Geometric Transformations - Translations

Summary

This 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

00:00

🔄 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.

05:01

📐 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.

10:02

📊 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.

Keywords

💡Transformation

A transformation in geometry refers to changing the position, size, or orientation of a shape. In the context of the video, transformations involve moving shapes or lines in various ways, including translations, rotations, reflections, and dilations. The video emphasizes that transformations either maintain congruency or similarity, depending on the type of transformation.

💡Translation

Translation, also called a slide, is a type of transformation that moves an object a fixed distance in a specified direction without changing its shape, size, or orientation. In the video, translation is compared to sliding an object across a surface, keeping the object in the same position but at a new location. An example discussed is sliding points 12 units to the right.

💡Reflection

Reflection, often called a flip, is a type of transformation where a shape is mirrored across a line, such as the x-axis or y-axis. This transformation maintains the shape's congruency while flipping it to create a mirror image. The video compares reflection to seeing yourself in a mirror, where the image appears reversed but remains the same in size and shape.

💡Rotation

Rotation is a transformation that involves turning a shape around a specific point without altering its shape or size. In the video, it is described as turning an object like a wheel, where the shape is rotated by a certain angle but remains congruent to its original form.

💡Dilation

Dilation is a transformation that changes the size of a shape while maintaining its proportions. Unlike the other transformations (translation, reflection, and rotation), dilation affects the shape's size, making it either larger or smaller. The video mentions that this transformation keeps the shape similar but not congruent since the size is altered.

💡Congruency

Congruency refers to two shapes having the same size and shape, even if they are transformed through translation, reflection, or rotation. The video highlights that transformations like translation, reflection, and rotation preserve congruency, meaning the shape does not change its dimensions or overall structure during these transformations.

💡Prime Notation

Prime notation is a way to label the transformed points in a figure to distinguish them from the original points. In the video, the prime symbol is introduced as a tally-like mark that is placed next to the letters representing the points after a transformation. For instance, a point 'A' may become 'A' (prime) after being translated to a new position.

💡Similarity

Similarity refers to two shapes having the same shape but different sizes. The video explains that dilation transformations preserve similarity because the shape remains proportionate, but the size changes. This contrasts with transformations that maintain congruency, where both shape and size are preserved.

💡Fixed Distance

A fixed distance is the unchanging distance by which a shape is moved during a translation. In the video, this concept is explained in relation to translations, where shapes are shifted by a set distance without altering their direction, shape, or size. An example given is moving a shape 12 units to the right.

💡Coordinates

Coordinates are numerical values that represent a point's position on a graph, usually in the form of (x, y). In the video, coordinates are used extensively to describe transformations, as the speaker explains how to add or subtract from the x or y values to move shapes. The process of translating points involves modifying their coordinates to reflect their new position.

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