08:42.378
maps to that one right over there.

00:08:42.378 --> 00:08:44.177
But now we have in our image our

00:08:44.177 --> 00:08:45.776
S prime, our A prime, and our M prime,

00:08:45.776 --> 00:08:47.017
and we can connect them.

00:08:47.017 --> 00:08:48.736
So, it will look like this.

00:08:48.736 --> 00:08:49.982
So, let's just connect them.

00:08:49.982 --> 00:08:51.539
Connect the dots.

00:08:51.539 --> 00:08:56.539
So, that, and that, and that.

00:08:58.337 --> 00:09:03.007
And so, you see, you might have
lost our original triangle.

00:09:03.007 --> 00:09:05.536
I'm just gonna shade it
in so you can see it clear

00:09:05.536 --> 00:09:07.958
although what we really care
about is mapping the points.

00:09:07.958 --> 00:09:09.456
Mapping, actually, let me.

00:09:09.456 --> 00:09:11.787
I don't want you to
think that we're mapping,

00:09:11.787 --> 00:09:14.896
that we're mapping everything in between.

00:09:14.896 --> 00:09:17.376
We're just mapping the points
that define the triangle.

00:09:17.376 --> 00:09:19.238
So, our original triangle
that we cared about

00:09:19.238 --> 00:09:22.107
was this thing that I'm
gonna draw really bold.

00:09:22.107 --> 00:09:24.119
It was this thing.

00:09:24.119 --> 00:09:27.275
And I'm gonna draw it really, really bold.

00:09:27.275 --> 00:09:29.116
And you rotate it by negative 90 degrees

00:09:29.116 --> 00:09:30.995
which is equivalent to positive 270.

00:09:30.995 --> 00:09:34.217
You could've rotated it positive 270.

00:09:34.217 --> 00:09:36.198
You could've rotated
it positive 270 to get

00:09:36.198 --> 00:09:38.438
to this other one right over here

00:09:38.438 --> 00:09:40.057
but it maps to this.

00:09:40.057 --> 00:09:41.678
This is its image.

00:09:41.678 --> 00:09:44.457
Now we just have to go
back onto the exercise.

00:09:44.457 --> 00:09:47.336
Now we have to go back onto
the exercise and put it in.

00:09:47.336 --> 00:09:48.355
So, let's remember these points.

00:09:48.355 --> 00:09:52.115
We have the point eight comma five.

00:09:52.115 --> 00:09:57.017
So, we have the point eight
comma five in our image.

00:09:57.717 --> 00:10:02.596
We have the point nine comma seven,

00:10:03.426 --> 00:10:08.426
nine comma seven, and we have the point,

00:10:09.204 --> 00:10:14.155
and we have the point one comma five.

00:10:14.925 --> 00:10:18.551
So, we have the point one comma five

00:10:18.551 --> 00:10:19.512
and now I just have to finish it

00:10:19.512 --> 00:10:21.333
by going to this point again

00:10:21.333 --> 00:10:22.075
and we're all done.

00:10:22.075 --> 00:10:24.874
We've rotated it by 270 degrees

00:10:24.874 --> 00:10:27.514
which is equivalent to a
negative 90 degree rotation.

00:10:27.514 --> 00:10:31.072
We can check that we indeed got it right.


Points after rotation

WEBVTT
Kind: captions
Language: en

00:00:00.375 --> 00:00:02.326
- [Voiceover] We're told that triangle PIN

00:00:02.326 --> 00:00:06.411
is rotated negative 270
degrees about the origin.

00:00:06.411 --> 00:00:09.518
So this is the triangle PIN
and we're gonna rotate it

00:00:09.518 --> 00:00:12.419
negative 270 degrees about the origin.

00:00:12.419 --> 00:00:16.390
Draw the image of this rotation
using the interactive graph.

00:00:16.390 --> 00:00:18.666
The direction of rotation
by a positive angle

00:00:18.666 --> 00:00:20.291
is counter-clockwise.

00:00:20.291 --> 00:00:22.405
So positive is counter-clockwise,

00:00:22.405 --> 00:00:24.052
which is a standard convention,

00:00:24.052 --> 00:00:26.553
and this is negative, so a negative degree

00:00:26.553 --> 00:00:28.387
would be clockwise.

00:00:28.387 --> 00:00:29.897
And we want to use this tool here.

00:00:29.897 --> 00:00:32.492
And this tool, I can put points in,

00:00:32.492 --> 00:00:33.815
or I could delete points.

00:00:33.815 --> 00:00:35.674
I can draw a point by clicking on it.

00:00:35.674 --> 00:00:36.996
So what we want to do is think about,

00:00:36.996 --> 00:00:40.271
well look, if we rotate
the points of this triangle

00:00:40.271 --> 00:00:43.986
around the origin by negative 270 degrees,

00:00:43.986 --> 00:00:45.890
where is it gonna put these points?

00:00:45.890 --> 00:00:47.191
And to help us think about that,

00:00:47.191 --> 00:00:49.940
I have copied and pasted
this on our scratch pad.

00:00:49.940 --> 00:00:51.658
So actually let me go over here

00:00:51.658 --> 00:00:53.539
so I can actually draw on it.

00:00:53.539 --> 00:00:55.187
So let's just first think
about what a negative

00:00:55.187 --> 00:00:58.415
270 degree rotation actually is.

00:00:58.415 --> 00:01:01.201
So if I were to start,

00:01:01.201 --> 00:01:02.315
if I were to,

00:01:02.315 --> 00:01:04.568
let me draw some coordinate axes here.

00:01:04.568 --> 00:01:06.843
So that's an x-axis and that's a y-axis.

00:01:06.843 --> 00:01:10.489
If you were to start right over here

00:01:10.489 --> 00:01:12.788
and you were to rotate around
the origin by negative,

00:01:12.788 --> 00:01:14.158
so this is the origin here,

00:01:14.158 --> 00:01:15.643
by negative 270 degrees,

00:01:15.643 --> 00:01:16.874
what would that be?

