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https://www.youtube.com/watch?v=ouNp8FtgiEE
This video is explaining how axis of rreflection is used on the x and y axis

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Reflections Over the X-Axis and Y-Axis Explained!

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https://www.youtube.com/watch?v=NhtTKhP3d6s

this helped with rotations

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Geometry Rotations Explained (90, 180, 270, 360)

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https://www.khanacademy.org/math/geometry/hs-geo-transformations/hs-geo-transformations-intro/v/introduction-to-transformations

This video explains rigid transformation.

Question
What is dilation.

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- [Voiceover] What I hope to introduce you to in this video is the notion of a transformation in mathematics, and you're probably used to the word in everyday language. Transformation means something is changing, it's transforming from one thing to another. What would transformation mean in a mathematical context? Well, it could mean that you're taking something mathematical and you're changing it into something else mathematical, that's exactly what it is. It's talking about taking a set of coordinates or a set of points, and then changing them into a different set of coordinates or a different set of points. For example, this right over here, this is a quadrilateral we've plotted it on the coordinate plane. This is a set of points, not just the four points that represent the vertices of the quadrilateral, but all the points along the sides too. There's a bunch of points along this. You could argue there's an infinite, or there are an infinite number of points along this quadrilateral. This right over here, the point X equals 0, y equals negative four, this is a point on the quadrilateral. Now, we can apply a transformation to this, and the first one I'm going to show you is a translation, which just means moving all the points in the same direction, and the same amount in that same direction, and I'm using the Khan Academy translation widget to do it. Let's translate, let's translate this, and I can do it by grabbing onto one of the vertices, and notice I've now shifted it to the right by two. Every point here, not just the orange points has shifted to the right by two. This one has shifted to the right by two, this point right over here has shifted to the right by two, every point has shifted in the same direction by the same amount, that's what a translation is. Now, I've shifted, let's see if I put it here every point has shifted to the right one and up one, they've all shifted by the same amount in the same directions. That is a translation, but you could imagine a translation is not the only kind of transformation. In fact, there is an unlimited variation, there's an unlimited number different transformations. So, for example, I could do a rotation. I have another set of points here that's represented by quadrilateral, I guess we could call it CD or BCDE, and I could rotate it, and I rotate it I would rotate it around the point. So for example, I could rotate it around the point D, so this is what I started with, if I, let me see if I can do this, I could rotate it like, actually let me see. So if I start like this I could rotate it 90 degrees, I could rotate 90 degrees, so I could rotate it, I could rotate it like, that looks pretty close to a 90-degree rotation. So, every point that was on the original or in the original set of points I've now shifted it relative to that point that I'm rotating around. I've now rotated it 90 degrees, so this point has now mapped to this point over here. This point has now mapped to this point over here, and I'm just picking the vertices because those are a little bit easier to think about. This point has mapped to this point. The point of rotation, actually, since D is actually the point of rotation that one actually has not shifted, and just 'til you get some terminology, the set of points after you apply the transformation this is called the image of the transformation. So, I had quadrilateral BCDE, I applied a 90-degree counterclockwise rotation around the point D, and so this new set of points this is the image of our original quadrilateral after the transformation. I don't have to just, let me undo this, I don't have to rotate around just one of the points that are on the original set that are on our quadrilateral, I could rotate around, I could rotate around the origin. I could do something like that. Notice it's a different rotation now. It's a different rotation. I could rotate around any point. Now let's look at another transformation, and that would be the notion of a reflection, and you know what reflection means in everyday life. You imagine the reflection of an image in a mirror or on the water, and that's exactly what we're going to do over here. If we reflect, we reflect across a line, so let me do that. This, what is this one, two, three, four, five, this not-irregular pentagon, let's reflect it. To reflect it, let me actually, let me actually make a line like this. I could reflect it across a whole series of lines. Woops, let me see if I can, so let's reflect it across this. Now, what does it mean to reflect across something? One way I imagine is if this was, we're going to get its mirror image, and you imagine this as the line of symmetry that the image and the original shape they should be mirror images across this line we could see that. Let's do the reflection. There you go, and you see we have a mirror image. This is this far away from the line. This, its corresponding point in the image is on the other side of the line but the same distance. This point over here is this distance from the line, and this point over here is the same distance but on the other side. Now, all of the transformations that I've just showed you, the translation, the reflection, the rotation, these are called rigid transformations. Once again you could just think about what does rigid mean in everyday life? It means something that's not flexible. It means something that you can't stretch or scale up or scale down it kind of maintains its shape, and that's what rigid transformations are fundamentally about. If you want to think a little bit more mathematically, a rigid transformation is one in which lengths and angles are preserved. You can see in this transformation right over here the distance between this point and this point, between points T and R, and the difference between their corresponding image points, that distance is the same. The angle here, angle R, T, Y, the measure of this angle over here, if you look at the corresponding angle in the image it's going to be the same angle. The same thing is true if you're doing a translation. You could imagine these are acting like rigid objects. You can't stretch them, they're not flexible they're maintaining their shape. Now what would be examples of transformations that are not rigid transformations? Well you could imagine scaling things up and down. If I were to scale this out where it has maybe the angles are preserved, but the lengths aren't preserved that would not be a rigid transformation. If I were to just stretch one side of it, or if I were to just pull this point while the other points stayed where they are I'd be distorting it or stretching it that would not be a rigid transformation. So hopefully this gets you, it's actually very, very interesting. When you use an art program, or actually you use a lot of computer graphics, or you play a video game, most of what the video game is doing is actually doing transformations. Sometimes in two dimensions, sometimes in three dimensions, and once you get into more advanced math, especially things like linear algebra, there's a whole field that's really focused around transformations. In fact, some of the computers with really good graphics processors, a graphics processor is just a piece of hardware that is really good at performing mathematical transformations, so that you can immerse yourself in a 3D reality or whatever else. This is really really interesting stuff.

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Intro to geometric transformations (video) | Khan Academy

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Voiceover] What I hope to introduce you to in this video is the notion of a transformation in mathematics, and you're probably used to the word in everyday language. Transformation means something is changing, it's transforming from one thing to another. What would transformation mean in a mathematical context?

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https://www.youtube.com/watch?v=7y4FGCvNgB8
This is a video that will help explain how rigid trasformation works.

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Finding measures using rigid transformations

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https://www.youtube.com/watch?v=7y4FGCvNgB8
Another video and expanded rigid transformation.

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Finding measures using rigid transformations

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This helped me me understand rotations more.

