Geometry Translations

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For the next lesson, we are going to go over translations.0002

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.0006

So, it is just the same distance and the same direction--the same image; it just moves.0016

Remember: another word for translations would be "sliding"; slide, shift, glide--those are all words for translation.0026

The first image, the initial image, is called the pre-image, and then it goes to the new image.0039

I like to say "new image," because that is the image created from the pre-image.0050

If the pre-image has coordinates (x,y) for point A, then the new image is going to be A',0058

and it is the same (x,y) coordinates, but shifted up or down, and that would be a...0069

I'm sorry; a would be left and right, because x is moving horizontally.0078

So then, a would be how many it is moving either left or right; and b,0085

because it is the y-coordinate plus the b, is how many the y-coordinate is moving up and down.0090

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).0098

Now, if this whole image, the pre-image, shifted up 1, or let's just say it shifted right 4 and up 1;0111

we know that, if it shifts right, it is a positive 4; if it shifts up, then it is a positive 1.0126

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.0132

These are positive numbers; so if A is shifting to the right 4 and up 1, then I can say that the x-coordinate0144

(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,0155

just so that you know that it is not the same as this coordinate, A, and then y + b.0170

How many did it move left or right? It moved 4 to the right, so this point, (2,1), became...0177

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).0188

All it is: (2,1) became 2 + 4...this is how much it moved...and 1 + 1; so it is the same coordinates,0222

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'.0231

(6,2) will be the new coordinates for A.0250

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 3.0257

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.0296

How many did it move left and right? It moved left 2.0325

Again, if we move this way on the coordinate plane, then it is a negative number, because we are moving towards a negative.0328

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 3.0336

If I go down, this is going towards the negative numbers; this is positive; this is negative; this is negative; and this is positive.0353

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).0367

Now, translations are actually a composite of two reflections.0390

If this is the pre-image, and you reflect it once, you get this image; this is reflection, like the mirror reflection.0399

When we reflect it again (and again, the lines of reflection have to be parallel), reflect it a second time,0412

then this image right here, this pre-image, to this image right here, is actually a translation.0420

Two reflections is equal to one translation, but of course, only when the two lines are parallel.0429

If the two lines that you reflected along are not parallel, then it is not going to be a translation.0446

Only if the two lines are parallel when you make your two reflections, then it becomes a translation.0455

That is a composite of reflections, or a composition of reflections.0463

We are going to use this translation for each ordered pair; that is the rule.0470

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.0481

Now, what this is saying is that you are going to take the x-coordinate, and we are going to add 3.0497

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,0503

because x's are only along this line; you can't count up and down and call that x.0515

It is the x-coordinate, which is the 4, and we are going to add 3 to that.0523

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).0528

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,0552

because that is a positive"; this becomes A', and then you are going to go down 1, 2, 3, 4, down the y-coordinate;0559

it was at 0; now it is going to go to -4; so go 3 to the right and 4 down.0581

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.0590

There are your new coordinates for the prime.0597

Now, for this one, we are going backwards, because we have the prime number, so we want to just find B.0608

That means that, for this B, we have (x,y), and then B' became (x + a, y + b).0615

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.0637

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 -5.0656

So, I know that x + a equals -5; isn't this equal to this?0666

And then, the same thing happens here: y + b equals -2.0678

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 -8.0685

Here, this is y + -4 = -2; add the 4, so y is 2.0702

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 -5,0718

which is right here; and then, make y - 4 equal to -2, because this is our B', and then this is our B.0740

You want to use this to solve for that, to find the x and the y.0752

OK, given the coordinates of the image and the pre-image, find the rule for the translation.0760

This is the pre-image, and this is the new image; find the rule for the translation.0767

I want to know how to get from R (we are going to call that (x,y); that is the pre-image),0774

and then it went to R', when you add a to the x and you add b to the y.0783

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.0796

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?0822

I can say that -2 + a is equal to 4; and the same thing works here: 8 + b is equal to -4.0839

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 -12.0850

They want to know the rule for the translation: the rule would be back up here; this is what we used for the rule.0866

We are going to keep everything the same; we are just going to replace a and b.0875

The rule for R (we have coordinates (x,y)), to make it into R', is x + 6, comma, y - 12.0879

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.0894

This would be the final answer.0902

OK, and the next one: Find the translation image using a composite of reflections.0910

Remember: we need two reflections along parallel lines to make those two reflections equal to one translation.0916

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.0926

And then, I will get one translation, meaning that, for my second reflection, it should look exactly like this.0935

So the, the first reflection: remember: to draw reflections, you are just going to reflect the points.0945

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.0951

It is going to be the first reflection, and then the second reflection, like this; and then, that is maybe right there.0966

And then, the same thing happens for this; it is going to go (I'll draw the line better) here.0986

See, if you only look at this black and this red image, it is a translation.1003

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.1014

Again, two reflections equals one translation, as long as the two lines are parallel.1025

If they are not parallel, then it is not going to be a translation.1033

In the fourth example, find the value of each variable in the translation.1040

Here, we have that one of these is the pre-image and one of them is the translated image.1046

And because it is a congruence transformation, they have to be exactly the same.1053

They are congruent, meaning that all corresponding parts are congruent.1059

We just want to find the value of each variable; we have x here, and we have y here.1063

That means that this angle and that angle are corresponding, and they are congruent, so I can make them equal to each other.1071

So, 2x is equal to 80; to solve for x, I just divide the 2; so x is 40.1081

And then, for my y, to make this angle and that angle congruent, y is equal to 110.1096

And again, the whole point of this is to just keep in mind that translation is a congruence transformation.1107

So, always remember that all corresponding parts are congruent.1115

That is it for this lesson; thank you for watching Educator.com.1121