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 rotations.
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Now, rotations, remember, are a type of congruence transformation.
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This is the third one, in which an image moves in a circular motion to a new position.
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Here, this is our pre-image; the center of rotation is a point (that is this right here--this is the center of rotation);
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it is a fixed point which the image is rotating around.
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Remember: rotation is when we are rotating it; we are going either this way, clockwise, or counterclockwise, to create a congruent image.
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This image that we start with is always called the pre-image; this is the fixed point that is the center of rotation;
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this is the point that we are going to rotate around.
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The way we do that, and the easiest way to do rotations, is to pick a point; pick one of the vertices here,
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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.
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Now, the center of rotation, when we draw a line like this, is going to create an angle.
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So, if we are going to draw a line from this fixed point to the center of rotation, and then draw another line
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to the corresponding point in the new image, which is this right here--this is corresponding to this point right here--
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that is going to have an angle; so this is the angle of rotation.
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You are moving a certain angle amount, a certain degree, either clockwise or counterclockwise, from the pre-image to the image.
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Remember: this is a congruence transformation, so both images are congruent; they are the same.
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There are two ways in which you can perform a rotation.
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The first is a composite of two successive reflections over two intersecting lines.
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Remember: for translations, we also performed two reflections over two parallel lines; don't get that confused.
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For translation, it is also two reflections, but with two parallel lines; here, for rotation, it is also two reflections,
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but the lines are intersecting; they have to be intersecting.
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If this is our pre-image, it is first reflected over this line right here to get this.
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So, if I call this, let's say, A, then this corresponding point, which is right here (because it is a reflection;
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remember that a reflection is like the mirror, where you are making this act as a mirror,
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and you are reflecting this image)--if this is A, then this would be A'.
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This is the first reflection, and then you are going to do a second reflection over this line.
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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.
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We rotated this image to this image right here.
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So again, to do a rotation, one rotation can be the same as two reflections, but with intersecting lines.
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The lines of reflection have to be intersecting; that is the first way.
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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;
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so then, you are going to pick a point from the pre-image, draw a line to that center of rotation,
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and then find the corresponding point in the other image, right there; and then, you are going to draw a line to that point.
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And this angle that is formed is called the angle of rotation.
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Whenever you do angle of rotation, I will always do it in red, so that you know that that is what we are doing.
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We are finding the angle from the corresponding point to the center of rotation, and then back to the image that was formed.
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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.
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In a given rotation, the angle of rotation is twice the measure of the angle formed by the intersecting lines of reflection.
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This shows you both methods in one diagram: remember: with the first method, we did two reflections;
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this one reflected to this; this reflected to this; and this became A''.
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And then, the second method: we did, right here, the angle of rotation.
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This shows you both methods in one diagram, and it is saying that this angle right here,
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the angle between the two intersecting lines of reflection--if this is 90 (let's say that it is a right angle),
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then the angle of rotation is going to be double that; so the angle between the lines of reflection,
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the angle formed by method 1, is going to be half the measure of the angle formed from the angle of rotation,
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this whole thing right here, which was method 2.
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So then again, the angle of rotation, this right here, the red, is going to be double the blue, between the lines of reflection.
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The angle of rotation is twice the measure of the angle formed from the lines of reflection.
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Then, this angle of rotation is going to be 180; and that is the postulate on rotation.
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Then, our first example: Find the rotated image by reflecting the image twice over the intersecting lines.
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Remember: one rotation is going to be two reflections only if the lines of reflection are intersecting.
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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.
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So, we are going to get our new image here.
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Remember: to reflect, you are going to go on the other side like that, here, something like that right there.
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And then, for this one, put it maybe right there; it is going to be that.
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If this is A, then this is A'; that is the first reflection.
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Then, for the second reflection, we are going to reflect over this one right here; it is going to be right there.
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And then, let me just do that one in red, so that you know that this is the second reflection...right there.
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And then, for this one, make sure that, when you draw your line of reflection, it is perpendicular.
