Geometry Rotations

Welcome back to Educator.com.0000

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

Now, rotations, remember, are a type of congruence transformation.0006

This is the third one, in which an image moves in a circular motion to a new position.0011

Here, this is our pre-image; the center of rotation is a point (that is this right here--this is the center of rotation);0017

it is a fixed point which the image is rotating around.0034

Remember: rotation is when we are rotating it; we are going either this way, clockwise, or counterclockwise, to create a congruent image.0037

This image that we start with is always called the pre-image; this is the fixed point that is the center of rotation;0052

this is the point that we are going to rotate around.0058

The way we do that, and the easiest way to do rotations, is to pick a point; pick one of the vertices here,0063

on the pre-image (I am going to pick that point right there), and you are going to draw a line to the center of rotation.0076

Now, the center of rotation, when we draw a line like this, is going to create an angle.0087

So, if we are going to draw a line from this fixed point to the center of rotation, and then draw another line0094

to the corresponding point in the new image, which is this right here--this is corresponding to this point right here--0101

that is going to have an angle; so this is the angle of rotation.0111

You are moving a certain angle amount, a certain degree, either clockwise or counterclockwise, from the pre-image to the image.0116

Remember: this is a congruence transformation, so both images are congruent; they are the same.0125

There are two ways in which you can perform a rotation.0135

The first is a composite of two successive reflections over two intersecting lines.0140

Remember: for translations, we also performed two reflections over two parallel lines; don't get that confused.0151

For translation, it is also two reflections, but with two parallel lines; here, for rotation, it is also two reflections,0165

but the lines are intersecting; they have to be intersecting.0175

If this is our pre-image, it is first reflected over this line right here to get this.0182

So, if I call this, let's say, A, then this corresponding point, which is right here (because it is a reflection;0197

remember that a reflection is like the mirror, where you are making this act as a mirror,0206

and you are reflecting this image)--if this is A, then this would be A'.0211

This is the first reflection, and then you are going to do a second reflection over this line.0218

This point now becomes this point way over here; so this is A''; that means that this pre-image to this image right here, A'', is a rotation.0224

We rotated this image to this image right here.0242

So again, to do a rotation, one rotation can be the same as two reflections, but with intersecting lines.0247

The lines of reflection have to be intersecting; that is the first way.0265

The second way to perform a rotation is kind of how we did on the previous slide, where you are going to have an angle;0271

so then, you are going to pick a point from the pre-image, draw a line to that center of rotation,0282

and then find the corresponding point in the other image, right there; and then, you are going to draw a line to that point.0290

And this angle that is formed is called the angle of rotation.0299

Whenever you do angle of rotation, I will always do it in red, so that you know that that is what we are doing.0307

We are finding the angle from the corresponding point to the center of rotation, and then back to the image that was formed.0312

That is the angle of rotation; and then, this is the same diagram from the previous slide, and this is where we get our postulate.0323

In a given rotation, the angle of rotation is twice the measure of the angle formed by the intersecting lines of reflection.0341

This shows you both methods in one diagram: remember: with the first method, we did two reflections;0350

this one reflected to this; this reflected to this; and this became A''.0359

And then, the second method: we did, right here, the angle of rotation.0364

This shows you both methods in one diagram, and it is saying that this angle right here,0378

the angle between the two intersecting lines of reflection--if this is 90 (let's say that it is a right angle),0386

then the angle of rotation is going to be double that; so the angle between the lines of reflection,0396

the angle formed by method 1, is going to be half the measure of the angle formed from the angle of rotation,0408

this whole thing right here, which was method 2.0416

So then again, the angle of rotation, this right here, the red, is going to be double the blue, between the lines of reflection.0424

The angle of rotation is twice the measure of the angle formed from the lines of reflection.0433

Then, this angle of rotation is going to be 180; and that is the postulate on rotation.0442

Then, our first example: Find the rotated image by reflecting the image twice over the intersecting lines.0455

Remember: one rotation is going to be two reflections only if the lines of reflection are intersecting.0462

So then, here is our pre-image, and we are going to reflect it along here, and then, for the second reflection, reflect it along this one.0479

So, we are going to get our new image here.0488

Remember: to reflect, you are going to go on the other side like that, here, something like that right there.0492

And then, for this one, put it maybe right there; it is going to be that.0508

If this is A, then this is A'; that is the first reflection.0529

Then, for the second reflection, we are going to reflect over this one right here; it is going to be right there.0536

And then, let me just do that one in red, so that you know that this is the second reflection...right there.0546

And then, for this one, make sure that, when you draw your line of reflection, it is perpendicular.0557

So then, moving straight across...maybe it is somewhere right there...it is going to be something like that.0570

This is corresponding to that point right there; that is A''.0585

Make sure that, if you are going to draw these lines to help you, to guide you in where to draw, that they are kind of faint.0594

You can either draw them dotted, or maybe you can erase them after, because they are not really supposed to be there.0604

This is the image right here; this would be the reflected image.0611

So, from this pre-image, the rotation occurs to this image right here; from this, and then to this, is the rotation.0616