00:01:16.874 --> 00:01:20.389
Well let's see, this would
be rotating negative 90.

00:01:20.389 --> 00:01:23.826
This would be rotating
another negative 90,

00:01:23.826 --> 00:01:25.985
which would, together, be negative 180.

00:01:25.985 --> 00:01:28.238
And then this would be
another negative 90,

00:01:28.238 --> 00:01:29.886
which would give you in total,

00:01:29.886 --> 00:01:31.905
negative 270 degrees.

00:01:31.905 --> 00:01:34.367
That's negative 270 degrees.

00:01:34.367 --> 00:01:36.527
Now notice, that would
get that point here,

00:01:36.527 --> 00:01:38.663
which we could have also gotten there

00:01:38.663 --> 00:01:42.007
by just rotating it by
positive 90 degrees.

00:01:42.007 --> 00:01:43.817
We could have just said
that this is equivalent

00:01:43.817 --> 00:01:47.068
to a positive 90 degree rotation.

00:01:47.068 --> 00:01:49.064
So if they want us to
rotate the points here

00:01:49.064 --> 00:01:52.339
around the origin by negative 270 degrees,

00:01:52.339 --> 00:01:54.220
that's equivalent to just rotating

00:01:54.220 --> 00:01:56.728
all of the points, and I'll
just focus on the vertices,

00:01:56.728 --> 00:01:58.817
because those are the
easiest ones to think about,

00:01:58.817 --> 00:02:00.443
to visualize.

00:02:00.443 --> 00:02:03.786
I can just rotate each of
those around the origin

00:02:03.786 --> 00:02:06.038
by positive 90 degrees.

00:02:06.038 --> 00:02:08.197
But how do we do that?

00:02:08.197 --> 00:02:11.146
And to do that, what I am going to do,

00:02:11.146 --> 00:02:12.447
to do that what I'm gonna do is

00:02:12.447 --> 00:02:15.674
I'm gonna draw a series
of right triangles.

00:02:15.674 --> 00:02:18.043
So let's first focus on,

00:02:18.043 --> 00:02:22.176
actually let's first focus
on point I right over here.

00:02:22.176 --> 00:02:24.149
And let me draw right triangle.

00:02:24.149 --> 00:02:26.146
And I could draw it several ways.

00:02:26.146 --> 00:02:28.341
But let me draw it like this.

00:02:29.183 --> 00:02:31.968
So, it's a right triangle

00:02:31.968 --> 00:02:34.337
where the line between the origin and I

00:02:34.337 --> 00:02:35.986
is its hypotenuse.

00:02:35.986 --> 00:02:38.098
So let me see if I can,

00:02:38.098 --> 00:02:40.189
It's I could probably draw,

00:02:40.189 --> 00:02:42.487
actually I can use a line tool for that.

00:02:42.487 --> 00:02:44.026
So let me...

00:02:45.204 --> 00:02:48.048
So that's the hypotenuse of the line.

00:02:48.048 --> 00:02:51.693
Now if I'm gonna rotate I 90 degrees

00:02:51.693 --> 00:02:55.478
about the origin, that's
equivalent to rotating

00:02:55.478 --> 00:02:58.032
this right triangle 90 degrees.

00:02:58.032 --> 00:02:59.913
So what's going to happen there?

00:02:59.913 --> 00:03:02.351
Well this side, right over here,

00:03:02.351 --> 00:03:06.229
if I rotate this 90 degrees,

00:03:06.229 --> 00:03:07.692
where is that going to go?

00:03:07.692 --> 00:03:09.874
Well instead of going
seven along the x-axis,

00:03:09.874 --> 00:03:11.638
it's going to go seven along the y-axis.

00:03:11.638 --> 00:03:15.331
So it's going to, if you
rotate it positive 90 degrees,

00:03:15.331 --> 00:03:18.789
that side it going to look like this.

00:03:18.789 --> 00:03:21.786
So that's rotating it 90 degrees,

00:03:21.786 --> 00:03:22.923
just like that.

00:03:22.923 --> 00:03:25.454
Now what about this side over here?

00:03:25.454 --> 00:03:27.567
What about this side?

00:03:27.567 --> 00:03:29.796
Let me do this in a different color.

00:03:29.796 --> 00:03:31.329
What about,

00:03:31.329 --> 00:03:34.997
what about this side right over here?

00:03:34.997 --> 00:03:36.530
Well this side over here, notice

00:03:36.530 --> 00:03:38.294
we've gone down from the origin,

00:03:38.294 --> 00:03:40.222
we've gone down seven.

00:03:40.222 --> 00:03:42.311
But if you were to rotate it up,

00:03:42.311 --> 00:03:44.471
notice this forms a right angle

00:03:44.471 --> 00:03:45.980
between this magenta
side and this blue side,

00:03:45.980 --> 00:03:48.395
so you're gonna form a right angle again.

00:03:48.395 --> 00:03:51.413
And so from this point,
you're gonna go straight

00:03:51.413 --> 00:03:54.756
as a right angle and
instead of going down seven,

00:03:54.756 --> 00:03:56.219
you're going to go to the right seven.

00:03:56.219 --> 00:03:59.401
So you're gonna go to the right seven,

00:03:59.401 --> 00:04:01.816
just like this.

00:04:01.816 --> 00:04:05.206
And so the point I, or
the corresponding point

00:04:05.206 --> 00:04:07.179
in the image after the rotation

00:04:07.179 --> 00:04:08.967
is going to be right over here.

00:04:08.967 --> 00:04:12.393
So that green line, let me
draw the hypotenuse now,

00:04:13.165 --> 00:04:16.508
it's gonna look like, oops,

00:04:16.508 --> 00:04:17.832
I wanted to do that in a different color.

