Geometry Dilation

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For the next lesson, we are going to go over the fourth and final transformation, and that is dilation.0002

Dilation is a transformation that alters the size of a geometric figure, but does not change the shape.0009

The first three transformations that we went over (those were translation, reflection, and rotation) are all congruence transformations,0017

meaning that when you perform that transformation, the pre-image and the image are exactly the same; they are congruent.0026

Dilation is the only transformation that is not a congruence transformation.0034

Now, it says that it alters the size, but not the shape; so that means that the shape is the same, but the size can change.0041

That, we know, is similarity; so for dilation, the pre-image and the image are going to be similar.0052

They are not going to be congruent; they could be, if they have the same ratio;0062

but otherwise, the pre-image and the image are going to be similar.0068

If this is the pre-image and this is the dilated image, then it got smaller from the pre-image to the image.0075

That is what is called a reduction: it got smaller.0084

If this is the pre-image, and this is the dilated image, then it got bigger, so it is an enlargement.0088

So, we are going to go over that next, which is scale factor.0095

Now, the scale factor (we are going to use k as the scale factor) is the ratio between the image and the pre-image.0100

The image is the dilated image; it is the new image; the pre-image is the original.0119

So, any time you see "prime"--here we see A'--that has to do with the new image, the dilated image.0126

And then, we know that this is the pre-image.0138

For a dilation, we are going to have a center; this is the center, C.0141

And we are going to base our dilation (meaning our enlargement/reduction) on this center.0148

Now, again, the scale factor is the image to the pre-image.0156

We can also think of it as from the center to...and then, which one is the image, this one or this one?0169

We know that it is the prime; whenever you see the prime, that is the clear indication that it is going to be part of the image.0178

So, if I want to measure the length from C, the center, to the image, that point right there, that is going to be CA', that segment.0184

That has to do with the image; so that is going to be the numerator, over...C to the pre-image is that point right there, so CA.0204

So, it is going to be CA' to CA.0216

Now, if I were to draw a line from C to A' to show the length, this center, the image, and the pre-image are all going to line up.0222

It is always going to line up; that is what we are going to base it on.0246

So, we are going to use that to draw some dilations: so again, it is the center to the image0249

--anything that says "prime"--that length, over the center to the pre-image.0261

This, we know, is an enlargement, because this was the pre-image, and this is the dilated image; it got bigger.0271

So, from here to here, it is bigger; so my scale factor for this one...if I say that k is 2, then it is actually 2/1.0279

So then, this number up here has to do with my image, and this number down here has to do with my pre-image.0296

That means that my image is twice as big, or as long, as my pre-image; that is scale factor--that is what it is talking about.0302

It is comparing the image to the pre-image; so I can also use this ratio for the length, the distance, from the center.0314

If all of this, from here to here, is 2, then from my center, the distance away from the center of the pre-image is going to be 1.0330

That is also talking about the scale factor; not only is it talking about the length or the size of the image and the pre-image,0346

but it is talking about the distance away from the center.0356

So, if this distance, from here in the image to the center, is 2; then from the pre-image to the center is 1.0359

And they are always going to line up; so the center, to B, to B'...they are all going to line up.0368

This one is an enlargement: it got bigger--it is twice as big as that.0377

Now, for the second diagram here, this is my center; this is A'; and this is A.0383

That means that this is my pre-image, and this is my dilated image.0392

Again, if I find the distance from my pre-image to that, it is all going to line up--the center, A', and A will line up.0404

Now, here, because the image right here, P to A'...let's say this is 1, and this is 1; that means that0422

from my center to my image is 1; and then, from the center to the pre-image has to be 2,0440

even though this part from here to here is 1; remember: it is from the center, so this point, all the way to the pre-image, is 2.0458

My scale factor is 1/2; now, if you notice, this is the pre-image (this is the original), and then this is my dilated image.0466

See how it got smaller: it is like saying, "Well, the image is half the size of the pre-image."0473

This is half the size of my pre-image; that is what the scale factor is saying--it is comparing these two.0483

And so, keep that in mind: this is image over pre-image, I/P.0491

Now, going over scale factor some more: if the scale factor is positive, then it is on the same side of the center point.0501

The two diagrams that we just went over were both on the same side, meaning that they were on the right side of the center.0514

Here is the center; this is to the right; and this is to the right...because the scale factor could be negative.0523

When it is positive, they are just on the same side; so then, the center to right here (let me always do that in red)...they are on the same side.0531

They are kind of going in the same direction; it is CA', over CA--it is always starting out from C, and it is going to there, and then C to A.0550

Now, if it is negative, then it is going to go to the opposite side of the center point; they are going to be on opposite sides.0568