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https://youtu.be/zIImkJd3iMo

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https://youtu.be/BCllaARDOWI

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https://youtube.com/watch?v=m73VMHLzt-4&feature=shareuu

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- [Instructor] What we're going to do in this video is think about what properties of a shape are preserved or not preserved, as they undergo a transformation. In particular, we're gonna think about rotations and reflections in this video. And both of those are rigid transformations which means that the length between corresponding points do not change. So for example, let's say we take this circle A, it's centered at Point A. And we were to rotate it around Point P. Point P is the center of rotation. And just for the sake of argument we rotate it clockwise a certain angle. So let's say we end up right over so we're gonna rotate that way. And let's say our center ends up right over here. So our new circle, the image after the rotation might look something like this. And I'm hand drawing it. So you got to forgive that it's not that well hand drawn of a circle. But the circle might look something like this. And so, the clear things that are preserved or maybe it's not so clear, we're gonna hope we make them clear right now. Things that are preserved under a rigid transformation like this rotation right over here. This is clearly a rotation. Things that are preserved well, you have things like the radius of the circle. The radius length, I could say, to be more particular. The radius here is two. The radius here is also is also two, right over there. You have things like the perimeter. Well, if the radius is preserved the perimeter of a circle which we call a circumference well, that's just a function of the radius. We're talking about two times pi times the radius. So the perimeter, of course, is going to be preserved. In fact, that follows from the fact that the length of the radius is preserved. And of course, if the radius is preserved and then the area is also going to be preserved. The area is just pi times the radius squared. So they have the same radius. They're gonna have all of these in common. And you could also that feels intuitively right. So what is not preserved? Not preserved. And this is in general true of rigid transformations is that they will preserve the distance between corresponding points if we're transforming a shape they'll preserve things like perimeter and area. And this case, I can set a perimeter. I can say circumference. Circumference. So they'll preserve things like that. They'll preserve angles. We don't have clear angles in this picture. But, they'll preserve things like angles. But what they won't preserve is the coordinates. Coordinates of corresponding points. They might sometimes, but not always. So for example, the coordinate of the center here is for sure, going to change. We go from the coordinate negative three comma zero. To here we went to the coordinate we went to the coordinate negative one comma two. So the coordinates are not preserved. Coordinates of the center. Let's do another example with a non-circular shape. And we'll do a different type of transformation. In this situation let us do a reflection. So, we have a quadrilateral here. Quadrilateral ABCD. And we want to think about what is preserved, or not preserved as we do a reflection across the line L. So let me write that down. We're gonna have a reflection in this situation. And we can even think about this without even doing the reflection ourselves. But let's just do the reflection really fast. So we're reflecting across the line XYZ equal to X. So what it essentially does to the coordinates is it swaps the X and Y coordinates. But you don't have to know that for the sake of this video. So, B prime would be right over here. A prime would be right over there. D prime would be right over here. And since C is right on the line now its image, C prime, won't change. And so our new when we reflect over the line L. And you don't have to know for the sake of this video, exactly how I did that fairly quickly. I really just want you to see what the reflection looks like. The real appreciation here is think about, well, what happens with rigid transformations. So, it's gonna look something like this. The reflection. The reflection looks something like this. So what's preserved? And in general, this is good to know for any rigid transformation what's preserved. Well, side lengths. That's actually one way that we even use to define what a rigid transformation is. A transformation that preserves the lengths between corresponding points. Angle measures. Angle measures. So, for example, this angle here, the angle A, is gonna be the same as the angle A prime over here. Side lengths, the distance between A and B is going to be the same as the distance between A prime and B prime. Perimeter. If you have the same side lengths and the same angles, the perimeter and area are also going to be preserved. Just like we saw with the rotation example. These are rigid transformations. These are the types of things that are preserved. Well, what is not preserved? Not preserved. And this just goes back to the example we just looked at. Well, coordinates are not preserved. So as we see, the image of A A prime has different coordinates than A. B prime has different coordinates than B. C prime, in this case, happens to have the same coordinates as C because C happens to sit on our, the line that we're reflecting over. But D prime definitely does not have the same coordinates as D. So, most of or let me say, coordinates of AB ABD. Coordinates of A B D not preserved. After transformation, or their images don't have the same coordinates. After transformation. The one coordinate that happened to be preserved here is Cs coordinates. Because it was right on the line of reflection. And you can also look at other properties so how it might relate how different segments might relate to lines that would not be that were not being transformed. So, for example, right over here before transformation, CD is parallel to the Y axis. You see this right over here. But after the transformation C prime D prime. So this could be C prime D prime is no longer parallel to the Y axis. In fact, now it is parallel to the X axis. So, when you have the relationship to things outside of the things we're transformed that relationship might not. Those relationships may no longer be true after the transformation.

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Rigid transformations: preserved properties (video) | Khan Academy

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Instructor] What we're going to do in this video is think about what properties of a shape are preserved or not preserved, as they undergo a transformation. In particular, we're gonna think about rotations and reflections in this video. And both of those are rigid transformations which means that the length between corresponding points do not change.

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This video is a song that helps understand translations and the 3 basic steps.

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Transformations Song | Translations, Rotations, Reflections, & Dilations

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This site informs you about different types of rigid transformation and teaches some new vocabulary.

Questions -
Is the center of rotation always the origin (0,0)?

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A rigid transformation (also called an isometry ) is a transformation of the plane that preserves length. , and combinations of these three transformations are "rigid transformations". Reflections, translations, rotations

While the pre-image and the image under a rigid transformation will be congruent,
they may not be facing in the same direction.

For a review of reflections, see the Refresher section Transformations: Reflections.
Now let's expand that knowledge of reflections in relation to geometry.

A reflection over line m (notation r m ) is a transformation in which each point of the original figure (pre-image) has an image that is the same distance from the reflection line as the original point, but is on the opposite side of the line. A reflection is called a rigid transformation or isometry because the image is the same size and shape as the pre-image.

f the segments joining each point to its image.
The reflection line, m, is the perpendicular bisector o
Notice that these segments are parallel , since they are perpendicular to the same line.

In this reflection that maps Δ (are not necessarily equal), but the segments representing these distances are parallel., in this diagram, is clockwise is lettered counterclockwise ABC to Δ A'B'C', the distances from the pre-image points to the image points vary
A-B-C, while the
A'-B'-C'. When lettering changes direction, in this manner, the transformation is referred to as a non-direct or opposite isometry.
Orientation (lettering): The lettering of the points of the pre-image image

Properties preserved under a line reflection from the pre-image to the image.
1. distance (lengths of segments remain the same)
2. angle measures (remain the same)
3. parallelism (parallel lines remain parallel)
4. collinearity (points remain on the same lines)
----------------------------------------------------------
The orientation (lettering around the outside of the figure), is not preserved . The order of the lettering in a reflection is reversed (from clockwise to counterclockwise or vice versa).

When formally defining a reflection, there are two situations (cases) that need to be considered for a precise definition. Is the point being reflected lying ON the line of reflection, or NOT ON the line of reflection?

A reflection is a rigid transformation (isometry) that maps every point P in the plane to point P', across a line of reflection, m, such that:

Case 1: if point P is ON line m, the
point is its own reflection ( P = P') and

(point P is "fixed").

Case 2: if point P is NOT on line m, then m is the perpendicular bisector of

where

For a review of translations, see the Refresher section Transformations: Translations.
Now let's expand that knowledge of translations in relation to geometry.

A translation (notation T a,b ) is a transformation which "slides" a figure a fixed distance in a given direction. In a translation, ALL of the points move the same distance in the same direction.
A translation is called a rigid transformation or isometry because the image is the same size and shape as the pre-image.

In this translation that maps Δ. , in this diagram, is counterclockwise is also lettered counterclockwise ABC to Δ A'B'C', the distances from the pre-image points to the image points are equal, and the segments representing these distances are parallel parallel.

In addition, the corresponding segment sides of the pre-image and image are

Orientation (lettering):The lettering of the points of the pre-image A-B-C, and the image A'-B'-C'. When lettering order remains the same, the transformation is referred to as a direct isometry.