00:04:17.832 --> 00:04:18.946
I want to do that in the green.

00:04:18.946 --> 00:04:20.595
I have trouble changing colors.

00:04:20.595 --> 00:04:22.592
Alright, so there you go.

00:04:22.592 --> 00:04:26.075
It's gonna look like that.

00:04:26.075 --> 00:04:27.143
So your new point I,

00:04:27.143 --> 00:04:31.276
if you rotate this triangle,
this right triangle 90 degrees,

00:04:31.276 --> 00:04:33.017
your new point I, maybe I shouldn't say,

00:04:33.017 --> 00:04:35.060
I'll call it I prime,

00:04:35.060 --> 00:04:38.543
which is what is the image of this point

00:04:38.543 --> 00:04:40.214
after I've done the 90 degree rotation

00:04:40.214 --> 00:04:42.049
is going to be right over here.

00:04:42.049 --> 00:04:45.091
And now we can do that
for each of the points.

00:04:45.091 --> 00:04:47.622
We can do that for N here.

00:04:47.622 --> 00:04:49.456
Let's draw a right triangle.

00:04:49.456 --> 00:04:50.918
Let's draw a right triangle.

00:04:50.918 --> 00:04:53.125
And I can do it a bunch of different ways.

00:04:53.125 --> 00:04:56.561
I could draw a right triangle like this.

00:04:56.561 --> 00:04:57.444
Let me draw it like this.

00:04:57.444 --> 00:04:59.532
So that's one side of my right triangle.

00:04:59.532 --> 00:05:01.345
Actually, let me draw
the hypotenuse first.

00:05:01.345 --> 00:05:03.456
So I have the hypotenuse.

00:05:03.456 --> 00:05:07.168
Connects the origin to
my point, just like that.

00:05:07.168 --> 00:05:09.698
And then I could either draw it up here,

00:05:09.698 --> 00:05:11.406
or I could draw it down here,

00:05:12.549 --> 00:05:14.430
like this.

00:05:14.430 --> 00:05:15.986
I could draw it like this.

00:05:15.986 --> 00:05:18.256
So if you rotated this 90 degrees,

00:05:18.256 --> 00:05:20.509
if you rotate this 90 degrees,

00:05:20.509 --> 00:05:23.992
this side, which is seven units long

00:05:23.992 --> 00:05:26.616
and we're going seven below the origin,

00:05:26.616 --> 00:05:28.381
it's now going to be seven to the right.

00:05:28.381 --> 00:05:30.006
Right? If I rotate it by 90 degrees,

00:05:30.006 --> 00:05:32.397
it's going to be right over here.

00:05:32.397 --> 00:05:33.442
It's gonna be here

00:05:33.442 --> 00:05:36.924
and this side, which has length,

00:05:36.924 --> 00:05:38.899
let me switch colors.

00:05:38.899 --> 00:05:40.546
This side, which has length two,

00:05:40.546 --> 00:05:41.871
it forms a right angle,

00:05:41.871 --> 00:05:43.333
so we're gonna forma a right angle

00:05:43.333 --> 00:05:45.632
and have length two right over here.

00:05:45.632 --> 00:05:49.696
And so the image of
point N is going to be,

00:05:49.696 --> 00:05:52.389
let me get the right color.

00:05:52.389 --> 00:05:56.196
It is going to be just like that.

00:05:56.196 --> 00:05:58.798
It is going to be the image of point N

00:05:58.798 --> 00:05:59.842
is going to be right here.

00:05:59.842 --> 00:06:01.329
I'll call that N prime.

00:06:01.329 --> 00:06:02.699
So we know where the new,

00:06:02.699 --> 00:06:04.904
where the image of I
is, the image of N is,

00:06:04.904 --> 00:06:06.228
so now we just have to
think about where does

00:06:06.228 --> 00:06:07.714
the image of P sit?

00:06:07.714 --> 00:06:11.916
And once again, we can do our
little right triangle idea.

00:06:11.916 --> 00:06:13.811
So let's draw a right triangle.

00:06:14.721 --> 00:06:16.927
So, just like that.

00:06:16.927 --> 00:06:19.087
I could draw that side.

00:06:19.087 --> 00:06:21.376
And I can do this side, right over here.

00:06:22.355 --> 00:06:23.632
So if I were to rotate,

00:06:23.632 --> 00:06:25.096
if I were to focus on this,

00:06:25.096 --> 00:06:26.907
let me do this in a color I haven't.

00:06:26.907 --> 00:06:28.463
So if I were to focus on,

00:06:28.463 --> 00:06:29.949
I've already used that color.

00:06:29.949 --> 00:06:32.520
If I were to focus on
this, right over here,

00:06:32.520 --> 00:06:34.633
and if I were to rotate it by 90 degrees.

00:06:34.633 --> 00:06:35.923
Instead of going two to the right,

00:06:35.923 --> 00:06:37.223
it's going to go two straight up.

00:06:37.223 --> 00:06:40.775
If I rotate this by 90 degrees
it's gonna be just like this.

00:06:40.775 --> 00:06:43.307
Now, this side on,

00:06:43.307 --> 00:06:45.337
let me pick another color.

00:06:46.388 --> 00:06:49.614
This side, right over
here, forms a right angle

00:06:49.614 --> 00:06:51.356
and it has a length of three.

00:06:51.356 --> 00:06:53.353
So we're gonna forma right angle here

00:06:53.353 --> 00:06:55.536
and have a length of three.

00:06:55.536 --> 00:06:56.790
And just like that, we know

00:06:56.790 --> 00:06:58.740
where the image of P is going to be.

00:06:58.740 --> 00:07:00.729
It is going to be,

00:07:00.729 --> 00:07:03.723
it is going to be right over here.

00:07:03.723 --> 00:07:04.978
So this is P prime.

