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### Mapping

- Mapping is a transformation of a pre-image to another congruent or similar image
- Types of transformations
- Rotation: turn
- Translation: slide
- Reflection: flip
- Dilation: enlarge/reduce
- In a plane, an isometry is a transformation that maps every segment to a congruent segment

### Mapping

Describe the transformation that occurred in the mapping.

Describe the transformation that occurred in the mapping.

Describe the transformation that occurred in the mapping.

Determine whether the transformation is an isometry.

- BC = EF = 300m
- AC = DF = 500m
- m∠ABC = m∠DEF = 90
- ∆ ABC ≅ ∆ DEF (HL)

If the transformation from an octagon to another octagon is isometry, then the two octagons are congruent.

describe the transformation that occurred in the mapping.

A triangle can _____ transform to a rectangle through dilation.

If the transformation from ∆ABC to ∆DEF is an isometry, then all the corresponding angles in these two triangles are congruent.

describe the transformation that occurred in the mapping.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Mapping

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Transformation 0:04
- Rotation
- Translation
- Reflection
- Dilation
- Transformations 1:45
- Examples
- Congruence Transformation 2:51
- Congruence Transformation
- Extra Example 1: Describe the Transformation that Occurred in the Mappings 3:37
- Extra Example 2: Determine if the Transformation is an Isometry 5:16
- Extra Example 3: Isometry 8:16

### Geometry Online Course

### Transcription: Mapping

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over mappings.*0002

*Mappings are transformations of a pre-image to another congruent or similar image.*0006

*When you have an image, and it has either a congruent or a similar relationship with another image,*0014

*then that is transformations, which is also mappings.*0023

*The different types of transformations are rotation, translation, reflection, and dilation--four types of transformations.*0030

*The first one, rotation, is when you take the pre-image (the pre-image is the original, the initial, the first image), and it rotates.*0041

*So, you turn it to make the second image; it is just rotating or turning.*0055

* Translation is when you take the image and you slide it, so it just moves; that is it.*0063

*It doesn't rotate; it doesn't do anything but just move--a slide or a glide.*0070

* Reflection is when you flip the image: you have two images, and they are just reflections of each other.*0079

*And dilation is when you enlarge or reduce the image.*0086

*Again, transformations are when you perform one of these four to a pre-image to create another image that is either congruent or similar.*0092

*Here are just some image: with rotation, you take this image (this is the pre-image), and to make this image, all I did was rotated it--just turned.*0107

*It is the same image, and it just rotated.*0121

*Translation: again, this is the pre-image, and it just slides or glides--just moves.*0124

*It stays the same; it just moves to a different location, a different place right here.*0132

*Reflection is, again, like a mirror reflection; they are reflections of each other.*0138

*And dilation is when an image gets larger or smaller; this is the same shape, but different size.*0146

*It just gets bigger, or it gets smaller; but it has to be the same shape.*0157

*And if two images have the same shape, but a different size, then we know that they are similar.*0161

*With dilation, it will be similar images; so then, the other three (rotation, translation, and reflection) are all congruence transformations,*0167

*because when you perform these transformations, they don't change; they are still congruent in size and shape.*0180

*Nothing changes; it is just the way you position it, or the way you rotate or reflect or translate the image; it is just going to stay the same.*0188

*And that is called an isometry; an isometry is a transformation that maps every segment to a congruent segment.*0200

*Again, when you either rotate, translate, or reflect, the images are congruent; they are the same.*0209

*Describe the transformation that occurred in the mappings.*0220

*Here, we want to know what happened with this image to get this image.*0223

*All that this did was to turn; so from this to this, it just turned a certain angle amount; and so, this is rotation, because it just rotated.*0233

*This right here, from this image to this image, the pre-image to the image, looks like a reflection; it looks like it is looking in a mirror.*0253

*It reflects, so this is reflection.*0264

*And then, for these two, see how one is bigger than the other.*0271

*So, even though it kind of looks like reflection, it can't be, because reflection has to be exactly the same.*0276

*It has to line up exactly the same way and be the same size; they have to be congruent.*0283

