Geometric Transformations - Translations
Summary
TLDRThis lesson focuses on transformations in geometry, explaining how shapes or lines can be shifted, reflected, rotated, or resized. The four main types of transformations covered are translation (sliding), reflection (flipping), rotation (turning), and dilation (resizing). The key concept is how congruency and similarity are maintained, except for dilation, which changes the size but not the shape. The video includes practical examples and explanations of how to perform translations on coordinate planes, emphasizing accuracy in plotting points and ensuring shapes retain their original properties.
Takeaways
- 📐 Transformations in geometry involve changing the position or size of a shape.
- 🔄 There are four main types of transformations: translation, reflection, rotation, and dilation.
- ➡️ Translations, also known as slides, involve moving a shape a fixed distance in a specific direction without changing its size or orientation.
- 🪞 Reflections, or flips, involve mirroring a shape across a line like the x-axis or y-axis.
- 🔄 Rotations involve turning a shape around a point, like a wheel rotates.
- 📏 Dilations expand or shrink a shape, changing its size but keeping the same proportions.
- ✅ The first three transformations (translation, reflection, and rotation) maintain congruency (same shape, same size).
- ↕️ A prime symbol (′) is used to indicate a transformed point, and double or triple primes are used for subsequent transformations.
- 📏 For translations, all points of the shape move the same distance and in the same direction, preserving congruency.
- ✏️ Transformations can be performed by adding/subtracting values from coordinates, counting on a graph, or using a reference point.
Q & A
What is a transformation in geometry?
-A transformation in geometry refers to changing the position, size, or orientation of a shape. There are four types of transformations: translation, reflection, rotation, and dilation.
What is the difference between a translation and a reflection?
-A translation, also known as a slide, moves a shape to a different position without changing its orientation. A reflection, also known as a flip, mirrors a shape across a line, such as the x-axis or y-axis.
What property is preserved in translations, reflections, and rotations?
-Translations, reflections, and rotations preserve congruency, meaning the shape maintains the same size and shape, as only its position changes.
How does a dilation differ from other transformations?
-A dilation changes the size of a shape, either shrinking or expanding it, while keeping the same shape. Unlike translations, reflections, and rotations, it does not preserve congruency, but it does preserve similarity.
What does the 'prime' symbol (') indicate in transformations?
-The 'prime' symbol (') is used to denote the new position of a shape or point after a transformation. For example, point A becomes A' after a translation. If there is another transformation, it could become A''.
How do you perform a translation by a specific number of units?
-To perform a translation, you move each point of a shape by a fixed distance in a given direction. For example, to move a shape 12 units to the right, you add 12 to the x-coordinates of all the points while leaving the y-coordinates unchanged.
What are two methods for performing a translation?
-One method is to add or subtract a specific value to the coordinates of each point. Another method is to visually count the units on a graph, shifting each point by the desired number of units.
What does it mean when a translation is described as (x + 12, y)?
-This notation indicates that the translation moves each point of the shape 12 units to the right along the x-axis, while the y-coordinate remains unchanged.
How can you verify that a translation has been done correctly?
-You can verify a translation by checking that all points of the shape have been moved the same distance in the same direction. If one point is off, the shape will no longer be congruent.
Can you perform a translation in any order when moving both horizontally and vertically?
-Yes, for simple translations that involve moving along both the x and y axes, the order doesn't matter. Whether you move horizontally first or vertically first, the result will be the same.
Outlines
🔄 Introduction to Transformations in Geometry
The speaker introduces the concept of transformations in geometry, defining it as the process of changing the position of shapes or lines. Transformations include translation (slide), reflection (flip), rotation (turn), and dilation (resize). The first three maintain congruency (same shape and size), while dilation alters the size. The speaker uses examples like mirrors and carnival mirrors to illustrate transformations that do or do not maintain congruency.
📐 Detailed Explanation of Translations
The focus shifts to translations, a type of transformation where an object is moved a fixed distance in a specific direction without changing its shape or size. The speaker explains how translations can be understood using a step-by-step method. Examples include shifting points by adding or subtracting from their coordinates, with various techniques for visualizing and performing the translation. A key point is that all parts of the shape must move uniformly to maintain congruency.
📊 Practical Approach to Plotting Translations
This paragraph provides a step-by-step example of plotting a translation on a graph, moving points 12 units to the right. The speaker discusses how to accurately count the units and check for consistency to avoid errors. Different methods are suggested, including manual counting, using right triangles, and adjusting coordinates mathematically. The speaker emphasizes checking the correctness by ensuring that all points have moved uniformly.
