N-Gen Math 8.Unit 3.Lesson 1.Introduction to Transformations

00:21:55
https://www.youtube.com/watch?v=YveE67oXWU4

Résumé

TLDRCette vidéo de cours, dirigée par Kirk Weiler pour emathinstruction, est la première leçon de l'unité sur les transformations géométriques. Les transformations sont définies comme des règles qui associent chaque point d'une figure géométrique à un autre point du plan, et peuvent être exprimées algébriquement ou verbalement, à l'exemple d'une translation simple. Le cours explique également la notion de transformations rigides, c'est-à-dire celles qui conservent la taille et la forme des figures, assurant ainsi la congruence entre l'objet d'origine et celui transformé. Plusieurs exemples sont travaillés, incluant des manipulations de carrés et de triangles dans le plan cartésien. Une distinction est faite entre les types de transformations qui préservent ou non la congruence géométrique, introduisant des notions de base de la géométrie moderne.

A retenir

  • 📚 Introduction aux transformations en géométrie.
  • 🔄 Une transformation associe chaque point d'une figure à un autre.
  • 🟦 Les transformations rigides préservent la taille et la forme.
  • 🔍 Utilisation de papier calque pour vérifier la congruence.
  • 📝 Explication via des exercices pratiques dans le plan cartésien.
  • 🔀 Exemples de translations, déchets et modifications.
  • 🟰 Importance de la congruence dans les transformations rigides.
  • 📏 Comparaison entre les figures préimage et image.
  • 🔄 Travail sur les réflexions prévu dans la prochaine leçon.
  • 📐 Essentiel pour comprendre les mouvements et changements en géométrie.

Chronologie

  • 00:00:00 - 00:05:00

    Dans cette leçon introductive sur les transformations, Kirk Weiler explique que celles-ci sont des règles assignant chaque point du plan à un autre point. L'objectif est de comprendre différents types de transformations en géométrie et leurs propriétés, ainsi que de les connecter aux figures congruentes. L'exercice initial consiste à déterminer les coordonnées des sommets d'un triangle ABC et à appliquer une règle de transformation donnée pour découvrir les nouveaux sommets A', B' et C'.

  • 00:05:00 - 00:10:00

    Kirk montre comment appliquer la transformation aux points du triangle original pour obtenir les nouveaux points A', B' et C'. Il explique que le nouvel ensemble de points forme une image qui est congruente au triangle original, soulignant l'importance de comprendre comment lire et appliquer les coordonnées. Les élèves doivent ensuite dessiner le nouveau triangle en utilisant ces nouveaux points.

  • 00:10:00 - 00:15:00

    Une nouvelle transformation est introduite : x, y est mappé à 2x, y+1. Il s'agit de déterminer les nouvelles positions des sommets d'un carré et de voir si cette transformation est un mouvement rigide. Kirk démontre que le nouvel objet n'est pas congruent à l'original car sa forme diffère, illustrant une transformation non rigide et expliquant l'importance de cette distinction.

  • 00:15:00 - 00:21:55

    Enfin, les étudiants explorent une réflexion avec du papier calque pour visualiser comment une image est réfléchie à travers un segment de ligne. Cette réflexion est un mouvement rigide qui conserve la forme et la taille de l'objet original. La leçon conclut sur l'importance des transformations rigides comme les réflexions, qui préservent la congruence, et prépare à des leçons futures approfondissant ces concepts.

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Carte mentale

Vidéo Q&R

  • Qu'est-ce qu'une transformation en géométrie ?

    Une transformation en géométrie est une règle qui associe chaque point d'une figure à un autre point dans le plan. Il existe différents types de transformations comme les translations, les rotations, et les réflexions.

  • Qu'est-ce qu'une transformation rigide ?

    Les transformations rigides sont celles qui ne changent ni la taille ni la forme de l'objet initial, rendant ainsi les objets de départ et d'arrivée congruents.

  • Est-ce que toutes les transformations rigides conservent la congruence entre les figures ?

    Oui, l'une des propriétés des transformations rigides est de maintenir les figures congruentes, donc les triangles de départ et d'arrivée sont identiques en taille et en forme.

  • Que se passe-t-il lors d'une translation d'un triangle sur le plan cartésien ?

    S'il s'agit d'une translation où les coordonnées de chaque point sont modifiées de manière à augmenter l'une et diminuer l'autre, le triangle pourrait être simplement déplacé, mais pas nécessairement déformé.

  • Comment peut-on vérifier si deux figures après transformation sont congruentes ?

    Pour tester la congruence après une transformation, on peut utiliser du papier calque pour trasposer et vérifier si les images coïncident parfaitement.

