WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For the next lesson, we are going to go over the fourth and final transformation, and that is dilation.
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Dilation is a transformation that alters the size of a geometric figure, but does not change the shape.
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The first three transformations that we went over (those were translation, reflection, and rotation) are all congruence transformations,
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meaning that when you perform that transformation, the pre-image and the image are exactly the same; they are congruent.
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Dilation is the only transformation that is not a congruence transformation.
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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.
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That, we know, is similarity; so for dilation, the pre-image and the image are going to be similar.
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They are not going to be congruent; they could be, if they have the same ratio;
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but otherwise, the pre-image and the image are going to be similar.
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If this is the pre-image and this is the dilated image, then it got smaller from the pre-image to the image.
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That is what is called a **reduction**: it got smaller.
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If this is the pre-image, and this is the dilated image, then it got bigger, so it is an **enlargement**.
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So, we are going to go over that next, which is scale factor.
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Now, the **scale factor** (we are going to use k as the scale factor) is the ratio between the image and the pre-image.
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The image is the dilated image; it is the new image; the pre-image is the original.
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So, any time you see "prime"--here we see A'--that has to do with the new image, the dilated image.
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And then, we know that this is the pre-image.
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For a dilation, we are going to have a center; this is the center, C.
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And we are going to base our dilation (meaning our enlargement/reduction) on this center.
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Now, again, the scale factor is the image to the pre-image.
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We can also think of it as from the center to...and then, which one is the image, this one or this one?
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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.
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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.
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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.
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So, it is going to be CA' to CA.
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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.
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It is always going to line up; that is what we are going to base it on.
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So, we are going to use that to draw some dilations: so again, it is the center to the image
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--anything that says "prime"--that length, over the center to the pre-image.
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This, we know, is an enlargement, because this was the pre-image, and this is the dilated image; it got bigger.
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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.
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So then, this number up here has to do with my image, and this number down here has to do with my pre-image.
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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.
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It is comparing the image to the pre-image; so I can also use this ratio for the length, the distance, from the center.
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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.
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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,
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but it is talking about the distance away from the center.
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So, if this distance, from here in the image to the center, is 2; then from the pre-image to the center is 1.
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And they are always going to line up; so the center, to B, to B'...they are all going to line up.
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This one is an enlargement: it got bigger--it is twice as big as that.
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Now, for the second diagram here, this is my center; this is A'; and this is A.
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That means that this is my pre-image, and this is my dilated image.
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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.
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Now, here, because the image right here, P to A'...let's say this is 1, and this is 1; that means that
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from my center to my image is 1; and then, from the center to the pre-image has to be 2,
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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.
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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.
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See how it got smaller: it is like saying, "Well, the image is half the size of the pre-image."
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This is half the size of my pre-image; that is what the scale factor is saying--it is comparing these two.
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And so, keep that in mind: this is image over pre-image, I/P.
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Now, going over scale factor some more: if the scale factor is positive, then it is on the same side of the center point.
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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.
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Here is the center; this is to the right; and this is to the right...because the scale factor could be negative.
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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.
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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.
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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.
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So, here is A'--here is my image--and here is the pre-image.
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Now, from the center point, C, if I go to the pre-image, it is going this way.
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Now, let's say that my scale factor is -2: k = -2; so it is -2/1.
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Because my scale factor is negative, I know that this is CA' over CA.
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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.
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Then, because CA' is negative, instead of going this way and then drawing it twice as big--
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instead of going this way, I have to go the opposite way--that is what it is saying.
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So then, if CA is going one way, that is your pre-image that is going this way, kind of to the left.
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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.
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Just think of that negative as opposite; so then, if it went this way, then CA' is going to go twice as long.
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From here to here is going to be 2; that is what it means to be negative.
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So, if you have to draw your dilated image, that means that you are not going to have this.
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You are just going to base it on this and this point; you are going in this direction--that is the pre-image.
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So, when you draw your dilated image, instead of continuing on like you would here (this was your center, to the pre-image,
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and then to draw your image, you kept going and had a line of C, A, A'--you are going to keep going
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in that same direction if it is positive), because it is not positive, instead of going in the same direction,
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we are going to turn around and go in the opposite direction.
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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.
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It is the same thing, but it is going in the opposite direction.
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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.
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C, A, and A', C, A, and A'--they are all going to line up.
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And then, next, we have enlargement and reduction--we kind of talked about this already.
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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,
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because it is talking about the image and the pre-image.
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So, if k, let's say, is 3, isn't it 3/1?
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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.
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So then, if the image's length is 3, then the pre-image is 1, so then it has to be bigger,
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because the number with the image is bigger than the number with the pre-image, so it is getting bigger--it is enlarging.
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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.