So, here is A'--here is my image--and here is the pre-image.0575

Now, from the center point, C, if I go to the pre-image, it is going this way.0587

Now, let's say that my scale factor is -2: k = -2; so it is -2/1.0599

Because my scale factor is negative, I know that this is CA' over CA.0609

This right here, that I just drew, is this right here; so that means that this length right here is 1, because that is what that shows me: CA is 1.0622

Then, because CA' is negative, instead of going this way and then drawing it twice as big--0639

instead of going this way, I have to go the opposite way--that is what it is saying.0651

So then, if CA is going one way, that is your pre-image that is going this way, kind of to the left.0655

Then, this has to be drawn twice as long; so if CA is 1, then I have to draw CA' with a length of 2, but going in the opposite direction.0666

Just think of that negative as opposite; so then, if it went this way, then CA' is going to go twice as long.0677

From here to here is going to be 2; that is what it means to be negative.0687

So, if you have to draw your dilated image, that means that you are not going to have this.0696

You are just going to base it on this and this point; you are going in this direction--that is the pre-image.0702

So, when you draw your dilated image, instead of continuing on like you would here (this was your center, to the pre-image,0710

and then to draw your image, you kept going and had a line of C, A, A'--you are going to keep going0719

in that same direction if it is positive), because it is not positive, instead of going in the same direction,0725

we are going to turn around and go in the opposite direction.0730

And you are still going to draw it so that CA' is 2; so this is still 2--CA', that length right here, is 2.0734

It is the same thing, but it is going in the opposite direction.0745

And notice how C, A, and A' still line up, no matter what; if the scale factor is positive or negative, they are still going to all line up.0750

C, A, and A', C, A, and A'--they are all going to line up.0760

And then, next, we have enlargement and reduction--we kind of talked about this already.0766

If the absolute value of k (meaning regardless of if it is positive or negative) is bigger than 1, then it is going to be an enlargement,0773

because it is talking about the image and the pre-image.0784

So, if k, let's say, is 3, isn't it 3/1?0788

Isn't that saying that the image is 3 times bigger than the pre-image?--because we know that it is the image over the pre-image, I/P.0792

So then, if the image's length is 3, then the pre-image is 1, so then it has to be bigger,0802

because the number with the image is bigger than the number with the pre-image, so it is getting bigger--it is enlarging.0811

And that is just what this is saying; this is the pre-image, and this is the dilated image; it is getting bigger--small to bigger.0823

And then here, if the absolute value of k (meaning with no regard to whether it is positive or negative)...0833

if you have a fraction between 0 and 1 (let's say 1/2 or 1/3, or whatever...any fraction that is smaller than 1,0842

and greater than 0--it is going to be greater than 0, because it is absolute value), then it is going to be reduction,0853

because then you are saying that, let's say, for example, if k is 1/2, then again, this is image;0861

this number is associated with the image, and then this number is with the pre-image.0869

You are saying that the pre-image number is bigger than the image number.0877

If the image is 1, then the pre-image will be double that; so then, the pre-image,0882

the image before the dilation, is bigger than the dilated image; it is actually getting smaller--that is called reduction.0886

From C (if we are going to say that this is C), this is the pre-image; it is still going to line up.0896

That means that we know that A' has to be on that line; but it can't go this way, so it is going to be halfway between.0910

That means that, because, again, image is going to be CA'/CA, then this number right here...0919

if that is 1, then it is going to be 1 over whatever this whole thing is here, CA.0931

A review: If your scale factor, k, is positive, then you are going to keep drawing it in the same direction from the center.0939

So, it is on the same side of the center.0949

If it is negative, well, C to the pre-image is going in one direction; then, to go to the dilated image, you are going to go in the opposite direction.0954

It is like you are going to turn around if it is negative.0964

And then, regardless of it is positive or negative, if the absolute value is greater than 1, then it is going to be an enlargement,0969

because that means that the dilated image is larger than the pre-image.0978

And then, if it is between 0 and 1, then the top number is going to be smaller than the bottom number.0986

That means that the image is going to be smaller than the pre-image.0993

Let's do our examples: Find the scale factor used for the dilation with center C and determine if it is an enlargement or a reduction.1002

Here are our two similar figures, STUV and...here is the other image, because this has T' and U'.1015

So, I know that this bigger one is the pre-image; remember: it is always image to pre-image.1028

We see that this has "prime," T', and that has to do with the image, the dilated image, the new image.1044

That is going to be this right here.1054

That means that we went from pre-image, which is STUV, to this prime.1057

Because it got smaller, we know that it is a reduction; for #1, it is a reduction.1068