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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
  • 00:10:10
    though they're in different orientations
  • 00:10:12
    they have the same size they have the
  • 00:10:14
    same shape on the other hand we can
  • 00:10:16
    certainly have transformations of the
  • 00:10:19
    Eiffel Tower
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    then aren't rigid motions so let's say I
  • 00:10:22
    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
  • 00:10:32
    it's not identical to it
  • 00:10:34
    this concept and this piece of
  • 00:10:36
    terminology is exceptionally important
  • 00:10:38
    rigid motions are transformations that
  • 00:10:41
    do not change the size of the shape of
  • 00:10:44
    the image and therefore the image is
  • 00:10:47
    congruent to the pre image let's work
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    with this a little bit more in the next
  • 00:10:52
    exercise
  • 00:10:53
    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
  • 00:11:13
    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
  • 00:13:14
    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
  • 00:13:22
    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
  • 00:13:39
    a prime is at negative 6 comma 4 now be
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    careful a little bit when you add 1 to
  • 00:13:44
    the y-coordinate especially when the
  • 00:13:46
    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
  • 00:14:04
    here we go so B this is probably the
  • 00:14:07
    easiest one 3 comma 3 right B prime when
  • 00:14:12
    I multiply the X by 2 I'm gonna get 6
  • 00:14:14
    and when I add 1 to the Y I'm gonna get
  • 00:14:16
    4 pretty easy there on the other hand C
  • 00:14:20
    which is that 3 comma negative 3 this
  • 00:14:22
    gets a little bit trickier because
  • 00:14:24
    although multiplying 3 by 2 and getting
  • 00:14:26
    6 is simple negative 3 plus positive 1
  • 00:14:30
    is actually just negative 2 right
  • 00:14:32
    there's where we got to be careful and
  • 00:14:34
    not say negative 4 on the other hand D
  • 00:14:36
    which is at negative 3 comma negative 3
  • 00:14:39
    right that will go to D prime which will
  • 00:14:42
    be negative 6 comma negative 2 we were
  • 00:14:47
    also asked to draw a prime B prime C
  • 00:14:50
    prime D prime so let's go ahead and do
  • 00:14:52
    that a prime is that negative 6 comma
  • 00:14:54
    positive 4 so there's a prime right B
  • 00:15:00
    prime is that positive 6 positive 4 C
  • 00:15:06
    prime is that positive 6 negative 2 and
  • 00:15:11
    D prime is that negative 6 negative 2
  • 00:15:14
    now I could take out a straightedge
  • 00:15:17
    right now and kind of connect these but
  • 00:15:19
    I'm just gonna freehand it to save
  • 00:15:20
    ourselves a little bit of time I really
  • 00:15:25
    again should use
  • 00:15:26
    a straightedge but good enough there all
  • 00:15:30
    right and there is our image letter C
  • 00:15:33
    should be an easy easy question to
  • 00:15:35
    answer now
  • 00:15:36
    is this transformation given by that
  • 00:15:39
    rule a rigid motion explained all right
  • 00:15:43
    well go for it pause the video now and
  • 00:15:46
    just answer that simple question yes no
  • 00:15:48
    is it a rigid motion and then explain
  • 00:15:50
    your choice and the answer is most
  • 00:15:57
    certainly no right now there is a lot of
  • 00:16:02
    different ways that you can justify No
  • 00:16:04
    you know one way you could justify knows
  • 00:16:06
    you could say well I traced out ABCD on
  • 00:16:08
    a piece of tracing paper and and it
  • 00:16:10
    doesn't lie directly on top of the image
  • 00:16:12
    you could go with that rationale you
  • 00:16:14
    could also go with a rationale and I
  • 00:16:15
    like this a lot
  • 00:16:17
    ABCD is a square and yet a-prime b-prime
  • 00:16:21
    c-prime d-prime
  • 00:16:23
    is clearly not a square right right
  • 00:16:27
    these two shapes are simply not the same
  • 00:16:29
    they're both rectangles but ABC is a
  • 00:16:32
    square where as a prime B prime C prime
  • 00:16:35
    D prime is most certainly not a square
  • 00:16:38
    and I think that's where I'm gonna go
  • 00:16:40
    with I'm gonna say a B C D is a square
  • 00:16:47
    while a prime B prime C prime D prime is
  • 00:16:56
    not by the way please be careful you
  • 00:17:00
    know it would be very easy to say
  • 00:17:02
    something like ABCD is a square while a
  • 00:17:05
    prime B prime C prime D prime is a
  • 00:17:07
    rectangle now that would be true but all
  • 00:17:10
    squares are also rectangles so saying
  • 00:17:13
    ABCD is a square and a prime B prime C
  • 00:17:16
    prime D prime is a rectangle that
  • 00:17:17
    doesn't explain them not being congruent
