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### Dilation

- Dilation: transformation that alters the size of the geometric figure, but does not change its shape
- Scale factor (k): the ratio between the image to the pre-image

### Dilation

A dilation is a transformation that alters both the size and the shape of the geometric figure.

The ratio between the image to the preimage is always larger than 1.

If the scale factor of a dilation is positive, then both of the preimage and the image are ______at the same side of center point.

k is the scale factor of a dilation, If 0 <|k|< 1, then it is an enlargement.

- k = [(CA′)/CA]
- k = [(3 + 4)/4] = [7/4]

- [radius of the image/radius of the preimage] = 2
- radius of the image = 2 * radius of the preimage

A triangle after dilation is still a triangle.

If the scale factor is 1, then the dilation of an image is congruent to the original image.

*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

### Dilation

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
- Dilations
- Scale Factor
- Scale Factor
- Extra Example 1: Find the Scale Factor
- Extra Example 2: Find the Measure of the Dilation Image
- Extra Example 3: Find the Coordinates of the Image with Scale Factor and the Origin as the Center of Dilation
- Extra Example 4: Graphing Polygon, Dilation, and Scale Factor

- Intro 0:00
- Dilations 0:06
- Dilations
- Scale Factor 1:36
- Scale Factor
- Example 1
- Example 2
- Scale Factor 8:20
- Positive Scale Factor
- Negative Scale Factor
- Enlargement
- Reduction
- Extra Example 1: Find the Scale Factor 16:39
- Extra Example 2: Find the Measure of the Dilation Image 19:32
- Extra Example 3: Find the Coordinates of the Image with Scale Factor and the Origin as the Center of Dilation 26:18
- Extra Example 4: Graphing Polygon, Dilation, and Scale Factor 32:08

### Geometry Online Course

### Transcription: Dilation

*Welcome back to Educator.com.*0000

*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 image*0249

*--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 that*0422

*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 going*0719

*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 case*1458

*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

1 answer

Last reply by: Professor Pyo

Thu Jan 2, 2014 4:36 PM

Post by Mirza Baig on December 6, 2013

I have question "do we always have draw a point ????"