00:07:04.978 --> 00:07:06.069
And I know it's kind of confusing,

00:07:06.069 --> 00:07:08.461
but now we just have to connect

00:07:08.461 --> 00:07:10.295
the P prime, the I prime, and the N prime

00:07:10.295 --> 00:07:13.057
to figure out what the
image of my triangle is

00:07:13.057 --> 00:07:13.934
after rotation.

00:07:13.934 --> 00:07:15.350
So let me do that.

00:07:15.350 --> 00:07:17.552
So if I connect these two, I get that.

00:07:17.552 --> 00:07:21.034
If I connect these two, I get that.

00:07:21.034 --> 00:07:23.704
And if I connect these two,

00:07:23.704 --> 00:07:25.423
if I connect those two, I have that.

00:07:25.423 --> 00:07:26.886
And there you have it, I have the image.

00:07:26.886 --> 00:07:28.440
And now I just have to input it

00:07:28.440 --> 00:07:30.856
on the actual problem.

00:07:30.856 --> 00:07:34.223
So, let's see, point
negative three comma two,

00:07:34.223 --> 00:07:36.057
let me get it out.

00:07:36.057 --> 00:07:39.076
So negative three comma two is there.

00:07:39.076 --> 00:07:41.723
I have seven comma two,

00:07:41.723 --> 00:07:43.681
so let me put that in there.

00:07:44.615 --> 00:07:46.457
Seven comma two.

00:07:47.390 --> 00:07:51.169
And then I have seven
comma seven is there.

00:07:52.174 --> 00:07:54.101
So let me get this out.

00:07:54.101 --> 00:07:56.377
So seven comma seven.

00:07:56.377 --> 00:07:58.699
And then I'll draw here
again to connect the lines.

00:07:58.699 --> 00:07:59.720
And I'm done.

00:07:59.720 --> 00:08:03.839
I've rotated it through
an angle of 90 degrees.

00:08:04.727 --> 00:08:06.676
Or negative 270 degrees, which is what

00:08:06.676 --> 00:08:09.440
they originally asked me for.

00:08:09.440 --> 00:08:11.653
And I can check to make
sure I got the right answer.


Defining rotation example

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:01.769
- [Voiceover] Use the rotation tool

00:00:01.769 --> 00:00:04.137
to determine the angle of rotation

00:00:04.137 --> 00:00:08.871
about the point (2, -3)

00:00:08.871 --> 00:00:13.438
that will map line segments HE and RO

00:00:13.438 --> 00:00:16.197
onto the other line segments.

00:00:16.197 --> 00:00:18.844
So, we wanna rotate HE,

00:00:18.844 --> 00:00:20.144
this is segment HE,

00:00:20.144 --> 00:00:22.327
and this is line segment RO,

00:00:22.327 --> 00:00:24.277
and we wanna rotate it around this point

00:00:24.277 --> 00:00:27.180
so that they map, so that their images

00:00:27.180 --> 00:00:29.683
after the rotation map to these two

00:00:29.683 --> 00:00:31.215
right over here.

00:00:31.215 --> 00:00:32.123
Let's think about it:

00:00:32.123 --> 00:00:33.499
which one should map to which?

00:00:33.499 --> 00:00:38.397
So, it looks like HE

00:00:38.397 --> 00:00:42.140
should map to this right over here.

00:00:42.140 --> 00:00:44.596
Well, actually, let me just play around.

00:00:44.596 --> 00:00:46.496
I can add a rotation.

00:00:46.496 --> 00:00:48.957
I could add a rotation here,

00:00:48.957 --> 00:00:51.325
and let's see, I could rotate around.

00:00:51.325 --> 00:00:55.319
If this point over here is going to,

00:00:55.319 --> 00:00:56.410
well, we already know we're gonna

00:00:56.410 --> 00:00:58.467
rotate around the point (2, -3),

00:00:58.467 --> 00:00:59.419
so let me write that.

00:00:59.419 --> 00:01:02.479
We're gonna rotate
around the point (2, -3),

00:01:02.479 --> 00:01:04.174
and let's see, if we rotate by 90 degrees,

00:01:04.174 --> 00:01:05.068
what happens?

00:01:06.068 --> 00:01:09.899
That was too much.

00:01:12.941 --> 00:01:14.265
Oh, actually that's too little,

00:01:14.265 --> 00:01:16.308
because a positive angle right over here,

00:01:16.308 --> 00:01:19.721
we're going counter-clockwise
by 90 degrees,

00:01:19.721 --> 00:01:21.671
so Point R mapped to this point

00:01:21.671 --> 00:01:23.691
right over here, Point O mapped

00:01:23.691 --> 00:01:25.363
to this point over here,

00:01:25.363 --> 00:01:26.478
but we want it to go even farther.

00:01:26.478 --> 00:01:28.237
We want it to map to maybe this point

00:01:28.237 --> 00:01:30.861
over here, so let me increase this.

00:01:30.861 --> 00:01:33.679
I don't know, let me try 135.

00:01:34.776 --> 00:01:37.674
All right, it looks
like we're almost there.

00:01:37.674 --> 00:01:42.485
Maybe let's try 140.

00:01:42.485 --> 00:01:43.952
Press "Enter."

00:01:43.952 --> 00:01:47.342
That still didn't get us there.

00:01:47.342 --> 00:01:48.687
Let me see 145.

00:01:49.900 --> 00:01:52.439
We're getting close, maybe 150.

00:01:53.466 --> 00:01:58.093
150, and that looks pretty good.

00:01:58.093 --> 00:01:59.931
It looks like we have done it.

