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 translations.
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Remember: a translation is a congruence transformation where all of the points of an image are moved from one place to another place without changing.
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So, it is just the same distance and the same direction--the same image; it just moves.
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Remember: another word for translations would be "sliding"; slide, shift, glide--those are all words for translation.
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The first image, the initial image, is called the pre-image, and then it goes to the new image.
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I like to say "new image," because that is the image created from the pre-image.
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If the pre-image has coordinates (x,y) for point A, then the new image is going to be A',
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and it is the same (x,y) coordinates, but shifted up or down, and that would be a...
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I'm sorry; a would be left and right, because x is moving horizontally.
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So then, a would be how many it is moving either left or right; and b,
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because it is the y-coordinate plus the b, is how many the y-coordinate is moving up and down.
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So then, this would be A'; so let's say this is the pre-image, and we have A (let's say this is A), and the coordinate for this are (2,1).
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Now, if this whole image, the pre-image, shifted up 1, or let's just say it shifted right 4 and up 1;
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we know that, if it shifts right, it is a positive 4; if it shifts up, then it is a positive 1.
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If it shifts to the left, then we know that it is a negative; so going this way, left is negative, and down is negative; these are negative numbers.
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These are positive numbers; so if A is shifting to the right 4 and up 1, then I can say that the x-coordinate
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(this has to do with the x, if it is moving left and right)...for A, it is (x,y); A' is x + a...lowercase,
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just so that you know that it is not the same as this coordinate, A, and then y + b.
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How many did it move left or right? It moved 4 to the right, so this point, (2,1), became...
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for A', x is 2; we went 4, and then y moved up 1, so that is y, which is 1, plus 1; so then, my A' is (6,2).
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All it is: (2,1) became 2 + 4...this is how much it moved...and 1 + 1; so it is the same coordinates,
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but then we just count how many we shift: 4 to the right and 1 up--so our new coordinates are going to be right here, so it will be A'.
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(6,2) will be the new coordinates for A.
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If we do another one, let's say that this A is (-2,-1); let's say it shifted; it moved left 2 and down 3.
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So then, my A', we know, is -2 + a, and -1 + b; see how it is the same coordinates here, -1 and 2, and we just add how many we move to this coordinate.
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How many did it move left and right? It moved left 2.
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Again, if we move this way on the coordinate plane, then it is a negative number, because we are moving towards a negative.
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So then, it is going to be -2 -2, because for a, how many did I move?...-2, and then -1 - 3, because I moved down 3.
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If I go down, this is going towards the negative numbers; this is positive; this is negative; this is negative; and this is positive.
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A' is (-4,-4); if you take this ordered pair, and we move 2 to the left and down 3, then I will be at (-4,-4).
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Now, translations are actually a composite of two reflections.
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If this is the pre-image, and you reflect it once, you get this image; this is reflection, like the mirror reflection.
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When we reflect it again (and again, the lines of reflection have to be parallel), reflect it a second time,
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then this image right here, this pre-image, to this image right here, is actually a translation.
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Two reflections is equal to one translation, but of course, only when the two lines are parallel.
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If the two lines that you reflected along are not parallel, then it is not going to be a translation.
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Only if the two lines are parallel when you make your two reflections, then it becomes a translation.
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That is a composite of reflections, or a composition of reflections.
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We are going to use this translation for each ordered pair; that is the rule.
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Now, just to show you, if you are still confused about this, A is (4,0), so 1, 2, 3, 4...0; this is where A is at.
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Now, what this is saying is that you are going to take the x-coordinate, and we are going to add 3.
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We are going to move 3; now, it is the x number, so if it is the x number, and we are going to move 3, then it has to go left or right,
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because x's are only along this line; you can't count up and down and call that x.
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It is the x-coordinate, which is the 4, and we are going to add 3 to that.
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So, if you add 4, that will be x, which is 4, plus 3; and then, the y, which is 0, minus 4; then A' becomes (7,-4).
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And again, it is just moving; that means that it is saying, "Well, we are going to add 3; so we are going to go 1, 2, 3 to the right,
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because that is a positive"; this becomes A', and then you are going to go down 1, 2, 3, 4, down the y-coordinate;
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it was at 0; now it is going to go to -4; so go 3 to the right and 4 down.