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So then, moving straight across...maybe it is somewhere right there...it is going to be something like that.
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This is corresponding to that point right there; that is A''.
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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.
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You can either draw them dotted, or maybe you can erase them after, because they are not really supposed to be there.
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This is the image right here; this would be the reflected image.
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So, from this pre-image, the rotation occurs to this image right here; from this, and then to this, is the rotation.
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The next example: Triangle ABC is rotated 180 degrees on the coordinate plane; draw out a form of triangle A'B'C'.
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So, we are going to rotate it 180 degrees.
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Now, it doesn't tell us which way to go, clockwise or counterclockwise.
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But for this problem, it doesn't matter, because 180 degrees is just a straight line.
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So, whether you go this way or the other way, it doesn't matter; you are going to end up in the same spot.
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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.
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Since we know that this is the angle of rotation, that is how much we are going to be moving.
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Your fixed point (and again, I am doing this in red, so that you know that this is the angle of rotation, method #2):
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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.
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And make sure that this angle is the angle of rotation, which is 180.
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So then, this is going to end up right there; so this is C'.
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And then, for B, right there, it is going to go...you can also use slope to help you,
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because if it is 180 degrees, then you have to make sure that it is a straight line;
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so this angle of rotation has to be a straight line; so then, the slope of this right here is 1/4;
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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.
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This right here is B'; and again, you have to make sure that it is a straight line.
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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,
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then you can end up anywhere but there; so make sure that it is a straight line, 180 degrees.
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That is B'; and then, for A, again, you are going to draw like that; my slope of this is 5/2.
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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.
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This right here is A': this is C', this is B', and this is A'.
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Usually, rotation is probably going to be the most difficult to draw, but instead of looking at this whole diagram,
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instead of trying to draw that image all at once, use the points.
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And then, all you are doing is plotting a new place for this point: the rotated image of this point,
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the rotated image of that point, and the same thing for that point.
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And then, you just connect them to create your new image.
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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.
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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.
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The first equation that I can set is with this right here, the x.
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This is corresponding, on the pre-image, to that right there; so x equals y - 5.
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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.
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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.
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x equals...y is 15, minus 5; so x is 10.
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If you get a problem like this, where you have to show that they are congruent, then just set your equations.
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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.
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It has to be the corresponding sides and angles.
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Draw the polygon rotated 90 degrees clockwise about P; this is P right here.
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And we are going to draw it clockwise, meaning that we are going to go this way, 90 degrees.
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If you want to, you can use a protractor; we are going to just sketch it.
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I am going to do one point at a time, one vertex at a time.
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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',
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so that the angle formed is going to be 90 degrees.
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So then, maybe it is right there; so then, here, make sure that it is the same distance; that is C'.
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Let's do B: here, it is something like that; and then, maybe it is right there somewhere; this is going to be B'.
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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.
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Just use that as a reference; you can also use maybe the distance between this line and then to the C,
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so then this line to this line--how far apart are they?--and then make it that same distance when we draw this line.
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For this one, it is going to go somewhere like that.
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And again, make sure that this line right here, to this line, AP to PA', is a 90-degree angle.
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It would be somewhere like right there; it doesn't look straight--let me just draw that again.
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I am sure that my screen is a lot bigger than your paper, so it should be easier for you.
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I am going to label that A', and then for D...I just made that wrong; let's see,
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it should actually be going somewhere in front of B, somewhere there: A'...
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And then, for D, it should be somewhere there; so this is D', and then this is A'.
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When you draw it, you should have the same figure as this; it should look congruent.
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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.
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Just make sure that when you draw the line from the point to the center of rotation,
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and then from the center of rotation to that prime, that it is a right angle; it has to be 90 degrees,
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or whatever the angle of rotation is; if it is 100 degrees, then make sure that it is about 100 degrees.
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And then, your pre-image and your image have to look the same.
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This is just a sketch, so it is not exactly the same; but it should look very close to it.
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Well, that is it for this lesson; thank you for watching Educator.com.