00:01:59.931 --> 00:02:01.162
So, if you start with these,

00:02:01.162 --> 00:02:02.624
the set of points that represent

00:02:02.624 --> 00:02:04.969
line segments HE and RO,

00:02:04.969 --> 00:02:08.917
if you rotate them about the point (2, -3)

00:02:08.917 --> 00:02:12.098
by 150 degrees, and the convention is

00:02:12.098 --> 00:02:14.025
that that's going to be counter-clockwise,

00:02:14.025 --> 00:02:16.138
we see that we map, their images

00:02:16.138 --> 00:02:17.333
are going to be these two lines

00:02:17.333 --> 00:02:20.237
right over here, and we saw that

00:02:20.237 --> 00:02:23.000
Point O right over here maps

00:02:23.000 --> 00:02:24.416
to this point right over here.

00:02:24.416 --> 00:02:26.409
Point R maps to this point.

00:02:26.409 --> 00:02:30.170
Point E is going to map
to that point, I believe.

00:02:30.170 --> 00:02:32.371
Yup, that's right, and then Point H

00:02:32.371 --> 00:02:33.834
is going to map to that point,

00:02:33.834 --> 00:02:35.478
and obviously all of
these points in between,

00:02:35.478 --> 00:02:37.312
for example, this point right over here

00:02:37.312 --> 00:02:39.333
that's on the line, that would map

00:02:39.333 --> 00:02:41.157
to halfway between these two segments

00:02:41.157 --> 00:02:43.433
right over there, so there we have it.

00:02:43.433 --> 00:02:44.323
We've done it.


Drawing image of translation

WEBVTT
Kind: captions
Language: en

00:00:00.407 --> 00:00:04.099
- [Voiceover] Triangle ABC
undergoes a translation,

00:00:04.099 --> 00:00:08.917
and we're using the
notation capital T for it,

00:00:08.917 --> 00:00:12.087
and then we see what the
translation has to be.

00:00:12.087 --> 00:00:13.642
We're gonna move,

00:00:13.642 --> 00:00:14.803
it's kind of small, I hope you can see it

00:00:14.803 --> 00:00:15.999
on your video screen.

00:00:15.999 --> 00:00:17.218
We're gonna move positive eight.

00:00:17.218 --> 00:00:18.798
Every point here is
gonna move positive eight

00:00:18.798 --> 00:00:19.621
in the x direction.

00:00:19.621 --> 00:00:22.442
Its x coordinate is going
to increase by eight,

00:00:22.442 --> 00:00:24.404
or the corresponding point in the image,

00:00:24.404 --> 00:00:26.834
its x coordinate, is going
to increase by eight,

00:00:26.834 --> 00:00:27.868
and the corresponding point

00:00:27.868 --> 00:00:29.435
in the image's y coordinate

00:00:29.435 --> 00:00:32.616
is going to decrease by
one, so let's do that.

00:00:32.616 --> 00:00:34.322
And I'll focus on the vertices,

00:00:34.322 --> 00:00:36.609
whoops, let me drag that to the trash,

00:00:36.609 --> 00:00:37.992
I didn't mean to do that.

00:00:37.992 --> 00:00:41.760
I'm going to focus on the vertices

00:00:42.200 --> 00:00:43.561
well, that's just the
easiest thing for my brain

00:00:43.561 --> 00:00:45.057
to worth with.

00:00:45.057 --> 00:00:47.431
And actually, this is what
the tool expects as well.

00:00:47.431 --> 00:00:49.533
So the point B, is going to move eight

00:00:49.533 --> 00:00:51.065
to the right, or its corresponding point

00:00:51.065 --> 00:00:52.586
in the image is going
to have an x coordinate

00:00:52.586 --> 00:00:53.572
eight larger.

00:00:53.572 --> 00:00:55.047
So right now, the x
coordinate is negative four,

00:00:55.047 --> 00:00:56.209
if you added eight to that,

00:00:56.209 --> 00:00:58.123
it would be positive four,

00:00:58.123 --> 00:00:59.306
and its y coordinate is going

00:00:59.306 --> 00:01:00.422
to be one lower.

00:01:00.422 --> 00:01:03.040
Right now, point B's
y coordinate is eight,

00:01:03.040 --> 00:01:04.677
one lower than that is seven.

00:01:04.677 --> 00:01:06.454
So, in the image, the corresponding

00:01:06.454 --> 00:01:07.916
point of the image would
going to be right over there.

00:01:07.916 --> 00:01:09.711
And you see we moved eight to the right,

00:01:09.711 --> 00:01:11.568
and one down.

00:01:11.568 --> 00:01:13.043
Let's do that with point C.

00:01:13.043 --> 00:01:14.610
It's at x equals negative seven,

00:01:14.610 --> 00:01:16.038
if you move eight to the right,

00:01:16.038 --> 00:01:17.930
if you increase your
x coordinate by eight,

00:01:17.930 --> 00:01:19.973
you're gonna move to x equals one,

00:01:19.973 --> 00:01:21.715
and then if you change your y coordinate

00:01:21.715 --> 00:01:24.176
by negative one, you're
gonna move down one,

00:01:24.176 --> 00:01:24.930
then you're gonna get to that point

00:01:24.930 --> 00:01:26.533
right over there.

00:01:26.533 --> 00:01:29.934
Now, let's do it with point A.

00:01:29.934 --> 00:01:32.069
So point A's x coordinate is negative one.

00:01:32.069 --> 00:01:33.556
If you add eight to it,

00:01:33.556 --> 00:01:35.796
it's going to be positive seven,

00:01:35.796 --> 00:01:37.631
and its current y coordinate is two.

00:01:37.631 --> 00:01:39.035
If you take one away from it,

00:01:39.035 --> 00:01:41.218
you're gonna get to a y coordinate of one.

00:01:41.218 --> 00:01:43.702
And so there you have it.

00:01:43.702 --> 00:01:46.384
Let's see, how do I connect these two?

00:01:46.384 --> 00:01:47.406
Oh, there you go,

00:01:47.406 --> 00:01:48.822
and we can check our answer.

00:01:48.822 --> 00:01:50.239
And we got it right.

00:01:50.239 --> 00:01:53.193
We have performed the translation.