Properties preserved under a translation from the pre-image to the image.
1. distance (lengths of segments remain the same)
2. angle measures (remain the same)
3. parallelism (parallel lines remain parallel)
4. collinearity (points remain on the same lines)
5. orientation (lettering order remains the same)

A translation is a rigid transformation of the plane that moves every point of a pre-image a constant distance in a specified direction.

There are several ways to indicate that a translation is to occur:

mapping:
Example:
Read: "the x and y coordinates are mapped to x + 8 and y - 6 respectively".
( x, y) → ( x + 8, y - 6)

notations: The 8 tells you to "add 8" to all
Example: T 8,-6 or T 8,-6 (
x, y) = ( x-coordinates, while the -6 tells you to "subtract 6" from all of the y-coordinates. x + 8, y - 6)

description:"8 units to the right and 6 units down."
Example:

A verbal or written description of the translation is given.

Movement: Remember that adding a negative value (subtracting), indicates movement left and/or down , while adding a positive value indicates movement right and/or up .

Using vectors to show movement of a translation:

A vector is represented by a directed line segment, a segment with an arrow at one end indicating the direction of movement. Unlike a ray, a directed line segment has a specific length. The Pythagorean Theorem can be used to find the length of a vector in the coordinate plane.

Vectors used in translations are what are known as "free vectors", which are a set of parallel directed line segments. A vector translation moving "8 units to the right and 6 units down" can be written as

. You may also see

.

There is an "identity" translational transformation that maps all points onto themselves (no movement of the pre-image), expressed as ( x, y) → ( x, y). (You may see the identity transformation denoted by the letter I.)

For a review of rotations, see the Refresher section Transformations: Rotations.
Now let's expand that knowledge of rotations in relation to geometry.

A rotation of (notation ) is a transformation which "turns" a figure about a fixed point, θ degrees RC,θ C, called the center of rotation . When working in the coordinate plane, the center of rotation should be stated, and not assumed to be at the origin. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation . A rotation is called a rigid transformation or isometry because the image is the same size and shape as the pre-image.

An object and its rotation are the same shape and size, but the figures may be positioned differently.

During a rotation, every point is moved the exact same degree arc along the circle
defined by the center of the rotation and the angle of rotation.
(The dashed arcs in the diagram below represent the circles, with center P, through each of the triangle's vertices.
Note that PC=PC', for example, since they are the radii of the same circle.)

A positive angle of rotation turns a figure counterclockwise (CCW),negative angle of rotation turns the figure clockwise,
and a (CW).

In this rotation that maps Δ, in this diagram, is counterclockwise is also lettered counterclockwise ABC to Δ A'B'C', the distances from the pre-image points to the image points vary (as they are the radii of the circles). Unlike reflections and translations, the segments connecting pre-image and image points are NOT parallel. pre-image

Orientation (lettering):The lettering of the points of the image A-B-C, and the A'-B'-C'.

Properties preserved under a rotation from the pre-image to the image.
1. distance (lengths of segments remain the same)
2. angle measures (remain the same)
3. parallelism (parallel lines remain parallel)
4. collinearity (points remain on the same lines)
5. orientation (lettering order remains the same)

There are two possible directions to travel when rotating.

Look at point P in the diagram at the right. The positive x-axis is considered the "starting" location of 0º. You can have an angle of rotation of positive 135º counterclockwise to point P, or a negative angle of 225º clockwise to point P.
Notice that together, the angles (135º and 225º) form a complete circle (360º).

It is even possible to have one complete revolution (360º) PLUS and additional 135º to get to P (an angle of rotation of 495º).

When working in the coordinate plane, it is important to have a visual understanding of how the quadrants are divided through rotational angles. The most common angles seen on the grid are multiples of 15º. The "favorite" angles are usually 30º, 45º, 60º, 90º, 180º, and 270º.

There is an "identity" rotational transformation of 0º that maps all points onto themselves (no movement of the pre-image), written RC,0 ( P) = P. (You may also see the identity transformation denoted by the letter I.)

and is not considered "fair use" for educators. Please read the " NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
Terms of Use".

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Rigid Transformations (Isometries) - MathBitsNotebook(Geo - CCSS Math)

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A rigid transformation (also called an isometry ) is a transformation of the plane that preserves length. , and combinations of these three transformations are "rigid transformations". Reflections, translations, rotations While the pre-image and the image under a rigid transformation will be congruent, they may not be facing in the same direction.

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https://youtu.be/lxIbuXURm9w

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Transformations | Translations, Enlargements, Reflections & Rotations | Addvance Maths 😊➕

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This video shows you how to do rigid transformation.

Question
What is dilation.

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- [Voiceover] What I hope to introduce you to in this video is the notion of a transformation in mathematics, and you're probably used to the word in everyday language. Transformation means something is changing, it's transforming from one thing to another. What would transformation mean in a mathematical context? Well, it could mean that you're taking something mathematical and you're changing it into something else mathematical, that's exactly what it is. It's talking about taking a set of coordinates or a set of points, and then changing them into a different set of coordinates or a different set of points. For example, this right over here, this is a quadrilateral we've plotted it on the coordinate plane. This is a set of points, not just the four points that represent the vertices of the quadrilateral, but all the points along the sides too. There's a bunch of points along this. You could argue there's an infinite, or there are an infinite number of points along this quadrilateral. This right over here, the point X equals 0, y equals negative four, this is a point on the quadrilateral. Now, we can apply a transformation to this, and the first one I'm going to show you is a translation, which just means moving all the points in the same direction, and the same amount in that same direction, and I'm using the Khan Academy translation widget to do it. Let's translate, let's translate this, and I can do it by grabbing onto one of the vertices, and notice I've now shifted it to the right by two. Every point here, not just the orange points has shifted to the right by two. This one has shifted to the right by two, this point right over here has shifted to the right by two, every point has shifted in the same direction by the same amount, that's what a translation is. Now, I've shifted, let's see if I put it here every point has shifted to the right one and up one, they've all shifted by the same amount in the same directions. That is a translation, but you could imagine a translation is not the only kind of transformation. In fact, there is an unlimited variation, there's an unlimited number different transformations. So, for example, I could do a rotation. I have another set of points here that's represented by quadrilateral, I guess we could call it CD or BCDE, and I could rotate it, and I rotate it I would rotate it around the point. So for example, I could rotate it around the point D, so this is what I started with, if I, let me see if I can do this, I could rotate it like, actually let me see. So if I start like this I could rotate it 90 degrees, I could rotate 90 degrees, so I could rotate it, I could rotate it like, that looks pretty close to a 90-degree rotation. So, every point that was on the original or in the original set of points I've now shifted it relative to that point that I'm rotating around. I've now rotated it 90 degrees, so this point has now mapped to this point over here. This point has now mapped to this point over here, and I'm just picking the vertices because those are a little bit easier to think about. This point has mapped to this point. The point of rotation, actually, since D is actually the point of rotation that one actually has not shifted, and just 'til you get some terminology, the set of points after you apply the transformation this is called the image of the transformation. So, I had quadrilateral BCDE, I applied a 90-degree counterclockwise rotation around the point D, and so this new set of points this is the image of our original quadrilateral after the transformation. I don't have to just, let me undo this, I don't have to rotate around just one of the points that are on the original set that are on our quadrilateral, I could rotate around, I could rotate around the origin. I could do something like that. Notice it's a different rotation now. It's a different rotation. I could rotate around any point. Now let's look at another transformation, and that would be the notion of a reflection, and you know what reflection means in everyday life. You imagine the reflection of an image in a mirror or on the water, and that's exactly what we're going to do over here. If we reflect, we reflect across a line, so let me do that. This, what is this one, two, three, four, five, this not-irregular pentagon, let's reflect it. To reflect it, let me actually, let me actually make a line like this. I could reflect it across a whole series of lines. Woops, let me see if I can, so let's reflect it across this. Now, what does it mean to reflect across something? One way I imagine is if this was, we're going to get its mirror image, and you imagine this as the line of symmetry that the image and the original shape they should be mirror images across this line we could see that. Let's do the reflection. There you go, and you see we have a mirror image. This is this far away from the line. This, its corresponding point in the image is on the other side of the line but the same distance. This point over here is this distance from the line, and this point over here is the same distance but on the other side. Now, all of the transformations that I've just showed you, the translation, the reflection, the rotation, these are called rigid transformations. Once again you could just think about what does rigid mean in everyday life? It means something that's not flexible. It means something that you can't stretch or scale up or scale down it kind of maintains its shape, and that's what rigid transformations are fundamentally about. If you want to think a little bit more mathematically, a rigid transformation is one in which lengths and angles are preserved. You can see in this transformation right over here the distance between this point and this point, between points T and R, and the difference between their corresponding image points, that distance is the same. The angle here, angle R, T, Y, the measure of this angle over here, if you look at the corresponding angle in the image it's going to be the same angle. The same thing is true if you're doing a translation. You could imagine these are acting like rigid objects. You can't stretch them, they're not flexible they're maintaining their shape. Now what would be examples of transformations that are not rigid transformations? Well you could imagine scaling things up and down. If I were to scale this out where it has maybe the angles are preserved, but the lengths aren't preserved that would not be a rigid transformation. If I were to just stretch one side of it, or if I were to just pull this point while the other points stayed where they are I'd be distorting it or stretching it that would not be a rigid transformation. So hopefully this gets you, it's actually very, very interesting. When you use an art program, or actually you use a lot of computer graphics, or you play a video game, most of what the video game is doing is actually doing transformations. Sometimes in two dimensions, sometimes in three dimensions, and once you get into more advanced math, especially things like linear algebra, there's a whole field that's really focused around transformations. In fact, some of the computers with really good graphics processors, a graphics processor is just a piece of hardware that is really good at performing mathematical transformations, so that you can immerse yourself in a 3D reality or whatever else. This is really really interesting stuff.