To find my scale factor, I want to find the ratio (because it is proportional, because these are similar;1081

dilation is always similar): so, do I have corresponding parts?1095

I have this right here, with this right here; so I do have the lengths of corresponding sides.1103

This one has to do with my new image, my dilated image, 4; and then, this right here is my pre-image; that is 9.1112

It is going to be proportional; so, my image length is 4; in the pre-image, the corresponding side is 9.1122

Now, even though this also has to do with the length of my pre-image, I can't use that,1136

because I don't have the other corresponding side; I don't have the measure of that side right there, which is corresponding.1142

So, I have to use the corresponding pair, 4 and 9.1148

And be careful: it is not 9/4, because the image number has to go on top.1153

This is the image; this is the pre-image; so it is image over pre-image, 4/9; so this is the scale factor.1158

The next example: If AB is 16, find the measure of the dilation image of AB with a scale factor of 3/2.1174

AB is a line segment; let's say that that is A, and that is B; and this has a measure of 16.1189

Find the measure of the dilation image of AB with a scale factor of 3/2.1203

Now, remember: our scale factor is image over pre-image, or CA' over CA.1209

We are going to use this as a reference for our scale factor; we know that it is 3/2.1224

Since the number that has to do with the image, the new image, is greater than this number down here,1231

which is the pre-image, I know that it is an enlargement--it got bigger--because the new image is bigger than the pre-image.1238

This is enlargement; that means that this pre-image is going to get bigger; my new image is going to be bigger than this.1246

Let's draw a center point: if that is my center, C, this right here, CA, is this number.1260

So, CA (I should do that in red) is what? 3/2--that is the scale factor;1279

so, my CA, the number associated with my pre-image, is 2; that means that CA is 2.1296

That means that my CA' is going to be 3.1303

Now, I know, because it went from C to A in this direction, and my scale factor is positive...1310

that means that I am going to keep going in that same direction to draw A'.1317

That means that CA' is going to be 3, so I can't draw it twice as long as this--I can't draw another 2--1322

because I have to make sure that from C to A' is going to be 3.1332

So, if this is 2, well, let me just break this up into units, then; if this is 1, then this is 2.1339

So then, 1, 2, and then another one right here...it is 1, 2, 3 in the same direction; and then, this will be A',1347

because again, it is not from here to here; it is from C to A'; C to A' is 3.1363

That means that if this is 1, then this whole thing is 3; and I just found that from my scale factor.1370

So, CA' is 3; CA is 2; make sure that C, A, and A' all line up.1378

And then, the same thing works here: this is CB' over CB; this is also 3/2.1386

We have this, and then we are going to keep going in that same direction, because it is a positive.1412

So, CB, we know, is 2; that means that CB'...when I draw my B', it has to be 3.1417

So, if this is 1, and this is 2, then this is a little bit more...and that is C...another one more...that makes this whole thing 3, and this is B'.1427

From here to here is going to be my dilated image.1449

And then, to find the measure of it...now, it didn't say to draw it, but then, just in case1458

you would have to draw it on your homework, or you have problems where you have to draw it,1463

just keep in mind that if it is a positive scale factor, make sure that C, A, and A' all line up;1468

and it is all going to go in the same direction; and then just do that for each of the points.1476

And then, if this is 16, remember: the image to the pre-image...this is the image to the pre-image, so the scale factor is 3/2.1484

That is the ratio; it equals...and then again, the ratio between these two is going to be the same.1497

So then, AB, my new image, is going to go on the top, and that is what I am looking for--this x.1504

That is x, over my pre-image (is 16); so this is a proportion--I can solve this by cross-multiplying:1512

2 times x equals 3 times 16, or I can just do this in my head: this is 2 times 8 equals 16,1523

so 3 times 8 is going to be 24--it is just the equivalent fraction (3/2 is the same thing as 24/16).1535

If you want to just cross-multiply, then it would be 2 times x, 2x, is equal to 3 times 16, which is 48.1546

And then, divide the 2; x = 24.1556

So then, right here, this has a measure of 24; so AB is 24.1564

The next one: Find the coordinates of the image with a scale factor of 2 and the origin as the center of the dilation.1580

Here is the center that we are basing that on; and our scale factor is 2, which means it is 2/1.1589

And I want to write image over pre-image; and then, you can write center...1601

Well, we already have C, so let's label that as P: PA'/PA.1611

So then, the scale factor is 2/1 (let me just write that here, too, so that you know that this is 2, and then this is 1).1622

PA, going in that direction, is 1; that means that I have to draw PA' as 2--it is going to go 1, 2.1637