Determining a translation between points

WEBVTT
Kind: captions
Language: en

00:00:00.609 --> 00:00:03.144
Voiceover:A translation
acting on the coordinate plane

00:00:03.144 --> 00:00:07.472
takes a point negative 169 comma 434

00:00:07.472 --> 00:00:10.772
to point negative 203 comma negative 68.

00:00:10.772 --> 00:00:13.355
What are the coordinates of the image

00:00:13.355 --> 00:00:19.271
of point 31 comma negative
529 under this translation?

00:00:19.271 --> 00:00:20.604
I encourage you to pause this video

00:00:20.604 --> 00:00:22.408
and try to think about it on your own.

00:00:22.408 --> 00:00:25.469
Let's think about what the change in x

00:00:25.469 --> 00:00:26.947
that this translation does

00:00:26.947 --> 00:00:29.273
and the change in y that
this translation does.

00:00:29.273 --> 00:00:37.087
Our change in x is going
to be our ending point,

00:00:38.136 --> 00:00:42.939
so negative 203 minus our starting point.

00:00:42.939 --> 00:00:47.725
Minus negative 169.

00:00:47.996 --> 00:00:50.266
This is the same thing as,

00:00:50.266 --> 00:00:51.465
I'll just write it in yellow,

00:00:51.465 --> 00:00:55.869
negative 203 plus 169

00:00:55.869 --> 00:00:59.372
which is the same thing as 169 minus 203

00:00:59.372 --> 00:01:00.375
I'm just writing it this way

00:01:00.375 --> 00:01:02.351
for put my brain to process it,

00:01:02.351 --> 00:01:04.322
which is the same thing as the negative

00:01:04.322 --> 00:01:07.980
of 203 minus 169.

00:01:07.980 --> 00:01:10.182
Let's see this one is the easiest

00:01:10.182 --> 00:01:11.605
for my brain to process

00:01:11.605 --> 00:01:13.645
this is going to be 34.

00:01:13.645 --> 00:01:15.782
This is going to be negative 34

00:01:15.782 --> 00:01:17.052
that's our change in x.

00:01:17.052 --> 00:01:18.915
I was thinking about our change in y.

00:01:18.915 --> 00:01:23.312
Our change in y is our ending point

00:01:23.312 --> 00:01:27.397
is negative 68,

00:01:27.397 --> 00:01:30.810
and our starting point is 434.

00:01:30.810 --> 00:01:33.891
Our ending point minus our starting point,

00:01:33.891 --> 00:01:38.651
minus 434 is going to be equal to,

00:01:38.651 --> 00:01:40.512
so this is just going to be the same thing

00:01:40.512 --> 00:01:46.915
as the negative of 68 plus 434

00:01:46.915 --> 00:01:52.113
and let's see 430 plus 60 is 490,

00:01:52.113 --> 00:01:55.309
plus another eight plus
four, plus another 12

00:01:55.309 --> 00:01:57.611
so that's going to get us to 502.

00:01:57.611 --> 00:02:01.387
This is going to be negative 502.

00:02:01.387 --> 00:02:04.864
This translation, what it does is

00:02:04.864 --> 00:02:08.714
it shifts us in the x
direction by negative 34

00:02:08.714 --> 00:02:12.612
and it shifts us in the y
direction by negative 502.

00:02:12.612 --> 00:02:16.446
If we're starting at
31 comma negative 529,

00:02:16.446 --> 00:02:19.864
we're going to end up at,

00:02:19.864 --> 00:02:21.005
let me write it this way,

00:02:21.005 --> 00:02:22.313
if we're starting at this point,

00:02:22.313 --> 00:02:26.677
we're starting at 31 comma negative 529

00:02:26.677 --> 00:02:31.950
we're going to end up at 31 minus 34

00:02:31.950 --> 00:02:33.369
that's how much we're going to shift it

00:02:33.369 --> 00:02:39.091
and negative 529,

00:02:39.091 --> 00:02:45.927
so negative 529 minus 502.

00:02:45.927 --> 00:02:47.334
What's this going to be?

00:02:47.334 --> 00:02:50.565
31 minus 34 is negative 3

00:02:50.565 --> 00:02:54.998
and then if we have negative 529 minus 502

00:02:54.998 --> 00:02:59.566
that's going to be negative 1031.

00:02:59.566 --> 00:03:02.146
That is where we end up,

00:03:02.146 --> 00:03:05.002
we end up at the point negative 3 comma

00:03:05.002 --> 00:03:07.563
negative 1031,

00:03:07.563 --> 00:03:09.732
under this point gets mapped to this point

00:03:09.732 --> 00:03:11.664
under this translation.


Formal translation tool example

WEBVTT
Kind: captions
Language: en

00:00:00.248 --> 00:00:02.802
- [Voiceover] We're asked
to use the translation tool

00:00:02.802 --> 00:00:05.890
to determine the translation that will map

00:00:05.890 --> 00:00:08.559
triangle WIN, so this right over here,

00:00:08.559 --> 00:00:10.801
onto the other triangle.

00:00:10.801 --> 00:00:14.004
So let's see if we can
translate this thing.

00:00:14.004 --> 00:00:16.047
So I'm gonna use the translation tool.

00:00:16.047 --> 00:00:17.522
So I'm gonna start with the translation.

00:00:17.522 --> 00:00:20.042
So it says translate by,

00:00:20.042 --> 00:00:21.365
and this is gonna say so
how much do we translate

00:00:21.365 --> 00:00:22.606
the x coordinates, and how much

00:00:22.606 --> 00:00:24.462
do we translate the y coordinates?

00:00:24.462 --> 00:00:25.913
Let's see, if I want to map,

00:00:25.913 --> 00:00:29.187
if I want to get point W to correspond

00:00:29.187 --> 00:00:30.674
to this point right over here,

00:00:30.674 --> 00:00:32.392
which it seems like it should,

00:00:32.392 --> 00:00:35.109
I would have to go from x equals two,

00:00:35.109 --> 00:00:38.916
to x equals negative five.