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Intro to geometric transformations (video) | Khan Academy

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Voiceover] What I hope to introduce you to in this video is the notion of a transformation in mathematics, and you're probably used to the word in everyday language. Transformation means something is changing, it's transforming from one thing to another. What would transformation mean in a mathematical context?

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https://www.youtube.com/watch?v=m73VMHLzt-4

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https://youtu.be/XiAoUDfrar0

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Introduction to transformations | Transformations | Geometry | Khan Academy

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https://www.youtube.com/watch?v=7y4FGCvNgB8

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Finding measures using rigid transformations

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Transformations Song | Translations, Rotations, Reflections, & Dilations

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This video showed me a basic understanding of transformations and it helped me with the reflections and rotations.

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Translations Reflections and Rotations - Geometric Transformations!

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- [Instructor] We're told, "Kason was curious "if triangle ABC and triangle GFE were congruent, "so he tried to map one figure onto the other "using a rotation." So let's see, this is triangle ABC, and it looks like, at first, he rotates triangle ABC about point C, to get it right over here, so that's what they're depicting in this diagram. And then they say, "Kason concluded: "It is not possible to map triangle ABC "onto triangle GFE using a sequence "of rigid transformations, "so the triangles are not congruent." So what I want you to do is pause this video and think about, is Kason correct that they are not congruent, because you can not map ABC, triangle ABC onto triangle GFE with rigid transformations? All right, so the way I think about it, he was able to do the rotation that got us right over here, so it is rotation about point C, and so this point right over here, let me make sure I get this right, this would've become B prime, and then this is A prime, and then C is mapped to itself, so C is equal to C prime. But he's not done, there's another rigid transformation he could do, and that would be a reflection about the line FG. So if he reflects about the line FG, then this point is going to be mapped to point E, just like that. And then if you did that, you would see that there is a series of rigid transformations that maps triangle ABC onto triangle GFE. So Kason is not correct, he missed one more transformation he could've done, which is a reflection.

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Congruent shapes & transformations (video) | Khan Academy

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Instructor] We're told, "Kason was curious "if triangle ABC and triangle GFE were congruent, "so he tried to map one figure onto the other "using a rotation." So let's see, this is triangle ABC, and it looks like, at first, he rotates triangle ABC about point C, to get it right over here, so that's what they're depicting in this diagram.

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https://m.youtube.com/watch?v=dUboC2qvXuw

This video helped me have more of a understanding of rotating and flipping.

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https://www.khanacademy.org/math/geometry/xff63fac4:hs-geo-transformation-properties-and-proofs/hs-geo-rigid-transformations-overview/v/rigid-transformations-preserved-properties