  • 00:17:20
    given that all squares are rectangles
  • 00:17:21
    but saying that ABCD is a square wallis
  • 00:17:25
    it's image a prime B prime C prime D
  • 00:17:27
    prime is not that's perfectly good
  • 00:17:30
    alright let's take a look at one last
  • 00:17:34
    problem now there are lots of
  • 00:17:35
    transformations that we're going to
  • 00:17:36
    expose you to in the next
  • 00:17:38
    few lessons one of them is going to be a
  • 00:17:40
    reflection so we just thought we'd throw
  • 00:17:42
    in a reflection right at the end given
  • 00:17:44
    that we're going to be doing those in
  • 00:17:45
    the next lesson so that we could get a
  • 00:17:47
    little work with tracing paper which
  • 00:17:48
    you're definitely going to need on this
  • 00:17:49
    one and we're also not in the coordinate
  • 00:17:52
    plane here right everything that we've
  • 00:17:54
    been doing so far with those
  • 00:17:55
    transformation rules have been in the
  • 00:17:56
    coordinate plane but all of these
  • 00:17:58
    transformations actually can also occur
  • 00:18:01
    in what's called the Euclidean plane so
  • 00:18:02
    no no coordinates whatsoever let's take
  • 00:18:05
    a look at exercise number three right
  • 00:18:08
    triangle d EF is shown below along with
  • 00:18:11
    segment a-b use tracing paper to help
  • 00:18:13
    with this problem letter a trace both
  • 00:18:16
    triangle d EF and triangle am sorry line
  • 00:18:21
    segment a B onto the paper all right
  • 00:18:23
    so I literally want you to take your
  • 00:18:25
    tracing paper hopefully it's not this
  • 00:18:27
    big that's how big I would need and I
  • 00:18:29
    want you to lay it down on top of this
  • 00:18:31
    picture
  • 00:18:31
    I then want you to trace out d EF and i
  • 00:18:34
    would suggest labeling d EF on you're
  • 00:18:37
    tracing paper and also trace out line
  • 00:18:40
    segment a B and also also label a and B
  • 00:18:44
    ok take a moment to do that all right
  • 00:18:52
    now what we're gonna do to do the
  • 00:18:54
    reflection is really kind of cool take
  • 00:18:56
    your tracing paper and literally flip it
  • 00:19:01
    right just flip it over and then if you
  • 00:19:06
    flip it over it's gonna look something
  • 00:19:07
    like that and then what I want you to do
  • 00:19:09
    is lie a B on top of itself again
  • 00:19:15
    probably easier for you to do than for
  • 00:19:17
    me to do right once you lie a B on top
  • 00:19:20
    of itself then letter C draw triangle D
  • 00:19:23
    prime B prime F prime after a reflection
  • 00:19:26
    across a B well the whole point is right
  • 00:19:28
    this now is where D EF is gonna lie so
  • 00:19:32
    you know I could I could take my my my
  • 00:19:36
    pencil out trace this thing out right
  • 00:19:42
    this would definitely be prime this
  • 00:19:45
    would be D prime this would
  • 00:19:49
    f-prime and in fact one nice way to do
  • 00:19:52
    this is because if you're using pencil
  • 00:19:55
    not pen pencil works really well on this
  • 00:19:57
    if you just kind of trace over that
  • 00:20:00
    image then the the carbon from the
  • 00:20:03
    pencil will actually kind of make a
  • 00:20:05
    light image of that on your paper and
  • 00:20:08
    then when you pull this away you can
  • 00:20:10
    always kind of draw it in a little bit
  • 00:20:12
    darker but there is that image not
  • 00:20:14
    surprisingly right a reflection is a
  • 00:20:17
    rigid motion right it's a rigid motion
  • 00:20:19
    if we now took this thing we would find
  • 00:20:21
    that these two triangles are absolutely
  • 00:20:24
    identical to each other in terms of
  • 00:20:26
    their angle measures and their side
  • 00:20:28
    length measures we're gonna do more work
  • 00:20:31
    with reflections in the next lesson but
  • 00:20:34
    let's let's summarize this one alright
  • 00:20:36
    so today we looked at transformations
  • 00:20:38
    and all transformations are our rules
  • 00:20:40
    that take a figure in the plane and they
  • 00:20:44
    map it to another figure in the plane
  • 00:20:46
    the original figure is called the
  • 00:20:48
    preimage and the final figure is called
  • 00:20:50
    the image oftentimes they have vertices
  • 00:20:52
    that have the same letters but whereas
  • 00:20:54
    the first one might be a BC the second
  • 00:20:56
    one the image might be a-prime b-prime
  • 00:20:58
    c-prime
  • 00:20:59
    right we saw that some transformations
  • 00:21:01
    preserve both the size of the object and
  • 00:21:04
    its shape those are called rigid body
  • 00:21:07
    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
Tags
  • mathématiques
  • transformation
  • géométrie
  • mouvements rigides
  • congruence
  • plan cartésien
  • éducation
  • mathématiques collégiales