*But here, because this image and this image are different sizes, but the same shape, this has to be dilation.*0289

*Think of dilation as...when something dilates, it gets bigger; so it is getting bigger or getting smaller.*0309

*The next example: Determine if the transformation is an isometry.*0318

*Remember: an isometry is when you have two images, and the pre-image and the image are congruent; that is for rotation, reflection, and translation.*0322

*We just want to see if these two are congruent.*0335

*Now, to determine if two triangles are congruent, remember: we have those theorems and postulate,*0340

*where it says Angle-Side-Angle (they are corresponding parts), Side-Angle-Side, Side-Angle-Angle, and Side-Side-Side.*0346

*Those are the different congruence theorems and postulate.*0356

*We want to see if this pair of triangles applies to any of those.*0364

*Now, here I see that an angle is congruent here, and a side, and they are corresponding parts.*0371

*Now, for this one, angle B is corresponding with angle E; this one is given, and this one is not.*0381

*And angle C is corresponding with angle F, but this one is not given, and this one is.*0390

*I want to find the measure of this angle, and I can do that by taking these two and subtracting it from 180; so it is 180 - (105 + 40).*0396

*This right here is 145; so if you subtract this from 180, you will get 35 degrees.*0418

*The measure of angle C is 35 degrees.*0432

*Now, the measure of angle E is going to be 180 - (35 + 40); now, we don't have to solve for that, because we know that this is 35;*0439

*this angle is congruent to this angle; and of course, that means that this angle has to be congruent to this angle.*0458

*So, I have Angle-Side-Angle, because this pair of angles is congruent; their sides are congruent; and the angles are congruent.*0467

*So, because of this, these two triangles are congruent, and therefore, this is an isometry; so it is "yes."*0481

*The next one: Show that triangle ABC and triangle DEF are an isometry (so it is the same type of problem).*0497

*Now, for this, we have the coordinates of each vertex for each triangle.*0508

*So, we can find the measure, or the length, of each side.*0520

*I can just find the measure of that side with the length of that side and compare them and see if they are congruent.*0528

*I have to use the distance formula: the distance formula is (x _{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}.*0536

*Remember: this means the second x; so it is the second x, minus the first x, and the second y, minus the first y.*0550

*A is (-6,1); B is (-4,6); and C is (-2,3); then, D is (1,-1), E is (3,4); and F is (5,1); find the distance of AB.*0559

*So, AB is, let's see, (-4 + 6) ^{2}, and then (6 - 1)^{2}.*0608

*I have that this is 2 squared, plus 5 squared, which is 4 + 25, which is √29.*0626

*And then, let's do DE: DE is (3 - 1) ^{2} + (4 + 1)^{2}; and it is plus because it is 4 - -1.*0649

*So, this is 2 squared, plus 5 squared, the same as that; so it is the square root of 29.*0668

*I know that AB is congruent to DE; now, BC is (-2 + 4) ^{2} + (3 - 6)^{2};*0678

*so, this is 2 squared, plus -3 squared, which is 4 +...this is 9; that is √13.*0704

*And then, what is corresponding with BC? EF.*0721

*EF is (5 - 3) ^{2} + (1 - 4)^{2}: 2 squared plus -3 squared is the same, √13; so those two are the same.*0725

*And then, AC (I am running out of room here) is (-2 + 6) ^{2} + (3 - 1)^{2}.*0753

*That is 4 ^{2} + 2^{2}; this is 16 + 4, is 20; √20...*0781

*And then, AC and DF...DF is (5 - 1) ^{2} + (1 + 1)^{2}, so this is 4^{2} + 2^{2},*0794

*16 + 4; that is 20, so that is √20; so then, these two are the same.*0827

*So, by the Side-Side-Side Congruence Theorem, they are congruent, which means that it is an isometry.*0838

*That is it for this lesson; thank you for watching Educator.com.*0850

0 answers

Post by Rafael Wang on August 29, 2016

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0 answers

Post by Shahram Ahmadi N. Emran on July 10, 2013

Thanks

0 answers

Post by abdulrahim ahmed on November 8, 2011

thanks