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- [Instructor] What we're going to do in this video is think about what properties of a shape are preserved or not preserved, as they undergo a transformation. In particular, we're gonna think about rotations and reflections in this video. And both of those are rigid transformations which means that the length between corresponding points do not change. So for example, let's say we take this circle A, it's centered at Point A. And we were to rotate it around Point P. Point P is the center of rotation. And just for the sake of argument we rotate it clockwise a certain angle. So let's say we end up right over so we're gonna rotate that way. And let's say our center ends up right over here. So our new circle, the image after the rotation might look something like this. And I'm hand drawing it. So you got to forgive that it's not that well hand drawn of a circle. But the circle might look something like this. And so, the clear things that are preserved or maybe it's not so clear, we're gonna hope we make them clear right now. Things that are preserved under a rigid transformation like this rotation right over here. This is clearly a rotation. Things that are preserved well, you have things like the radius of the circle. The radius length, I could say, to be more particular. The radius here is two. The radius here is also is also two, right over there. You have things like the perimeter. Well, if the radius is preserved the perimeter of a circle which we call a circumference well, that's just a function of the radius. We're talking about two times pi times the radius. So the perimeter, of course, is going to be preserved. In fact, that follows from the fact that the length of the radius is preserved. And of course, if the radius is preserved and then the area is also going to be preserved. The area is just pi times the radius squared. So they have the same radius. They're gonna have all of these in common. And you could also that feels intuitively right. So what is not preserved? Not preserved. And this is in general true of rigid transformations is that they will preserve the distance between corresponding points if we're transforming a shape they'll preserve things like perimeter and area. And this case, I can set a perimeter. I can say circumference. Circumference. So they'll preserve things like that. They'll preserve angles. We don't have clear angles in this picture. But, they'll preserve things like angles. But what they won't preserve is the coordinates. Coordinates of corresponding points. They might sometimes, but not always. So for example, the coordinate of the center here is for sure, going to change. We go from the coordinate negative three comma zero. To here we went to the coordinate we went to the coordinate negative one comma two. So the coordinates are not preserved. Coordinates of the center. Let's do another example with a non-circular shape. And we'll do a different type of transformation. In this situation let us do a reflection. So, we have a quadrilateral here. Quadrilateral ABCD. And we want to think about what is preserved, or not preserved as we do a reflection across the line L. So let me write that down. We're gonna have a reflection in this situation. And we can even think about this without even doing the reflection ourselves. But let's just do the reflection really fast. So we're reflecting across the line XYZ equal to X. So what it essentially does to the coordinates is it swaps the X and Y coordinates. But you don't have to know that for the sake of this video. So, B prime would be right over here. A prime would be right over there. D prime would be right over here. And since C is right on the line now its image, C prime, won't change. And so our new when we reflect over the line L. And you don't have to know for the sake of this video, exactly how I did that fairly quickly. I really just want you to see what the reflection looks like. The real appreciation here is think about, well, what happens with rigid transformations. So, it's gonna look something like this. The reflection. The reflection looks something like this. So what's preserved? And in general, this is good to know for any rigid transformation what's preserved. Well, side lengths. That's actually one way that we even use to define what a rigid transformation is. A transformation that preserves the lengths between corresponding points. Angle measures. Angle measures. So, for example, this angle here, the angle A, is gonna be the same as the angle A prime over here. Side lengths, the distance between A and B is going to be the same as the distance between A prime and B prime. Perimeter. If you have the same side lengths and the same angles, the perimeter and area are also going to be preserved. Just like we saw with the rotation example. These are rigid transformations. These are the types of things that are preserved. Well, what is not preserved? Not preserved. And this just goes back to the example we just looked at. Well, coordinates are not preserved. So as we see, the image of A A prime has different coordinates than A. B prime has different coordinates than B. C prime, in this case, happens to have the same coordinates as C because C happens to sit on our, the line that we're reflecting over. But D prime definitely does not have the same coordinates as D. So, most of or let me say, coordinates of AB ABD. Coordinates of A B D not preserved. After transformation, or their images don't have the same coordinates. After transformation. The one coordinate that happened to be preserved here is Cs coordinates. Because it was right on the line of reflection. And you can also look at other properties so how it might relate how different segments might relate to lines that would not be that were not being transformed. So, for example, right over here before transformation, CD is parallel to the Y axis. You see this right over here. But after the transformation C prime D prime. So this could be C prime D prime is no longer parallel to the Y axis. In fact, now it is parallel to the X axis. So, when you have the relationship to things outside of the things we're transformed that relationship might not. Those relationships may no longer be true after the transformation.

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Rigid transformations: preserved properties (video) | Khan Academy

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Instructor] What we're going to do in this video is think about what properties of a shape are preserved or not preserved, as they undergo a transformation. In particular, we're gonna think about rotations and reflections in this video. And both of those are rigid transformations which means that the length between corresponding points do not change.

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https://www.khanacademy.org/math/geometry/xff63fac4:hs-geo-transformation-properties-and-proofs/hs-geo-rigid-transformations-overview/v/rigid-transformations-preserved-properties

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- [Instructor] What we're going to do in this video is think about what properties of a shape are preserved or not preserved, as they undergo a transformation. In particular, we're gonna think about rotations and reflections in this video. And both of those are rigid transformations which means that the length between corresponding points do not change. So for example, let's say we take this circle A, it's centered at Point A. And we were to rotate it around Point P. Point P is the center of rotation. And just for the sake of argument we rotate it clockwise a certain angle. So let's say we end up right over so we're gonna rotate that way. And let's say our center ends up right over here. So our new circle, the image after the rotation might look something like this. And I'm hand drawing it. So you got to forgive that it's not that well hand drawn of a circle. But the circle might look something like this. And so, the clear things that are preserved or maybe it's not so clear, we're gonna hope we make them clear right now. Things that are preserved under a rigid transformation like this rotation right over here. This is clearly a rotation. Things that are preserved well, you have things like the radius of the circle. The radius length, I could say, to be more particular. The radius here is two. The radius here is also is also two, right over there. You have things like the perimeter. Well, if the radius is preserved the perimeter of a circle which we call a circumference well, that's just a function of the radius. We're talking about two times pi times the radius. So the perimeter, of course, is going to be preserved. In fact, that follows from the fact that the length of the radius is preserved. And of course, if the radius is preserved and then the area is also going to be preserved. The area is just pi times the radius squared. So they have the same radius. They're gonna have all of these in common. And you could also that feels intuitively right. So what is not preserved? Not preserved. And this is in general true of rigid transformations is that they will preserve the distance between corresponding points if we're transforming a shape they'll preserve things like perimeter and area. And this case, I can set a perimeter. I can say circumference. Circumference. So they'll preserve things like that. They'll preserve angles. We don't have clear angles in this picture. But, they'll preserve things like angles. But what they won't preserve is the coordinates. Coordinates of corresponding points. They might sometimes, but not always. So for example, the coordinate of the center here is for sure, going to change. We go from the coordinate negative three comma zero. To here we went to the coordinate we went to the coordinate negative one comma two. So the coordinates are not preserved. Coordinates of the center. Let's do another example with a non-circular shape. And we'll do a different type of transformation. In this situation let us do a reflection. So, we have a quadrilateral here. Quadrilateral ABCD. And we want to think about what is preserved, or not preserved as we do a reflection across the line L. So let me write that down. We're gonna have a reflection in this situation. And we can even think about this without even doing the reflection ourselves. But let's just do the reflection really fast. So we're reflecting across the line XYZ equal to X. So what it essentially does to the coordinates is it swaps the X and Y coordinates. But you don't have to know that for the sake of this video. So, B prime would be right over here. A prime would be right over there. D prime would be right over here. And since C is right on the line now its image, C prime, won't change. And so our new when we reflect over the line L. And you don't have to know for the sake of this video, exactly how I did that fairly quickly. I really just want you to see what the reflection looks like. The real appreciation here is think about, well, what happens with rigid transformations. So, it's gonna look something like this. The reflection. The reflection looks something like this. So what's preserved? And in general, this is good to know for any rigid transformation what's preserved. Well, side lengths. That's actually one way that we even use to define what a rigid transformation is. A transformation that preserves the lengths between corresponding points. Angle measures. Angle measures. So, for example, this angle here, the angle A, is gonna be the same as the angle A prime over here. Side lengths, the distance between A and B is going to be the same as the distance between A prime and B prime. Perimeter. If you have the same side lengths and the same angles, the perimeter and area are also going to be preserved. Just like we saw with the rotation example. These are rigid transformations. These are the types of things that are preserved. Well, what is not preserved? Not preserved. And this just goes back to the example we just looked at. Well, coordinates are not preserved. So as we see, the image of A A prime has different coordinates than A. B prime has different coordinates than B. C prime, in this case, happens to have the same coordinates as C because C happens to sit on our, the line that we're reflecting over. But D prime definitely does not have the same coordinates as D. So, most of or let me say, coordinates of AB ABD. Coordinates of A B D not preserved. After transformation, or their images don't have the same coordinates. After transformation. The one coordinate that happened to be preserved here is Cs coordinates. Because it was right on the line of reflection. And you can also look at other properties so how it might relate how different segments might relate to lines that would not be that were not being transformed. So, for example, right over here before transformation, CD is parallel to the Y axis. You see this right over here. But after the transformation C prime D prime. So this could be C prime D prime is no longer parallel to the Y axis. In fact, now it is parallel to the X axis. So, when you have the relationship to things outside of the things we're transformed that relationship might not. Those relationships may no longer be true after the transformation.

