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[Music]
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hello and welcome to another engine math
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8 lesson by emathinstruction my name is
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Kirk Weiler and today we're going to be
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doing unit 3 lesson 1 on introduction to
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transformations
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now transformation is kind of a big word
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alright and we're gonna be getting into
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it in this lesson and entire and this
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entire unit is all about transformations
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they're amazingly important in geometry
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you've already played around with them
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just a little bit maybe in some earlier
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grades but this is the first time or the
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first course at least we're really gonna
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kind of dive into them full force so
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without further ado let me bring this
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all the way over here and let's kind of
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get into what a transformation is all
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right so very simply what is a
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transformation a transformation is a
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rule that assigns each point in the
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plane or each point in a geometric
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figure to another point in the plane and
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it could even be the same point all
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right so let's just talk about this
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right so transformations are typically
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some kind of a rule and that rule could
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be given to you algebraically we'll look
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at a few of those today or it could be
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almost kind of verbal like take every
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point in the plane and move it five
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units to the right and two units down
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something like that but the idea is that
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we would take all the points in the
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plane and we will be concerned about
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ones that form some kind of object like
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triangle ABC here and we will map them
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using some rule to somewhere else
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alright and that's a transformation and
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transformations you know depending on
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what kind of a transformation it is have
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various properties and that's what this
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entire units gonna be about is looking
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at different types of transformations
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learning their names learning their
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properties and then connecting them with
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something we saw in the last unit which
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was congruent figures we're gonna be
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doing a little bit of that connection
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even today but let's get right into it
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with exercise number one by taking a
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look at it exercise one triangle ABC is
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shown in the diagram below the original
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geometric object is known as the
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pre-image letter a asks us to state the
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coordinates of the vertices of the
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preimage below alright so the original
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object that you begin with is called the
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preimage the one that you get after you
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transform it not surprisingly is called
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the image what I want to know just right
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away is where our points a B and C so I
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just have to read the coordinates off of
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this grid let's do a together then we'll
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have you do B and C quickly on your own
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but point a right is two units to the
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right of the origin and then nine units
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up so it's it's simple enough for us to
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just say all right we've got a at the
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point 2 comma 9 what I'd like you to do
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is figure out where point B is and where
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Point C is and write their coordinates
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down that should be pretty easy why
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don't you go ahead take a few moments
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and read off their coordinate points all
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right let's do it
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well point B right we're going to go 4
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units to the right and we're gonna go 13
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units up so point B is that for 13 point
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C right we're going to move 8 units to
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the right and 8 units up so Point C is
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one of those points where the X and the
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y coordinate of the same eight comma
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eight now again we talked about this in
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the last unit isn't amazingly important
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for you to be able to very accurately
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read off these coordinates if you're
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thinking well I had the right things I
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just flipped the X and the y that is not
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a minor mistake you've got to get on top
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of that and you've got to get on top of
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it early given that you've been plotting
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points in the coordinate plane since
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fifth grade fifth grade that's when you
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start doing it so we got to have these
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things down now let's look at a
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transformation all right I'm gonna bring
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this just up a little bit letter B the
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following transformation rule is used to
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map triangle ABC and here we have X
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comma Y with a little arrow X plus five
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comma Y minus seven state the
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coordinates of the vertices of the new
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triangle a prime B prime C prime all
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right so let's first talk about cuz
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there's a lot of stuff that's new in
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this like little problem and I don't
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want to gloss any of it over first let's
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talk about the transformation rule this
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is a nice algebraic rule that
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basically just says hey no matter what
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your XY coordinate point is I want you
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to transform it by taking its
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x-coordinate and adding five and taking
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its y-coordinate and subtracting seven
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now what that means is that a given
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point like point a right which had
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coordinates at two comma nine will get
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mapped that's what that arrow means
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point a is going to move to point a