📉 Performing and Verifying Two-Step Translations
The speaker introduces a two-step translation, where the object is moved both horizontally (12 units to the right) and vertically (8 units down). They explain how performing these movements sequentially results in the same final position, regardless of the order. Emphasis is placed on maintaining congruency throughout the translation process, ensuring the shape stays the same despite the movement.
🔢 Listing Coordinates and Plotting the New Figure
This section walks through the process of listing coordinates for points on a figure, performing a translation by adding 8 to each x-coordinate, and plotting the new figure. The speaker highlights the relationship between coordinates and the visual graph, reinforcing the concept of congruency and the impact of coordinate changes on the figure's position.
🧮 Subtracting from Coordinates for Vertical Translation
A new translation task is introduced, where the shape is moved vertically by subtracting 5 from each y-coordinate. The speaker provides instructions for performing the translation, ensuring that the figure maintains its shape and size after the vertical movement. The exercise highlights how consistent transformations preserve congruency.
🗺️ Translating a Right Triangle by Adjusting Coordinates
The speaker presents an example of translating a right triangle by subtracting 7 from the x-coordinates and adding 10 to the y-coordinates. This translation moves the triangle to a new position on the graph. Different approaches are suggested, including calculating new coordinates and manually plotting the points. The speaker emphasizes accuracy and consistency in performing the translation.
🐒 Translating the 'Chimp' Shape with Coordinate Adjustments
In this paragraph, the speaker introduces the shape labeled 'CHIMP' and walks through a translation process that involves subtracting 1 from the x-coordinates and 10 from the y-coordinates. The goal is to move the entire shape while maintaining its structure. The speaker discusses verifying the new coordinates and ensuring that the translation was performed correctly, with no distortion of the figure.
Mindmap
Keywords
💡Transformation
💡Translation
💡Reflection
💡Rotation
💡Dilation
💡Congruency
💡Prime Notation
💡Similarity
💡Fixed Distance
💡Coordinates
Highlights
Transformation in geometry means changing the position of a shape from one location to another.
There are four main types of transformations: translation (slide), reflection (flip), rotation (turn), and dilation (resize).
Translation is like moving a shape to a different place without changing its size, shape, or orientation.
Reflections flip a shape over a line, such as across the x-axis, y-axis, or other lines, like a mirror image.
Rotations involve turning the shape around a specific point, without changing its size or shape.
Dilations change the size of the shape, either expanding or shrinking, while keeping the same proportions.
Translations maintain congruency, meaning the shape's size and structure remain identical.
In transformations, the notation uses the prime symbol (like a tally mark) to show the movement of points.
When translating points, simply add or subtract values to the coordinates to move them horizontally or vertically.
It's important to ensure all points move equally to maintain congruency in the shape.
In two-step translations, you can move points in two directions, such as right and down, and it doesn't matter which is done first.
Students should carefully check their steps, ensuring each point moves exactly the right amount to preserve congruency.
A transformation using x + 12 moves all points 12 units to the right, while keeping their vertical positions.
In reflection or dilation transformations, the order of steps might matter, but not for translations.
Using coordinate points makes it easier to track translations accurately, avoiding small errors like moving points only 11 instead of 12 units.
Transcripts
all right everybody what we're going to
be talking about today are
transformations again Transformations
are things that we do to shapes or lines
in geometry
class first off what is a transformation
so a transformation can be one of four
different kinds of Transformations we're
going to talk about in here but mainly
what it
is is it basically transform means to
change and in Geometry it means we're
changing the position of a shape um from
one position to another so it could also
mean we are um expanding the shape or
shrinking it but again that would still
change the position of all the
points so first off we're going to talk
about translations also known as a slide
I like to think of it as um you know
when you're translating a word from one
to another it means the same thing it's
just in a different place like a
different language so that's what we're
going to be talking about today which is
translations you have Reflections which
you I think you know reflection in the
mirror so that's a common one um we also
call that sometimes a flip because we're
flipping over the um the shape and we're
doing it across something like across
the x axis across the y- axis across a
line can be done a lot of different
ways a rotation which is also known as a
turn and as you know you know Wheels
rotate rotation means to
turn and the last last one is a dilation
the first three translation reflection
and rotation those ones they maintain
congruency because all we're doing is
moving the thing turning the thing
flipping the thing but we don't change
the shape and the size which is of
course the definition of congruent same
shape same size for dilation we actually
do shrink or expand the figure so it
keeps the same shape but not the same
size um for example when you look in a
mirror it is a reflection so it's the
same image as you which just been
flipped around um an example of
something that wouldn't be like one of
these four things would be like one of
those mirrors at the uh the fun house at