00:00:38.916 --> 00:00:42.166
So my x would have to decrease by seven,

00:00:42.166 --> 00:00:43.688
so let me type that in.

00:00:43.688 --> 00:00:45.615
Negative seven,

00:00:45.615 --> 00:00:47.054
and then, let's see,

00:00:47.054 --> 00:00:49.979
on the y side, and we saw that,

00:00:49.979 --> 00:00:51.136
by just typing in negative seven,

00:00:51.136 --> 00:00:53.702
here we moved it to the left by seven.

00:00:53.702 --> 00:00:57.128
And now, in the y axis,

00:00:57.128 --> 00:00:59.183
I need to move it down by, let's see,

00:00:59.183 --> 00:01:01.237
one, two, three, four.

00:01:01.237 --> 00:01:04.720
So my y coordinates, I
move it down by four.

00:01:04.720 --> 00:01:06.136
And let's see.

00:01:06.136 --> 00:01:07.123
What happens?

00:01:07.123 --> 00:01:09.689
And it looks like I was able
to successfully translate it

00:01:09.689 --> 00:01:11.650
by translating x by negative seven,

00:01:11.650 --> 00:01:13.310
every point here, every point

00:01:13.310 --> 00:01:15.981
on this has been shifted
to the left by seven,

00:01:15.981 --> 00:01:18.071
and has been shifted down by four.

00:01:18.071 --> 00:01:20.381
And I was able to get onto this triangle.

00:01:20.381 --> 00:01:21.947
I was able to, through the translation,

00:01:21.947 --> 00:01:24.630
the image is now in kind
of that target triangle

00:01:24.630 --> 00:01:27.624
that they asked me to map to,

00:01:27.624 --> 00:01:29.180
to translate to.

00:01:29.180 --> 00:01:30.637
We got it right.


Using reflection tool

WEBVTT
Kind: captions
Language: en

00:00:00.301 --> 00:00:03.446
- We're asked to use the
reflect tool to find the image

00:00:03.446 --> 00:00:05.767
of quadrilateral PQRS,

00:00:05.767 --> 00:00:08.125
that's this quadrilateral right over here,

00:00:08.125 --> 00:00:10.085
for a reflection over the line

00:00:10.085 --> 00:00:12.269
y is equal to x plus two.

00:00:12.269 --> 00:00:14.149
All right, so let's use the reflect tool.

00:00:14.149 --> 00:00:15.751
So let me scroll down.

00:00:15.751 --> 00:00:17.551
So let me click on Reflect,

00:00:17.551 --> 00:00:19.732
it brings up this tool,

00:00:19.732 --> 00:00:21.729
and I want to reflect across the line

00:00:21.729 --> 00:00:25.015
y is equal to x plus two.

00:00:25.015 --> 00:00:27.560
So let me move this so it is

00:00:27.560 --> 00:00:29.305
the line y equals x plus two

00:00:29.305 --> 00:00:30.502
and to think about it,

00:00:30.502 --> 00:00:33.337
let's see, it's going to have

00:00:33.337 --> 00:00:34.897
a slope of one, the coefficient

00:00:34.897 --> 00:00:36.348
on the x term is one,

00:00:36.348 --> 00:00:37.706
so it's going to have a slope of one,

00:00:37.706 --> 00:00:38.577
so let me see if I can give this

00:00:38.577 --> 00:00:40.156
a slope of one.

00:00:40.156 --> 00:00:41.190
Is this a slope of one?

00:00:41.190 --> 00:00:42.106
Let me put it a little bit --

00:00:42.106 --> 00:00:43.291
yep, it looks like a slope of one

00:00:43.291 --> 00:00:45.392
as the line moves one to the right.

00:00:45.392 --> 00:00:46.448
We go from one point of the line

00:00:46.448 --> 00:00:48.131
to the other, you have
to go one to the right,

00:00:48.131 --> 00:00:49.572
and one up or two to the right,

00:00:49.572 --> 00:00:50.952
and two up, however much you move

00:00:50.952 --> 00:00:52.612
to the right in the x direction,

00:00:52.612 --> 00:00:54.052
you have to move that same amount

00:00:54.052 --> 00:00:55.666
in the vertical direction.

00:00:55.666 --> 00:00:57.569
So now it has a slope of one,

00:00:57.569 --> 00:00:59.334
and the y intercept is going to be

00:00:59.334 --> 00:01:01.992
the point x equals zero, y equals two,

00:01:01.992 --> 00:01:02.829
we see that right over here.

00:01:02.829 --> 00:01:04.314
When x is equal to zero,

00:01:04.314 --> 00:01:05.534
y is going to be equal to two,

00:01:05.534 --> 00:01:07.078
so let me move this.

00:01:07.078 --> 00:01:09.330
So we see that we now
go through that point.

00:01:09.330 --> 00:01:11.344
When x is equal to zero,

00:01:11.344 --> 00:01:13.015
y is equal to two.

00:01:13.015 --> 00:01:15.692
And now, we just need to reflect

00:01:15.692 --> 00:01:18.749
PQRS, this quadrilateral,

00:01:18.749 --> 00:01:21.362
over this line, so let's do that.

00:01:21.362 --> 00:01:22.964
There you go, we did it.

00:01:22.964 --> 00:01:25.731
The things, the point, like
point S right over here

00:01:25.731 --> 00:01:28.190
that was to the top and left of the line,

00:01:28.190 --> 00:01:30.732
its reflection, the corresponding point

00:01:30.732 --> 00:01:33.066
in the image is now to
the right and the bottom

00:01:33.066 --> 00:01:34.112
of the line.