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prime and specifically it's going to
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move to a point where it's two plus five
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right that's the x coordinate is 2 plus
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five and then 9 minus seven
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right so a prime is going to be at the
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point seven comma two very often when we
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map a pre-image to an image if the
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original sort of point is a then we have
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a with a little mark it looks like a
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little ' mark we call it a prime all
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right and if there's two of them we call
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it double prime and three of them we
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call it triple prime anyway don't worry
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about that for right now but a at two
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comma nine went to two plus five because
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that's the rule and nine minus seven
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because that's the rule what I'd like
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you to do now is use the rule to figure
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out where B goes where C goes plot them
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right in the plane right well actually
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that's Part C so just figure out where B
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and C go right and then in letter C
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we'll actually plot them and draw the
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new triangle go ahead and spend a few
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moments doing that all right let's go
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through well first let me kind of go
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back up to here point B was at 4 comma
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13 right so that preimage point gets
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mapped to B Prime all right 4 plus 5 is
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going to be 9 and 13 minus 7 is going to
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be 6 so B prime is gonna be at 9 comma 6
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and then let's take a look C is at 8 8
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so let's do C which is that a 8 will get
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mapped to C Prime the first 8 we're
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going to add 5 2 and that's going to be
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13 the second eight we're gonna subtract
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7 from and
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gonna get one alright so we've got the
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vertices of our new triangle at seven
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two nine six and thirteen one so we're
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ready for letter C where it says draw
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triangle a prime B prime C prime in the
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coordinate plane this is known as the
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image of the transformation alright well
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this is as simple as it's going to be
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I'd like you to plot a prime B prime C
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prime make sure to label them without a
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and a little mark a be a little mark C a
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little mark then take your straight edge
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connect the three points and that will
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be our image triangle why don't you
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pause the video now and go ahead and do
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that all right well let me go through it
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I'm just gonna scroll up just a bit so
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we can see it let's see if we have a
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prime at seven comma two so that's gonna
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be right here right we've got B prime at
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nine comma six so that's gonna be right
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here and we have C prime at 13 comma one
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right then we're gonna take out our
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straightedge again a faster process for
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you than likely for me but I feel like I
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I really have to have these things
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straight let me go up here real quick
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and finally it almost worked there all
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right and that's my my image triangle
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this just drives me nuts I can never
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leave my my ruler kind of slanted like
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that I don't know why okay final
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question letter D are the two triangles
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congruent use tracing paper to decide
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alright so the question was you know
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really after I did this transformation a
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triangle got mapped to a triangle that's
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awesome the question is are they
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congruent are they identical triangles
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so what I'd like you to do is I'd like
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you to actually trace this one out like
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I've done in red and then see if it lies
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directly on top of this one pause the
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video now and do that test
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all right
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well hopefully you found the answer to
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be yes because when I actually trace
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that triangle out and I put it down here
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maybe it doesn't look the best but it
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definitely lies on top of it so the
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answer is most certainly yes all right
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and that's gonna bring us to a very
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important topic on the next page okay
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transformations that result in images
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identical in shape and size to their pre
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images are known as rigid motions think
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about what the word rigid means
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now maybe you don't know but maybe you
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got a teacher that's really rigid not
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your match or somebody else obviously
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but you know rigid means an unchanging
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not willing to change or kind of set in
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its ways right
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so a rigid motion is any kind of a
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transformation that doesn't change the
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size nor the shape of the object in
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other words it produces an image that is
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congruent to the pre image so you know
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here we've got the Eiffel Tower the two
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I fed and you know maybe I do something
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to it maybe I rotate it or something
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like that
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this would be a rigid motion because
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ultimately speaking this and this even
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though they're in different orientations
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they have the same size they have the
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same shape on the other hand we can
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certainly have transformations of the
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Eiffel Tower
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then aren't rigid motions so let's say I
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took this thing and I I stretched it
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horizontally right but maybe I didn't
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mess with it at all vertically all of a
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sudden we get an image of the Eiffel