the at the uh fair or the carnival
because the whole point of that thing is
to make parts of you look bigger than
other parts and so it looks strange it
looks not normal it's not similar or
congruent um for all of these we're
either going to keep congruency like the
first three or the last one will um keep
similarity and that the shape will stay
the
same so first of all anytime we talk
about transformations
meaning we take this shape and we move
the shape over to another spot whether
it's rotated or flipped or whatever um
we don't just give it three new letters
we don't just go DF or whatever um and
the reason for that is we want to make a
note that this is that figure just
translated to another place in a in a
mapping a oneto one mapping so it's gone
from one place to the next place so what
we do is we use the same three letters
but we put this symbol on them that's
called the prime symbol it kind of looks
like um kind of looks like a one um but
it's not not a one it's more like a
tally
mark so like that if I had a second
transformation another one that would
done from the original I might use two
tally marks like a double prime a triple
prime for three Etc and so you just put
extra marks on the thing and what that
shows is that's where a went when we
when we transformed it that's where B
went when we transformed it that's where
C went when we transformed
it okay what we're going to talk about
today is translations okay translations
remember are slides going from 1 Point
to the next so first of all a
translation moves the object a fixed
distance in a given Direction it does
not change size shape or Direction it
faces so you're not going to have it
turn you're not going to have it flip
over you're just going to literally like
it's like taking something on your table
and sliding it with your hand without
turning it without rotating it it's now
in a new position but it's facing the
exact same
way let's look at a specific example
okay and this is one's already in your
notes what we're going to do here is
we're going to translate or slide 12 to
X so what that means is for every one of
those four points you see and every
point on the line that's not an a
labeled point that whole thing is going
to slide 12 to the right okay and so it
looks a little bit something like
this okay so as you can see my7 now is
at 5 my -5 for T now is all the way over
at positive 7 at s i was at-4 and now
I'm at 8 and for r i was at8 now I'm at4
so everything is moved over 12 to the
right so a lot of people will ask how do
you do this how do you easily do this
well there's a couple ways first of all
you could write down the points R S L
and T write them out as points okay and
then you could simply add 12 to all the
X values and then draw your new figure
that would work just fine second thing
you could do is you could take each
point R STL and just count 12 over to
the right and plot the point that would
work fine too if you did that and then
connected the dots uh a third way the
way that I did it when I was a student
is to pick one point like for example if
I pick s here I go 12 over to the right
and I put my point okay and then all I
would do is to get R I notice that I go
four to the left and two up like almost
like make a little right triangle in my
head and then for maybe T I'd go down
four left one and then two to the left
of that would be l so I basically use
one point as a point of reference and
compare all the rest of them there is no
right wrong way any way is correct as
long as you understand that each piece
has a been moved the same amount even
the points that aren't labeled like the
one right in between RNs you can kind of
see that crosses at a at a point there
that if you notice is also exactly 12 to
the right if any one of them didn't move
the right amount we would lose
congruency meaning they wouldn't be the
same size and the same shape anymore and
probably what would happen is it would
look weird something would look off one
of the things I do I do notice my
students do sometimes is they do this
too quickly and they'll count 11 over
instead of 12 over for one of the points
point and the shape's pretty close to
the same but it's not quite the same so
they maybe don't notice it um
unfortunately that would be incorrect on
a test or something so what you want to
do is maybe have more than one step so
count to make sure they are 12 over but
also kind of count you know down to over
four to get from R to S and double check
those things still hold true as well so
the next thing I want to say is how
would this look so I may say hey Slide
by adding 12 to X or do a translation by
adding 12 to X however you may also see
sort of like this
I want you to do the translation
parentheses x + 2 comma y parenthesis
what that means is it says take X and
add 12 to it but leave y alone okay the
next thing we're going to do is is this
is sort of like a two-step translation
you can go ahead and put both of them on
your paper but if I were to do this
two-step translation usually what you
would do is have to go sort of from one
to the next and then down to the third
position so in this case I'm going to
subtract eight from Y which means I'm
going to take that thing and I'm going
to move every one of those points down
by eight
seeing it one more time I'm going to
take each of those points I'm starting
off at seven and I'm going to move it
down to neg1 for
R Prime
so at this point again same thing I
could have just taken each of those
points I just had and subtracted it now
this translation you see right now is
actually the translation x + 2 y - 8
because I did two things I moved to the
right 12 and I moved down eight some of
you might be wondering could I have
moved down eight first first sure think
about it if I move down eight and then I
move to the right 12 I'll be in the
exact same position so it doesn't matter
which one you do first um you might want
to be a little bit careful when we start
getting into flips and stuff like that