00:01:34.112 --> 00:01:35.527
The things that were to the right

00:01:35.527 --> 00:01:37.292
and the bottom of our line,

00:01:37.292 --> 00:01:38.593
like point P,

00:01:38.593 --> 00:01:40.775
it's the corresponding
point in the reflection

00:01:40.775 --> 00:01:43.852
is now on the other side of the line.

00:01:43.852 --> 00:01:45.534
So there you go, I think we're done,

00:01:45.534 --> 00:01:47.090
and we got it right.


Using rotation widget

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:02.151
- [Voiceover] Use the
"Rotate" tool to find

00:00:02.151 --> 00:00:05.506
the image of pentagon CDEFG, that's this

00:00:05.506 --> 00:00:07.491
irregular pentagon over here,

00:00:07.491 --> 00:00:10.951
when rotated 270 degrees about the point

00:00:10.951 --> 00:00:13.191
negative two comma three.

00:00:13.191 --> 00:00:15.316
The direction of rotation
by a positive angle

00:00:15.316 --> 00:00:16.930
is counter-clockwise.

00:00:16.930 --> 00:00:19.755
So it's counter-clockwise.

00:00:19.755 --> 00:00:21.848
All right. Rotate tool, to rotate around

00:00:21.848 --> 00:00:23.613
negative two comma three.

00:00:23.613 --> 00:00:26.445
So let's click on "Rotate," and then

00:00:26.445 --> 00:00:28.071
we want to go to the
point, they said rotate

00:00:28.071 --> 00:00:29.893
around the point negative two comma three.

00:00:29.893 --> 00:00:32.470
So I move this, to x equals negative two,

00:00:32.470 --> 00:00:34.072
y equals three.

00:00:34.072 --> 00:00:36.105
That's what we want to rotate around,

00:00:36.105 --> 00:00:38.251
and we want to rotate 270 degrees.

00:00:38.251 --> 00:00:39.691
So this is maybe the part that takes

00:00:39.691 --> 00:00:41.271
the most visualization.

00:00:41.271 --> 00:00:44.104
If we rotate 90 degrees, that would be

00:00:44.104 --> 00:00:46.842
going from, if we think about this arrow,

00:00:46.842 --> 00:00:48.213
actually, let me just do it.

00:00:48.213 --> 00:00:51.533
So, if I move this arrow up here,

00:00:51.533 --> 00:00:54.342
I have rotated 90 degrees,
positive 90 degrees,

00:00:54.342 --> 00:00:55.887
we're moving counter-clockwise.

00:00:55.887 --> 00:00:58.963
Now if I move it over
here, we have rotated

00:00:58.963 --> 00:01:02.945
180 degrees, and now I move it down here.

00:01:02.945 --> 00:01:07.669
I have now rotated it 270 degrees.

00:01:07.669 --> 00:01:12.111
If they said 360, we would
get back to where we started.

00:01:12.111 --> 00:01:17.111
So this is 270 degrees, just like that.

00:01:18.035 --> 00:01:20.229
And, we're done.

00:01:20.229 --> 00:01:23.219
Let's check our answer,
if we got it right.


Translation example

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.440
- Let's do an example

00:00:01.440 --> 00:00:04.544
on the performing translations exercise.

00:00:04.544 --> 00:00:07.121
Use the translate tool to find the image

00:00:07.121 --> 00:00:10.940
of triangle W I N for a
translation of six units,

00:00:10.940 --> 00:00:13.680
positive six units, in the X direction

00:00:13.680 --> 00:00:16.180
and negative three units
in the Y direction.

00:00:16.180 --> 00:00:18.174
Alright, so we wanna go positive six units

00:00:18.174 --> 00:00:20.809
in the X direction and
negative three units

00:00:20.809 --> 00:00:22.322
in the Y direction, alright.

00:00:22.322 --> 00:00:25.085
So, I click on the translate tool.

00:00:25.085 --> 00:00:28.185
Click on the translate
tool and I wanna go,

00:00:28.185 --> 00:00:31.751
so, I wanna go positive six
units in the X direction.

00:00:31.751 --> 00:00:34.055
So, I can pick any point
and go six to the right

00:00:34.055 --> 00:00:36.017
and everything else is gonna come with it.

00:00:36.017 --> 00:00:41.017
So, one, two, three, four, five, six.

00:00:42.229 --> 00:00:45.097
So, I did that part, I
translated positive six units

00:00:45.097 --> 00:00:47.593
in the X direction and
negative three units

00:00:47.593 --> 00:00:50.890
in the Y direction, so everything
needs to go down by three.

00:00:50.890 --> 00:00:55.650
One, two, three and notice,
I focused on point N

00:00:55.650 --> 00:00:58.691
and this is it's image now,
or the image of point N,

00:00:58.691 --> 00:01:01.698
this whole triangle is the
image of this entire triangle,

00:01:01.698 --> 00:01:05.258
the triangle W I N after
the transformation,

00:01:05.258 --> 00:01:08.982
but you see that every point
shifted six to the right,

00:01:08.982 --> 00:01:10.629
six to the right and three down.

00:01:10.629 --> 00:01:13.130
This point over here, six
to the right would take you,

00:01:13.130 --> 00:01:15.688
let's see, it's at one
and a half right now.

00:01:15.688 --> 00:01:17.220
It's X coordinate is one and a half.

00:01:17.220 --> 00:01:19.285
It's new X coordinate is seven and a half,

00:01:19.285 --> 00:01:21.414
so it's X coordinate increased by six

00:01:21.414 --> 00:01:23.867
and it's old Y coordinate,
or the original Y coordinate

00:01:23.867 --> 00:01:25.828
was six and now, in the image,

00:01:25.828 --> 00:01:28.300
the corresponding Y coordinate is three.

00:01:28.300 --> 00:01:31.261
So, it has, we have shifted it down three.

00:01:31.261 --> 00:01:34.773
So, we see that that's
happened to every point here

00:01:34.773 --> 00:01:36.084
and we're done.