The next example: Triangle ABC is rotated 180 degrees on the coordinate plane; draw out a form of triangle A'B'C'.0634

So, we are going to rotate it 180 degrees.0645

Now, it doesn't tell us which way to go, clockwise or counterclockwise.0648

But for this problem, it doesn't matter, because 180 degrees is just a straight line.0653

So, whether you go this way or the other way, it doesn't matter; you are going to end up in the same spot.0659

If we say that the origin is the center of rotation, this is our fixed point; this is where we are going to rotate around.0668

Since we know that this is the angle of rotation, that is how much we are going to be moving.0680

Your fixed point (and again, I am doing this in red, so that you know that this is the angle of rotation, method #2):0691

you are going to draw a line to your fixed point, your center of rotation, and then you are going to go the same distance.0696

And make sure that this angle is the angle of rotation, which is 180.0708

So then, this is going to end up right there; so this is C'.0717

And then, for B, right there, it is going to go...you can also use slope to help you,0726

because if it is 180 degrees, then you have to make sure that it is a straight line;0739

so this angle of rotation has to be a straight line; so then, the slope of this right here is 1/4;0744

so then, I have to make sure that, when I go this way, when I keep going, it is also going to be 1/4.0752

This right here is B'; and again, you have to make sure that it is a straight line.0760

If you want to use a ruler, you can use a ruler, because if you go here, and then you can maybe go a little bit sideways,0768

then you can end up anywhere but there; so make sure that it is a straight line, 180 degrees.0776

That is B'; and then, for A, again, you are going to draw like that; my slope of this is 5/2.0783

So then, I have to make sure that my slope is also going to be 5/2; it is as if it is going this way in a straight line, 5/2.0797

This right here is A': this is C', this is B', and this is A'.0813

Usually, rotation is probably going to be the most difficult to draw, but instead of looking at this whole diagram,0836

instead of trying to draw that image all at once, use the points.0845

And then, all you are doing is plotting a new place for this point: the rotated image of this point,0853

the rotated image of that point, and the same thing for that point.0860

And then, you just connect them to create your new image.0863

Find the value of each variable in the rotation: in this one, here is the pre-image--I know that because it rotated this way 90 degrees.0871

It is a rotation, and for this, we are just finding the value; it is just to show that these are congruent, that rotation is a congruence transformation.0884

The first equation that I can set is with this right here, the x.0900

This is corresponding, on the pre-image, to that right there; so x equals y - 5.0906

I have two variables that I need to make my other equation; here is y; y is corresponding to this right here, so that is 15.0917

So, that is one of my variables; and then, I need to plug this into this equation right here, so that I can find x.0930

x equals...y is 15, minus 5; so x is 10.0936

If you get a problem like this, where you have to show that they are congruent, then just set your equations.0947

Make sure that you make this equal to this; it is not x with the 15, so be careful on what you make equal to each other.0958

It has to be the corresponding sides and angles.0969

Draw the polygon rotated 90 degrees clockwise about P; this is P right here.0976

And we are going to draw it clockwise, meaning that we are going to go this way, 90 degrees.0987

If you want to, you can use a protractor; we are going to just sketch it.0998

I am going to do one point at a time, one vertex at a time.1007

Here, point C--I am going to draw it like this, and then I am going to draw my other line to my image, to C',1013

so that the angle formed is going to be 90 degrees.1027

So then, maybe it is right there; so then, here, make sure that it is the same distance; that is C'.1032

Let's do B: here, it is something like that; and then, maybe it is right there somewhere; this is going to be B'.1050

Then, my A: this one is going to be a little bit further, so it is better for you to use a ruler, because it is kind of far.1080

Just use that as a reference; you can also use maybe the distance between this line and then to the C,1095

so then this line to this line--how far apart are they?--and then make it that same distance when we draw this line.1101

For this one, it is going to go somewhere like that.1109

And again, make sure that this line right here, to this line, AP to PA', is a 90-degree angle.1119

It would be somewhere like right there; it doesn't look straight--let me just draw that again.1137

I am sure that my screen is a lot bigger than your paper, so it should be easier for you.1148

I am going to label that A', and then for D...I just made that wrong; let's see,1162

it should actually be going somewhere in front of B, somewhere there: A'...1192

And then, for D, it should be somewhere there; so this is D', and then this is A'.1212

When you draw it, you should have the same figure as this; it should look congruent.1228

Again, if you have a ruler, that will be a lot easier; I didn't have a ruler, so it was a little bit harder for me.1242

Just make sure that when you draw the line from the point to the center of rotation,1252

and then from the center of rotation to that prime, that it is a right angle; it has to be 90 degrees,1258

or whatever the angle of rotation is; if it is 100 degrees, then make sure that it is about 100 degrees.1264

And then, your pre-image and your image have to look the same.1272

This is just a sketch, so it is not exactly the same; but it should look very close to it.1276

Well, that is it for this lesson; thank you for watching Educator.com.1284