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Rigid transformations: preserved properties (video) | Khan Academy

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Instructor] What we're going to do in this video is think about what properties of a shape are preserved or not preserved, as they undergo a transformation. In particular, we're gonna think about rotations and reflections in this video. And both of those are rigid transformations which means that the length between corresponding points do not change.

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https://www.youtube.com/watch?v=6tPCNYAmXag

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https://youtu.be/XiAoUDfrar0

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Introduction to transformations | Transformations | Geometry | Khan Academy

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This video is just an intro to rigid transformation

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Moving Into Geometry Flips Slides and Turns

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https://www.youtube.com/watch?v=RlqtYZlh-iU

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This is an intro video to ridged transformation. This video talks about reflections, translations, and rotations. TheMathSocial demonstrates ridged transformation using a triangle.

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Orientation and Congruence of Transformations

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This video is just a basic understanding for the slide, flip, and rotate, for other people. That can help other people for better understanding.

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Example of rigid transformation and congruence | Congruence | Geometry | Khan Academy

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- [Instructor] We are told that triangle ABC, which is right over here, is reflected across line l, so it's reflected across line l right over here, to get to triangle A prime B prime C prime. Fair enough. So based on that, they're going to ask us some questions. And I encourage you to pause this video and see if you can figure out the answers to these questions on your own before I work through them. So the first question they say is well, what's A prime C prime? This is really what's the length of segment A prime C prime? So they want the length of this right over here. How do we figure that out? Well, the key realization here is a reflection is a rigid transformation. Rigid transformation, which is a very fancy word. But it's really just saying that it's a transformation where the length between corresponding points don't change. If we're talking about a shape like a triangle, the angle measures won't change, the perimeter won't change, and the area won't change. So we're gonna use the fact that the length between corresponding points won't change. So the length between A prime and C prime is gonna be the same as the length between A and C. So A prime C prime is going to be equal to AC, which is equal to they tell us right over there. That's this corresponding side of the triangle. That has a length of three. So we answered the first question. And maybe that gave you a good clue. And so I encourage you to keep pausing the video when you feel like you can have a go at it. Alright, the next question is what is the measure of angle B prime? So that's this angle right over here. And we're gonna use the exact same property. Angle B prime corresponds to angle B. It underwent a rigid transformation of a reflection. This would also be true if we had a translation, or if we had a rotation. And so right over here, the measure of angle B prime would be the same as the measure of angle B. But what is that going to be equal to? Well, we can use the fact that if we call that measure, let's just call that X. X plus 53 degrees, we'll do it all in degrees, plus 90 degrees, this right angle here. Well, the sum of the interior angles of a triangle add up to 180 degrees. And so what do we have? We could subtract, let's see, 53 plus 90 is X, plus 143 degrees is equal to 180 degrees. And so subtract 143 degrees from both sides. You will get X is equal to, let's see, 180 minus 40 would be 40. 80 minus 43 would be 37 degrees. X is equal to 37 degrees, so that is 37 degrees. If that's 37 degrees, then this is also going to be 37 degrees. Next, they ask us what is the area of triangle ABC? ABC. Well, it's gonna have the same area as A prime B prime C prime. And so a couple of ways we could think about it. We could try to find the area of A prime B prime C prime based on the fact that we already know that this length is three and this is a right triangle. Or we can use the fact that this length right over here, four, from A prime to B prime is gonna be the same thing as this length right over here, from A prime to B prime, which is four. And so the area of this triangle, especially this is a right triangle, it's quite straightforward, it's the base times the height times 1/2. So this area is gonna be 1/2 times the base, four, times the height, three, which is equal to 1/2 of 12, which is equal to six square units. And then last but not least, what's the perimeter of triangle A prime B prime C prime? Well, here we just used the Pythagorean Theorem to figure out the length of this hypotenuse. And we know that this is a length of three based on the whole rigid transformation and lengths are preserved. And so you might immediately recognize that if you have a right triangle where one side is three and the other side is four, that the hypotenuse is five. Three four five triangles. Or you could just the Pythagorean Theorem. You say three squared plus four squared, four squared is equal to let's just say the hypotenuse squared. Well, three squared plus four squared, that's nine plus 16. 25 is equal to the hypotenuse squared. And so the hypotenuse right over here will be equal to five. And so they're not asking us the length of the hypotenuse. They wanna know the perimeter. So it's gonna be four plus three plus five, which is equal to 12. The perimeter of either of those triangles, because it's just one's the image of the other under a rigid transformation. They're gonna have the same perimeter, the same area. The perimeter of either of the triangles is 12. The area of either of the triangles is six. And we're done.

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Finding measures using rigid transformations (video) | Khan Academy

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Instructor] We are told that triangle ABC, which is right over here, is reflected across line l, so it's reflected across line l right over here, to get to triangle A prime B prime C prime. Fair enough. So based on that, they're going to ask us some questions.

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Translations Reflections and Rotations - Geometric Transformations!

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https://youtu.be/XiAoUDfrar0
This video shows you what and how to do rigid transformation.
questions-
What is dilation ?

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Introduction to transformations | Transformations | Geometry | Khan Academy

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Congruent triangles and SSS | Congruence | Geometry | Khan Academy

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GCSE Maths - Congruent Triangle Rules (2026/27 exams)

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https://www.khanacademy.org/math/geometry/hs-geo-transformations/hs-geo-transformations-intro/v/identifying-transformations

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If you’re seeing this message, it means we’re having trouble loading external resources on our website.

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https://youtu.be/r3lN_BADmPQ

This will help you study on Translations, Rotations, and Reflections

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Math 8: 5-7: Similarity and Congruence Transformations

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https://youtu.be/yH4HleVQInU
The video is just a basic understanding and I feel basic things can help see the bigger picture.

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Uploaded by Andrew Kupisch on 2014-05-09.

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Congruent triangles and SSS | Congruence | Geometry | Khan Academy

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Translations Reflections and Rotations - Geometric Transformations!

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Maths Made Easy! Transformations #2: Rotation [O&U Learn]

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Which type of rigid transformation works the best?

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My first question is about dilation. Does it change its shape?
When using reflection can your axis of reflection be anywhere?

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Transformation involves moving an object from its original position to a new position. The object in the new position is called the image. Each point in the object is mapped to a corresponding point in the image.

The following figures show the four types of transformations: Translation, Reflection, Rotation, and Dilation. Scroll down the page for more examples and solutions using the transformations.

In these lessons, we will study the following types of transformations in math:

Translation

involves "sliding" the object from one position to another.

Reflection

involves "flipping" the object over a line called the line of reflection.

involves "turning" the object about a point called the center of rotation.

involves a resizing of the object. It could result in an increase in size (enlargement) or a decrease in size.

Translation, reflection and rotations are called isometric transformations because the image is the same size and shape as the original object. The original object and the image are congruent.

Dilation is not an isometric transformation because the size of the image is not the same as the object. The original object and the image are similar i.e. they have the same shape but different size.

Translation
How to translate a polygon on the coordinate plane? How to translate a shape using translation vectors or column vectors?
Translations GCSE foundation maths transformations translating shape using translation vectors or column vectors.