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Tower
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that is not congruent to the original
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it's not identical to it
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this concept and this piece of
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terminology is exceptionally important
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rigid motions are transformations that
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do not change the size of the shape of
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the image and therefore the image is
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congruent to the pre image let's work
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with this a little bit more in the next
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exercise
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all right here we go back to the
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coordinate plane exercise number two
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square ABCD is shown
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below a transformation is given by the
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rule X comma Y gets mapped to two x
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comma y plus one letter a asks us to
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state the coordinates of the vertices of
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the preimage all right so this is very
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similar to what we did before what I'd
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like you to do is pause the video write
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down the coordinates of points a b c and
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d right up here be careful all four
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points have very similar coordinates but
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subtly different that will make a
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difference once we start applying that
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rule take a few minutes to write down
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the coordinates of a b c and d alright
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here we go
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well Point a we get to by going left of
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the origin three and up from the origin
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three so point a has got coordinate
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negative three three point B on the
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other hand we get to by going three
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units to the right of the origin and
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three units up from the origin so point
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B has coordinates of three comma three
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Point C right we get there by going
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right three and down three so it's got
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coordinates of positive 3 negative three
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and D has coordinates of negative three
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negative three again it's very very
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important if you didn't get these
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completely right I want you to go back
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and I want you to really think about
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them you've got to be able to read off
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coordinate points from the coordinate
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grid now predictably let's take a look
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at letter B show where each of the
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vertices is mapped in the image a prime
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B prime C prime D prime and then draw a
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prime B prime C prime D prime alright
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let's do this together for point a and
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then we'll have you do points B C and D
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on your own so here we go a which is at
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negative 3 comma 3 where does it go well
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it goes to its image point a prime and
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what does the rule tell us let's let's
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really take a look at the rule the rule
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says here take the x-coordinate and map
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it to 2 times the x coordinate right
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that's what 2x means so if we start
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negative three and we do negative 3
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times positive 2 we will get negative 6
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right that's just what I get when I
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multiply a negative 3 by positive 2
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on the other hand the y-coordinate I'm
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gonna take the y-coordinate and just add
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1 to it well I'm starting at a
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y-coordinate of 3 and when I add 1 to it
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I should be at a y-coordinate of 4 so
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there it is there's the image of point a
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a prime is at negative 6 comma 4 now be
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careful a little bit when you add 1 to
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the y-coordinate especially when the
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y-coordinate is negative 3 hint negative
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3 plus 1 is not negative 4 anyway why
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don't you pause the video now and work
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through where points B C and D all get
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mapped to alright let's do it
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here we go so B this is probably the
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easiest one 3 comma 3 right B prime when
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I multiply the X by 2 I'm gonna get 6
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and when I add 1 to the Y I'm gonna get
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4 pretty easy there on the other hand C
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which is that 3 comma negative 3 this
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gets a little bit trickier because
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although multiplying 3 by 2 and getting
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6 is simple negative 3 plus positive 1
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is actually just negative 2 right
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there's where we got to be careful and
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not say negative 4 on the other hand D
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which is at negative 3 comma negative 3
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right that will go to D prime which will
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be negative 6 comma negative 2 we were
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also asked to draw a prime B prime C
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prime D prime so let's go ahead and do
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that a prime is that negative 6 comma
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positive 4 so there's a prime right B
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prime is that positive 6 positive 4 C
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prime is that positive 6 negative 2 and
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D prime is that negative 6 negative 2
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now I could take out a straightedge
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right now and kind of connect these but
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I'm just gonna freehand it to save
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ourselves a little bit of time I really
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again should use
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a straightedge but good enough there all
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right and there is our image letter C
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should be an easy easy question to
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answer now
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is this transformation given by that
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rule a rigid motion explained all right
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well go for it pause the video now and
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just answer that simple question yes no
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is it a rigid motion and then explain
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your choice and the answer is most
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certainly no right now there is a lot of
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different ways that you can justify No
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you know one way you could justify knows
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you could say well I traced out ABCD on
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a piece of tracing paper and and it