whether or not that still is is true but
for now since all we've doing is just
sliding twice it doesn't matter the
order what I want you to pay attention
to is like we said each point moves the
same amount which keeps the same shape
for the figure so why is this keep
congruency because when you move every
point at once it doesn't change the
shape and it's certainly doesn't change
the
size okay so this is the first example
so it may already be on your notes but
if not you need to plot those the kp&
points on your on your
paper and we're going to do a few
different things okay the first thing
I'm going to have you do is I'm going to
have you list the coordinates of the
points on your
figure so by that I mean the actual
points so if you want to pause the video
while you do that that
works okay so the three points
are5 comma -4 9 comma 99 that's e and -3
comma
ne8 second thing we're going to do is
we're going to do a translation the
translation here okay add 8 to X and
only X and not do anything to Y so when
you do that go ahead and dry that draw
that on your paper label it K Prime e
Prime P Prime um and then hit play and
we'll see if you have the correct figure
all right so what should have happened
is you should have ended up with that
figure right there and you should have
labeled it K Prime e Prime and P Prime
so as you can see I went from 9 on the E
point to negative 1 Etc so now what I
want you to do the last thing and you
can pause it again is list the
coordinates of the points on
K all right so what you have here is
hopefully what you realized is all I did
is basically add a to each of the X's so
the reason I had you do this is because
you can see what we did is we actually
drew the graph and then wrote down the
points but just as easily I could have
written down my new points just by
simply adding a to each of the x's and
then went ahead and plotted those points
so again that's just another way to do
it okay we're going to try another one
so go ahead and pause this for a second
and as just like before I'm going to
have you go ahead and draw the figure on
your paper and list the coordinates for
each point
okay so if you did the correctly your
point should have been 6A 8 9A 6 6A 3
and 3A 6 so you can see because the E
and the I and the T and the M are sort
of diagonal or sorry straight up and
down or left and right from each other
in that kite shape there you get some of
the same x's and
y's this time the translation I want you
to do is I want you to subtract five
from each of the Y values so we're going
to perform a slide that is X comma y - 5
and label it t Prime I I prime M Prime e
Prime all
right so you should have ended up with
the figure right there which is moving
every shape down by five so from eight I
went to three from six for E I went down
to
one and from I at six I went down to one
as
well just like before I want you to go
ahead and pause it if you haven't done
it already and list out the cordin for
each of those points again you should
notice that you're just subtracting five
from each of the
Y's all right so hopefully you got this
right we subtract five from each of the
Y's 6 3 91 6 -2 and 3 comma 1 and
hopefully you can verify that those
points do in fact work on the
graph okay let's try another one so here
you could see we have a what appears to
be a right triangle um and I want you to
go ahead first and draw the triangle and
then list the points r a t
as ordered
pairs okay so what we're going to do now
the points are 8 comma -3 8 comma ne8
and 4 comma
ne8 so now what we're going to do is
we're going to perform a slide or a
translation that has two parts so we're
going to subtract seven from all the x's
and add 10 to all of the Y's so go ahead
and do that make your translation when
you feel like you're good and done go
ahead and hit
play all right so if you did it
correctly you should end up here and
again that is moving up 10 on the y-
axis so fromg -3 to positive
7 and then moving over 1 2 3 4 five 6 7
so again to do this in two steps we need
draw the triangle and draw it a second
time to me is the hardest way to do it
you could look at it almost like slope
where you go up 10 left seven but I
still feel like the easiest thing to do
here is to Simply figure out the
coordinates of the RP prime a prime and
T Prime by subtracting seven and adding
10 and simply plotting your points on
the paper so go ahead and write those
three points verify that you did in fact
subtract seven from each of the X's add
10 to each of the Y's and hit play when
you're
ready all right so hopefully you noticed
we went from 8 to one which is
subtracting seven and from -3 to 7 which
is adding 10 so of course we did follow
the
pattern okay last one for you guys to
try I don't know if you picked up on The
Animal theme but here we have chimp CH i
m p so what I want you guys to do is
write the coordinates of those
points and then we'll do the
translation okay the the points are 9 6
98 -2 7 -3 3 and8
2 this time we're going to subtract one
from our x's and subtract 10 from our
y's and we're going to label C Prime H
Prime I prime M Prime P Prime as our new
figure okay so hopefully you got it
right it should be just like that so as
you can see we moved down 10 and left
one from our original
position lastly label my
points hopefully you guys labeled the
points now C Prime is -104 and you can
check your rest of your answers as
well so this is a quick activity
remember that these are required for
your notes um to have this completely
done and there is going to be a form
below that you can fill out if you were
in class today and you did the all you
were doing was finishing up the last few
examples uh you don't necessarily need
to fill that out because I'll see your
notes in in class but if you were absent
this is a great way to kind of make that
up and also if um in class today we
didn't do these notes and this is a
video homework then you'll want to make
sure you do that as
well for
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