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This is the left-right number; this is the up-down number; see how many you are going to move--that is all that it is saying.
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There are your new coordinates for the prime.
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Now, for this one, we are going backwards, because we have the prime number, so we want to just find B.
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That means that, for this B, we have (x,y), and then B' became (x + a, y + b).
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And we know that B' is (-5,-2); so, I want to find out what my (x,y) is, because if I have my (x,y), then that will give me B.
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How did I get -5? That is x + a; so the x-coordinate, including how many we move left and right, is going to become -5.
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So, I know that x + a equals -5; isn't this equal to this?
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And then, the same thing happens here: y + b equals -2.
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Now, I know my a and my b; my a is positive 3, so x + 3 equals -5; if I subtract 3, then I get that x is -8.
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Here, this is y + -4 = -2; add the 4, so y is 2.
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My B, then, becomes (-8,2); so again, I just took this rule: it is (x + 3, y - 4); and you just make this x + 3 equal to -5,
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which is right here; and then, make y - 4 equal to -2, because this is our B', and then this is our B.
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You want to use this to solve for that, to find the x and the y.
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OK, given the coordinates of the image and the pre-image, find the rule for the translation.
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This is the pre-image, and this is the new image; find the rule for the translation.
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I want to know how to get from R (we are going to call that (x,y); that is the pre-image),
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and then it went to R', when you add a to the x and you add b to the y.
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R is (-2,8), and it became R', which was...let me just replace this (x,y) with this (x,y) in here: it was -2 + a, and then 8 + b.
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The R', we know, is 4 and -4; so if -2 + a is the x-coordinate for R', and that is 4, then isn't that the same thing as that?
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I can say that -2 + a is equal to 4; and the same thing works here: 8 + b is equal to -4.
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So here, I am going to add 2; a is equal to 6; and then here, if I subtract the 8, b is equal to -12.
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They want to know the rule for the translation: the rule would be back up here; this is what we used for the rule.
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We are going to keep everything the same; we are just going to replace a and b.
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The rule for R (we have coordinates (x,y)), to make it into R', is x + 6, comma, y - 12.
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So, whatever this point is, we are going to move 6 to the right, and we are going to move 12 units down; that is the rule.
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This would be the final answer.
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OK, and the next one: Find the translation image using a composite of reflections.
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Remember: we need two reflections along parallel lines to make those two reflections equal to one translation.
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Here, we have two parallel lines; I want to reflect this image twice--the first one along this line and the second one along this line.
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And then, I will get one translation, meaning that, for my second reflection, it should look exactly like this.
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So the, the first reflection: remember: to draw reflections, you are just going to reflect the points.
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That is this point right here; for this one, you are going to go out this much; and for this point, you are going to go right there.
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It is going to be the first reflection, and then the second reflection, like this; and then, that is maybe right there.
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And then, the same thing happens for this; it is going to go (I'll draw the line better) here.
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See, if you only look at this black and this red image, it is a translation.
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All it did: it is as if this just glided over to this side; it is the same; it didn't rotate; it didn't flip; it didn't do anything else but just slide over there.
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Again, two reflections equals one translation, as long as the two lines are parallel.
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If they are not parallel, then it is not going to be a translation.
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In the fourth example, find the value of each variable in the translation.
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Here, we have that one of these is the pre-image and one of them is the translated image.
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And because it is a congruence transformation, they have to be exactly the same.
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They are congruent, meaning that all corresponding parts are congruent.
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We just want to find the value of each variable; we have x here, and we have y here.
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That means that this angle and that angle are corresponding, and they are congruent, so I can make them equal to each other.
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So, 2x is equal to 80; to solve for x, I just divide the 2; so x is 40.
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And then, for my y, to make this angle and that angle congruent, y is equal to 110.
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And again, the whole point of this is to just keep in mind that translation is a congruence transformation.
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So, always remember that all corresponding parts are congruent.
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That is it for this lesson; thank you for watching Educator.com.