Geometry 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 (x₂ - x₁)² + (y₂ - y₁)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 4² + 2²; this is 16 + 4, is 20; √20...0781

And then, AC and DF...DF is (5 - 1)² + (1 + 1)², so this is 4² + 2²,0794

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