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doesn't lie directly on top of the image
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you could go with that rationale you
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could also go with a rationale and I
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like this a lot
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ABCD is a square and yet a-prime b-prime
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c-prime d-prime
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is clearly not a square right right
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these two shapes are simply not the same
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they're both rectangles but ABC is a
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square where as a prime B prime C prime
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D prime is most certainly not a square
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and I think that's where I'm gonna go
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with I'm gonna say a B C D is a square
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while a prime B prime C prime D prime is
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not by the way please be careful you
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know it would be very easy to say
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something like ABCD is a square while a
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prime B prime C prime D prime is a
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rectangle now that would be true but all
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squares are also rectangles so saying
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ABCD is a square and a prime B prime C
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prime D prime is a rectangle that
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doesn't explain them not being congruent
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given that all squares are rectangles
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but saying that ABCD is a square wallis
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it's image a prime B prime C prime D
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prime is not that's perfectly good
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alright let's take a look at one last
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problem now there are lots of
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transformations that we're going to
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expose you to in the next
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few lessons one of them is going to be a
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reflection so we just thought we'd throw
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in a reflection right at the end given
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that we're going to be doing those in
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the next lesson so that we could get a
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little work with tracing paper which
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you're definitely going to need on this
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one and we're also not in the coordinate
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plane here right everything that we've
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been doing so far with those
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transformation rules have been in the
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coordinate plane but all of these
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transformations actually can also occur
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in what's called the Euclidean plane so
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no no coordinates whatsoever let's take
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a look at exercise number three right
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triangle d EF is shown below along with
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segment a-b use tracing paper to help
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with this problem letter a trace both
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triangle d EF and triangle am sorry line
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segment a B onto the paper all right
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so I literally want you to take your
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tracing paper hopefully it's not this
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big that's how big I would need and I
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want you to lay it down on top of this
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picture
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I then want you to trace out d EF and i
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would suggest labeling d EF on you're
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tracing paper and also trace out line
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segment a B and also also label a and B
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ok take a moment to do that all right
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now what we're gonna do to do the
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reflection is really kind of cool take
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your tracing paper and literally flip it
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right just flip it over and then if you
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flip it over it's gonna look something
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like that and then what I want you to do
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is lie a B on top of itself again
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probably easier for you to do than for
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me to do right once you lie a B on top
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of itself then letter C draw triangle D
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prime B prime F prime after a reflection
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across a B well the whole point is right
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this now is where D EF is gonna lie so
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you know I could I could take my my my
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pencil out trace this thing out right
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this would definitely be prime this
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would be D prime this would
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f-prime and in fact one nice way to do
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this is because if you're using pencil
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not pen pencil works really well on this
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if you just kind of trace over that
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image then the the carbon from the
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pencil will actually kind of make a
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light image of that on your paper and
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then when you pull this away you can
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always kind of draw it in a little bit
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darker but there is that image not
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surprisingly right a reflection is a
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rigid motion right it's a rigid motion
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if we now took this thing we would find
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that these two triangles are absolutely
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identical to each other in terms of
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their angle measures and their side
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length measures we're gonna do more work
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with reflections in the next lesson but
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let's let's summarize this one alright
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so today we looked at transformations
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and all transformations are our rules
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that take a figure in the plane and they
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map it to another figure in the plane
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the original figure is called the
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preimage and the final figure is called
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the image oftentimes they have vertices
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that have the same letters but whereas
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the first one might be a BC the second
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one the image might be a-prime b-prime
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c-prime
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right we saw that some transformations
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preserve both the size of the object and
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its shape those are called rigid body
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motions and when you have a rigid motion
00:21:09
then the image and the preimage are
00:21:12
congruent to each other we also saw a
00:21:14
transformation that took a square and
00:21:16
changed it into something that was no
00:21:18
longer a square so not every
00:21:19
transformation is a rigid motion but the
00:21:22
most important ones typically are and
00:21:25
we're gonna look at reflections which
00:21:27
are an example of a rigid motion a lot
00:21:28
more in the next exercise not in the
00:21:30
next exercise in the next lesson because
00:21:32
we're done with this one no more
00:21:34
exercises in this lesson today I'd like
00:21:37
to just thank you for joining me though
00:21:38
for another engine math 8 lesson by
00:21:42
emathinstruction my name is Kirk Weiler
00:21:44
and until next time keep thinking and
00:21:46
keep solving problems