And again, you are not starting from here and going 2 more; you are starting back at P and then going 2.1652

This right here is A'; and then, to PB...if that is 1, then to PB' is 1, and then 2; so this is B'.1658

And then, PD--that is this--is 1; then, PD' is 2; so then here, it is 1, and you go another--that is D'.1689

Make sure that they line up: P, D, and D'.1710

And then, go from P to C...like that; make sure that your lines are straight.1714

You can also use slope to help you: here, you know how we went down one, and then 1, 2, 3, 4: that is a -1/4 slope.1726

So then, I can go another 1, 2, 3...and then that would be right here; so it is going to keep going this way: this is C'.1737

Then, my new image is going to be from here, all the way down to here, to there, and there, and then there.1753

Make sure that your image has the same shape as your pre-image; it is just going to have a different size, but it is going to be the same shape.1769

That is my image; and then, I want to find the coordinates.1781

So then, A' is going to be (0,2); B' is going to be (4,4); C' is...this is 6, 7, 8, so (8,-2); and D' is (4,-4): those are my coordinates.1786

Now, if you had to find the coordinates without graphing--if you were just given the scale factor,1822

and you had to find the new coordinates for it--let's look at the original.1835

Let's look at the pre-image, just ABCD, the pre-image: the coordinates for the pre-image,1838

before we changed it, before we dilated it, were: (0,1); B was (2,2); C was...where is C?...right there: it is (4,-1); and D is (2,-2).1848

We went from the pre-image to image: notice how our scale factor is 2.1875

That means that our image is twice as big as our pre-image.1880

Look at this: your image, your coordinate points, are twice as big as your pre-image coordinates.1885

This one is (0,1), and this is (0,2); (2,2), (4,4); (4,-1), (8,-2); (2,-2), (4,-2).1896

It is like you just multiplied everything by 2, by the scale factor.1906

So then again, if you are given the coordinates of the pre-image, and you have a scale factor of 2,1910

that means that your image is going to be twice as big as your pre-image;1917

so then, you just have to multiply your pre-image by 2 to get your image; so then, those are your coordinates.1920

And the final example: Graph the polygon with the vertices A, B, C; use the origin as the center of dilation, and a scale factor of 1/2.1930

Let's copy these points: A is (1,-2); B is (6,-1); C is (4,-3); this was A, B, and C...so our polygon is a triangle.1942

And then, using the origin as the center of the dilation, and a scale factor of 1/2...again, the scale factor, k, is image over pre-image.1978

If this is a little confusing, you can always just, instead of "image," write "prime" or "new image" or something like that.1995

That way, you know what coordinates go with which one--image or pre-image.2003

This is our pre-image; our scale factor is 1/2.2011

I am going to use P for my center; what I can do is...for the image, it is PA', PB', PC', all for the image.2019

And then, the pre-image is just PA, to the original.2034

And our scale factor is 1/2: that means that our pre-image is twice the measure of our image--the pre-image is going to be bigger.2039

That means that, since the scale factor is 1/2 (which is smaller than 1), it is going to be a reduction.2050

The pre-image, the original, is larger than the new image, so the new image is smaller.2057

That means that our new image is going to be smaller than this.2063

Here, draw...again, from here, it is going to be like this; so then, PA (that is that) is 2.2072

That means that we are going to say that this whole thing is 2.2082

That means that PA' is going to be half that; it is going to be 1.2086

For PA', I am going to label it right there, halfway, because this whole thing is 2; then this has to be 1; that is going to be A'.2091

And then, for here to here, for C, our slope is 1, 2, 3 for 1, 2, 3, 4.2107

So then, here, we can just estimate where our halfway point is going to be, because this PC is 2; that means that PC' has to be 1.2122

So, if this whole thing is 2, then C' is going to be right there, halfway.2133

This is C', and then, for PB, it is going to go like that.2140

Our slope is down 1, over 6; so then, remember: our scale factor is going to be half that.2154

If this whole thing is 2, I have to find halfway; so if this is down 1, then it is only going to be down a half, because, remember, it is half of that.2160

So, go down 1/2; and then, going right was 6--we went down 1, right 6.2170

So then, instead of going all the way to 6, I have to go just halfway, which is 3; so it is going to be half, down half, and right 3.2176

There is my B'; so my new image is from here to here to here.2185

And then, let's see: all we had to do is just graph the polygon, and then use the origin as the center and a scale factor of 1/2.2199

Again, if the scale factor is smaller than 1, then you know that it is going to be a reduction; it is going to be smaller.2208

If it is greater than 1, then we know that it is going to be bigger than this pre-image.2216

That is it for this lesson; thank you for watching Educator.com.2223