Reflection

How to reflect a figure across the y-axis and the x-axis?
The following video shows examples of math reflection across the y-axis and x-axis.

How to reflect across horizontal and vertical lines?
GCSE Maths transformations: Reflections made easy in horizontal and vertical lines.

Rotation

How to rotate about the origin?
The following video shows examples of math rotation about the origin (0, 0).
Rotations in the Coordinate Plane
Dilation

The following video shows examples of math dilation (enlargement) with different scale factors.
Enlargement by a positive and negative scale factor
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

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Math Transformation (video lessons, examples and solutions)

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Transformation involves moving an object from its original position to a new position. The object in the new position is called the image. Each point in the object is mapped to a corresponding point in the image. The following figures show the four types of transformations: Translation, Reflection, Rotation, and Dilation.

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In this YouTube video, transformation, reflection, and rotation are explained in simple words.

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Geometry Rotations Explained (90, 180, 270, 360)

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This is a video showing real world problems to help you understand transformations.

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If you are confused about what Translations, Reflections are, they are just Slide and Flip worded differently. Also, just a reminder, the y-intercept isn't the only one that can be the axis of reflection, it can also be an x-intercept.

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Transformations in Geometry: Translations, Reflections, and Rotations

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If you were looking for the trailer for the new Michael Bay movie, you've got the wrong video. Transformations are rotations, reflections, and translations. After a transformation, the coordinates may be different, but the shape has the same size, area, angles, and line lengths. Shmoop-tron to the rescue!

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I don’t really have any questions. Other than are there any other movements than flip slide and rotate.

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Translations Reflections and Rotations - Geometric Transformations!

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I have two questions. 1. What does dilation mean? 2. If you have the axis of reflection, can one side be bigger or smaller than the other or do they have to be the same?

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Translations, Reflections and Rotations

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https://m.youtube.com/watch?v=XiAoUDfrar0

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Introduction to transformations | Transformations | Geometry | Khan Academy

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I want to know If certain symbols would represent slide, flip, or rotation because it would make most diagrams we make more interesting.

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Transformations video that gives definitions + examples to help understand subject. Plus website for further details.

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Introduction to transformations | Transformations | Geometry | Khan Academy

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Two question I still have are when they reflect off each other do the shapes always have to be equal with each other and my other question is what does dilation mean?

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The security system for this website has been triggered. Completing the challenge below proves you are a human and gives you temporary access.

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Directions: Read carefully.

2.

Seven line segments are shown on the graph below. Which of the segments listed could be the image of after a sequence of reflections, translations, and/or rotations?
( Check all that apply, and hit SUBMIT! )

The Bolt Soda Company is designing a logo. They start with the figure shown in the third quadrant. They then add the rotation of this figure 90º clockwise about the origin to the design.

What does the new design look like?

Choose:

Pentagon A is congruent to pentagon B. Which series of transformations can be used to verify this congruence?
Use pentagon A as the pre-image.

Choose:

Triangle A is congruent to triangle B. Which of the following transformations cannot transform triangle A to triangle B?

Choose:

Rectangle ABCD is congruent to rectangle QRST. Which transformation(s) to ABCD show it is congruent to QRST ?
Choose:

Triangles A, B and C are congruent.

a) Which sequence of transformations could verify that triangle A is congruent to triangle B?
Choose:

b) Which sequence of transformations could verify that triangle A is congruent to triangle C?
Choose:

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Grid Rigid Transformations Practice - MathBitsNotebook(JR)

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Directions: Read carefully. 2. Seven line segments are shown on the graph below. Which of the segments listed could be the image of after a sequence of reflections, translations, and/or rotations? ( Check all that apply, and hit SUBMIT! ) The Bolt Soda Company is designing a logo.

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https://youtu.be/6nfVe7GY9XA

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Desmos Transformation Golf - Rigid Motion

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Introduction to transformations | Transformations | Geometry | Khan Academy

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How can you realize that you are using rigid transformation in the graph ?

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https://www.youtube.com/watch?v=VJTxv-tRKj0

What stays the same in rigid transformations?

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3 Types of Transformations *Translations Reflections & Rotations *Math For Kids*

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This video explains Rotations, Reflections, and more. Two questions i have are can a congruent shape change? And why exactly are we learning this and were will it help us on are math journey.

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https://www.mathsisfun.com/geometry/transformations.html

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After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

When one shape can become another using only Turns, Flips and/or Slides, then the two shapes are Congruent

When a Resize is neeeded for one shape to become another (we may also Turn, Flip and/or Slide) the two shapes are

In other words:

When one shape becomes another ...

The shapes are ...

... using only Rotate, Reflect and/or Translate

... using a Resize (may also Rotate, Reflect
and/or Translate)

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Learn about the Four Transformations: Rotation, Reflection, Translation and Resizing

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Geometry Rotations Explained (90, 180, 270, 360)

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I was wondering how could ridged transformation be applied in our daily lives (or how can we see it)? Do we see it through out the day? Is it something we should pay more attention to? Do we need it in certain activities? Or does it not have any application to our lives at all?

<-the video linked helped me figure out how to identify the different transformations->

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These were our steps to completeing the Key in the Lock puzzle (using 2 different ways) 

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I would like to know if there is another way to show whether you are sliding, flipping, or rotating during a transformation. For example i understand using x and y coordinates shows what you did but i want to know if there is a symbol, variable, or expression that is used to show what type of rigid transformation you are doing to the object.

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https://youtu.be/216PtoDvu8o

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Maths Made Easy! Transformations #1: Translation [O&U Learn]

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1.7.1B Rigid Transformations of Functions:  Translations (Video #2)

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https://youtu.be/Tmdrjs9xufc

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This video helps remind you about translations, and even teaches you about Vector Notation, and how to use them correctly when translating. :>

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Maths Made Easy! Transformations #1: Translation [O&U Learn]

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IXL about reflection, rotation, and sliding/

translations

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IXL | Identify reflections, rotations, and translations | 7th grade math

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Improve your math knowledge with free questions in "Identify reflections, rotations, and translations" and thousands of other math skills.

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What does dilations mean? I understand everything else though need help on this

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Math Shorts Episode 14 - Understanding Dilations

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https://www.youtube.com/watch?v=A19TeoJcGXY

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Transformations - Rotate 90 Degrees Around The Origin

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Shifting functions introduction | Transformations of functions | Algebra 2 | Khan Academy

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I don't know how the axis of reflection and axis of symmetry are any different.

I think that the quizlet helped me understand the terms.

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Example of rigid transformation and congruence | Congruence | Geometry | Khan Academy

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Transformations Song | Translations, Rotations, Reflections, & Dilations

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Introduction to transformations | Transformations | Geometry | Khan Academy

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https://m.youtube.com/watch?v=XiAoUDfrar0

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Introduction to transformations | Transformations | Geometry | Khan Academy

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I wonder when in high-school if if 3d be implemented into rigid transformation and if so how.

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I wonder if in highschool there will be even more and complicated transitions and if so what are they called, and what do they do?

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Is there other terms for rigid transformation/ axis of reflection?

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Elise G. Rigid Transformation Information

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Elise G. Rigid Transformation Information

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https://mathbitsnotebook.com/Geometry/Transformations/TRRigidTransformations.html

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A rigid transformation (also called an isometry ) is a transformation of the plane that preserves length. , and combinations of these three transformations are "rigid transformations". Reflections, translations, rotations

While the pre-image and the image under a rigid transformation will be congruent,
they may not be facing in the same direction.