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And then here, if the absolute value of k (meaning with no regard to whether it is positive or negative)...
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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,
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and greater than 0--it is going to be greater than 0, because it is absolute value), then it is going to be reduction,
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because then you are saying that, let's say, for example, if k is 1/2, then again, this is image;
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this number is associated with the image, and then this number is with the pre-image.
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You are saying that the pre-image number is bigger than the image number.
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If the image is 1, then the pre-image will be double that; so then, the pre-image,
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the image before the dilation, is bigger than the dilated image; it is actually getting smaller--that is called reduction.
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From C (if we are going to say that this is C), this is the pre-image; it is still going to line up.
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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.
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That means that, because, again, image is going to be CA'/CA, then this number right here...
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if that is 1, then it is going to be 1 over whatever this whole thing is here, CA.
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A review: If your scale factor, k, is positive, then you are going to keep drawing it in the same direction from the center.
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So, it is on the same side of the center.
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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.
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It is like you are going to turn around if it is negative.
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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,
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because that means that the dilated image is larger than the pre-image.
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And then, if it is between 0 and 1, then the top number is going to be smaller than the bottom number.
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That means that the image is going to be smaller than the pre-image.
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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.
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Here are our two similar figures, STUV and...here is the other image, because this has T' and U'.
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So, I know that this bigger one is the pre-image; remember: it is always image to pre-image.
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We see that this has "prime," T', and that has to do with the image, the dilated image, the new image.
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That is going to be this right here.
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That means that we went from pre-image, which is STUV, to this prime.
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Because it got smaller, we know that it is a reduction; for #1, it is a reduction.
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To find my scale factor, I want to find the ratio (because it is proportional, because these are similar;
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dilation is always similar): so, do I have corresponding parts?
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I have this right here, with this right here; so I do have the lengths of corresponding sides.
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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.
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It is going to be proportional; so, my image length is 4; in the pre-image, the corresponding side is 9.
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Now, even though this also has to do with the length of my pre-image, I can't use that,
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because I don't have the other corresponding side; I don't have the measure of that side right there, which is corresponding.
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So, I have to use the corresponding pair, 4 and 9.
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And be careful: it is not 9/4, because the image number has to go on top.
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This is the image; this is the pre-image; so it is image over pre-image, 4/9; so this is the scale factor.
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The next example: If AB is 16, find the measure of the dilation image of AB with a scale factor of 3/2.
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AB is a line segment; let's say that that is A, and that is B; and this has a measure of 16.
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Find the measure of the dilation image of AB with a scale factor of 3/2.
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Now, remember: our scale factor is image over pre-image, or CA' over CA.
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We are going to use this as a reference for our scale factor; we know that it is 3/2.
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Since the number that has to do with the image, the new image, is greater than this number down here,
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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.
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This is enlargement; that means that this pre-image is going to get bigger; my new image is going to be bigger than this.
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Let's draw a center point: if that is my center, C, this right here, CA, is this number.
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So, CA (I should do that in red) is what? 3/2--that is the scale factor;
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so, my CA, the number associated with my pre-image, is 2; that means that CA is 2.
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That means that my CA' is going to be 3.
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Now, I know, because it went from C to A in this direction, and my scale factor is positive...
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that means that I am going to keep going in that same direction to draw A'.
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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--
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because I have to make sure that from C to A' is going to be 3.
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So, if this is 2, well, let me just break this up into units, then; if this is 1, then this is 2.
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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',
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because again, it is not from here to here; it is from C to A'; C to A' is 3.
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That means that if this is 1, then this whole thing is 3; and I just found that from my scale factor.
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So, CA' is 3; CA is 2; make sure that C, A, and A' all line up.
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And then, the same thing works here: this is CB' over CB; this is also 3/2.
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We have this, and then we are going to keep going in that same direction, because it is a positive.
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So, CB, we know, is 2; that means that CB'...when I draw my B', it has to be 3.
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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'.
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From here to here is going to be my dilated image.
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And then, to find the measure of it...now, it didn't say to draw it, but then, just in case
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you would have to draw it on your homework, or you have problems where you have to draw it,
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just keep in mind that if it is a positive scale factor, make sure that C, A, and A' all line up;
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and it is all going to go in the same direction; and then just do that for each of the points.
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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.
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That is the ratio; it equals...and then again, the ratio between these two is going to be the same.
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So then, AB, my new image, is going to go on the top, and that is what I am looking for--this x.
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That is x, over my pre-image (is 16); so this is a proportion--I can solve this by cross-multiplying:
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2 times x equals 3 times 16, or I can just do this in my head: this is 2 times 8 equals 16,
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so 3 times 8 is going to be 24--it is just the equivalent fraction (3/2 is the same thing as 24/16).