For a review of reflections, see the Refresher section Transformations: Reflections.
Now let's expand that knowledge of reflections in relation to geometry.

A reflection over line m (notation r m ) is a transformation in which each point of the original figure (pre-image) has an image that is the same distance from the reflection line as the original point, but is on the opposite side of the line. A reflection is called a rigid transformation or isometry because the image is the same size and shape as the pre-image.

f the segments joining each point to its image.
The reflection line, m, is the perpendicular bisector o
Notice that these segments are parallel , since they are perpendicular to the same line.

In this reflection that maps Δ (are not necessarily equal), but the segments representing these distances are parallel., in this diagram, is clockwise is lettered counterclockwise ABC to Δ A'B'C', the distances from the pre-image points to the image points vary
A-B-C, while the
A'-B'-C'. When lettering changes direction, in this manner, the transformation is referred to as a non-direct or opposite isometry.
Orientation (lettering): The lettering of the points of the pre-image image

Properties preserved under a line reflection from the pre-image to the image.
1. distance (lengths of segments remain the same)
2. angle measures (remain the same)
3. parallelism (parallel lines remain parallel)
4. collinearity (points remain on the same lines)
----------------------------------------------------------
The orientation (lettering around the outside of the figure), is not preserved . The order of the lettering in a reflection is reversed (from clockwise to counterclockwise or vice versa).

When formally defining a reflection, there are two situations (cases) that need to be considered for a precise definition. Is the point being reflected lying ON the line of reflection, or NOT ON the line of reflection?

A reflection is a rigid transformation (isometry) that maps every point P in the plane to point P', across a line of reflection, m, such that:

Case 1: if point P is ON line m, the
point is its own reflection ( P = P') and

(point P is "fixed").

Case 2: if point P is NOT on line m, then m is the perpendicular bisector of

where

For a review of translations, see the Refresher section Transformations: Translations.
Now let's expand that knowledge of translations in relation to geometry.

A translation (notation T a,b ) is a transformation which "slides" a figure a fixed distance in a given direction. In a translation, ALL of the points move the same distance in the same direction.
A translation is called a rigid transformation or isometry because the image is the same size and shape as the pre-image.

In this translation that maps Δ. , in this diagram, is counterclockwise is also lettered counterclockwise ABC to Δ A'B'C', the distances from the pre-image points to the image points are equal, and the segments representing these distances are parallel parallel.

In addition, the corresponding segment sides of the pre-image and image are

Orientation (lettering):The lettering of the points of the pre-image A-B-C, and the image A'-B'-C'. When lettering order remains the same, the transformation is referred to as a direct isometry.

Properties preserved under a translation from the pre-image to the image.
1. distance (lengths of segments remain the same)
2. angle measures (remain the same)
3. parallelism (parallel lines remain parallel)
4. collinearity (points remain on the same lines)
5. orientation (lettering order remains the same)

A translation is a rigid transformation of the plane that moves every point of a pre-image a constant distance in a specified direction.

There are several ways to indicate that a translation is to occur:

mapping:
Example:
Read: "the x and y coordinates are mapped to x + 8 and y - 6 respectively".
( x, y) → ( x + 8, y - 6)

notations: The 8 tells you to "add 8" to all
Example: T 8,-6 or T 8,-6 (
x, y) = ( x-coordinates, while the -6 tells you to "subtract 6" from all of the y-coordinates. x + 8, y - 6)

description:"8 units to the right and 6 units down."
Example:

A verbal or written description of the translation is given.

Movement: Remember that adding a negative value (subtracting), indicates movement left and/or down , while adding a positive value indicates movement right and/or up .

Using vectors to show movement of a translation:

A vector is represented by a directed line segment, a segment with an arrow at one end indicating the direction of movement. Unlike a ray, a directed line segment has a specific length. The Pythagorean Theorem can be used to find the length of a vector in the coordinate plane.

Vectors used in translations are what are known as "free vectors", which are a set of parallel directed line segments. A vector translation moving "8 units to the right and 6 units down" can be written as

. You may also see

.

There is an "identity" translational transformation that maps all points onto themselves (no movement of the pre-image), expressed as ( x, y) → ( x, y). (You may see the identity transformation denoted by the letter I.)

For a review of rotations, see the Refresher section Transformations: Rotations.
Now let's expand that knowledge of rotations in relation to geometry.

A rotation of (notation ) is a transformation which "turns" a figure about a fixed point, θ degrees RC,θ C, called the center of rotation . When working in the coordinate plane, the center of rotation should be stated, and not assumed to be at the origin. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation . A rotation is called a rigid transformation or isometry because the image is the same size and shape as the pre-image.

An object and its rotation are the same shape and size, but the figures may be positioned differently.

During a rotation, every point is moved the exact same degree arc along the circle
defined by the center of the rotation and the angle of rotation.
(The dashed arcs in the diagram below represent the circles, with center P, through each of the triangle's vertices.
Note that PC=PC', for example, since they are the radii of the same circle.)

A positive angle of rotation turns a figure counterclockwise (CCW),negative angle of rotation turns the figure clockwise,
and a (CW).

In this rotation that maps Δ, in this diagram, is counterclockwise is also lettered counterclockwise ABC to Δ A'B'C', the distances from the pre-image points to the image points vary (as they are the radii of the circles). Unlike reflections and translations, the segments connecting pre-image and image points are NOT parallel. pre-image

Orientation (lettering):The lettering of the points of the image A-B-C, and the A'-B'-C'.

Properties preserved under a rotation from the pre-image to the image.
1. distance (lengths of segments remain the same)
2. angle measures (remain the same)
3. parallelism (parallel lines remain parallel)
4. collinearity (points remain on the same lines)
5. orientation (lettering order remains the same)

There are two possible directions to travel when rotating.

Look at point P in the diagram at the right. The positive x-axis is considered the "starting" location of 0º. You can have an angle of rotation of positive 135º counterclockwise to point P, or a negative angle of 225º clockwise to point P.
Notice that together, the angles (135º and 225º) form a complete circle (360º).

It is even possible to have one complete revolution (360º) PLUS and additional 135º to get to P (an angle of rotation of 495º).

When working in the coordinate plane, it is important to have a visual understanding of how the quadrants are divided through rotational angles. The most common angles seen on the grid are multiples of 15º. The "favorite" angles are usually 30º, 45º, 60º, 90º, 180º, and 270º.

There is an "identity" rotational transformation of 0º that maps all points onto themselves (no movement of the pre-image), written RC,0 ( P) = P. (You may also see the identity transformation denoted by the letter I.)

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Rigid Transformations (Isometries) - MathBitsNotebook(Geo - CCSS Math)

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A rigid transformation (also called an isometry ) is a transformation of the plane that preserves length. , and combinations of these three transformations are "rigid transformations". Reflections, translations, rotations While the pre-image and the image under a rigid transformation will be congruent, they may not be facing in the same direction.

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What is dilation?
What does congruent mean?
I do know what rigid transformation mean and it when an object doesnt change shape or size.

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https://www.youtube.com/watch?v=XiAoUDfrar0

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Introduction to transformations | Transformations | Geometry | Khan Academy