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If you want to just cross-multiply, then it would be 2 times x, 2x, is equal to 3 times 16, which is 48.
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And then, divide the 2; x = 24.
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So then, right here, this has a measure of 24; so AB is 24.
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The next one: Find the coordinates of the image with a scale factor of 2 and the origin as the center of the dilation.
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Here is the center that we are basing that on; and our scale factor is 2, which means it is 2/1.
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And I want to write image over pre-image; and then, you can write center...
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Well, we already have C, so let's label that as P: PA'/PA.
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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).
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PA, going in that direction, is 1; that means that I have to draw PA' as 2--it is going to go 1, 2.
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And again, you are not starting from here and going 2 more; you are starting back at P and then going 2.
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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'.
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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'.
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Make sure that they line up: P, D, and D'.
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And then, go from P to C...like that; make sure that your lines are straight.
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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.
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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'.
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Then, my new image is going to be from here, all the way down to here, to there, and there, and then there.
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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.
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That is my image; and then, I want to find the coordinates.
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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.
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Now, if you had to find the coordinates without graphing--if you were just given the scale factor,
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and you had to find the new coordinates for it--let's look at the original.
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Let's look at the pre-image, just ABCD, the pre-image: the coordinates for the pre-image,
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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).
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We went from the pre-image to image: notice how our scale factor is 2.
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That means that our image is twice as big as our pre-image.
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Look at this: your image, your coordinate points, are twice as big as your pre-image coordinates.
00:31:35.800 --> 00:31:45.800
This one is (0,1), and this is (0,2); (2,2), (4,4); (4,-1), (8,-2); (2,-2), (4,-2).
00:31:45.800 --> 00:31:50.200
It is like you just multiplied everything by 2, by the scale factor.
00:31:50.200 --> 00:31:56.900
So then again, if you are given the coordinates of the pre-image, and you have a scale factor of 2,
00:31:56.900 --> 00:32:00.200
that means that your image is going to be twice as big as your pre-image;
00:32:00.200 --> 00:32:09.800
so then, you just have to multiply your pre-image by 2 to get your image; so then, those are your coordinates.
00:32:09.800 --> 00:32:22.100
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.
00:32:22.100 --> 00:32:57.900
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.
00:32:57.900 --> 00:33:15.500
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.
00:33:15.500 --> 00:33:23.000
If this is a little confusing, you can always just, instead of "image," write "prime" or "new image" or something like that.
00:33:23.000 --> 00:33:30.800
That way, you know what coordinates go with which one--image or pre-image.
00:33:30.800 --> 00:33:38.700
This is our pre-image; our scale factor is 1/2.
00:33:38.700 --> 00:33:53.800
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.
00:33:53.800 --> 00:33:59.200
And then, the pre-image is just PA, to the original.
00:33:59.200 --> 00:34:10.100
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.
00:34:10.100 --> 00:34:17.200
That means that, since the scale factor is 1/2 (which is smaller than 1), it is going to be a reduction.
00:34:17.200 --> 00:34:23.500
The pre-image, the original, is larger than the new image, so the new image is smaller.
00:34:23.500 --> 00:34:32.100
That means that our new image is going to be smaller than this.
00:34:32.100 --> 00:34:42.100
Here, draw...again, from here, it is going to be like this; so then, PA (that is that) is 2.
00:34:42.100 --> 00:34:45.700
That means that we are going to say that this whole thing is 2.
00:34:45.700 --> 00:34:50.700
That means that PA' is going to be half that; it is going to be 1.
00:34:50.700 --> 00:35:07.100
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'.
00:35:07.100 --> 00:35:21.700
And then, for here to here, for C, our slope is 1, 2, 3 for 1, 2, 3, 4.
00:35:21.700 --> 00:35:32.900
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.
00:35:32.900 --> 00:35:40.200
So, if this whole thing is 2, then C' is going to be right there, halfway.
00:35:40.200 --> 00:35:54.600
This is C', and then, for PB, it is going to go like that.
00:35:54.600 --> 00:36:00.400
Our slope is down 1, over 6; so then, remember: our scale factor is going to be half that.
00:36:00.400 --> 00:36:09.800
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.
00:36:09.800 --> 00:36:16.100
So, go down 1/2; and then, going right was 6--we went down 1, right 6.
00:36:16.100 --> 00:36:24.900
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.
00:36:24.900 --> 00:36:39.300
There is my B'; so my new image is from here to here to here.
00:36:39.300 --> 00:36:48.400
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.
00:36:48.400 --> 00:36:56.000
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.
00:36:56.000 --> 00:37:02.700
If it is greater than 1, then we know that it is going to be bigger than this pre-image.
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That is it for this lesson; thank you for watching Educator.com.