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 reflections.
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Remember: a reflection is a type of transformation whose image and pre-image mirror each other.
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It is a congruence transformation, meaning that, when you have the pre-image,
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and you reflect it to create the new image, they are going to be congruent; those two images are going to be exactly the same.
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And again, reflection is like a mirror; think of them as reflecting each other.
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That line that acts as a mirror, the line that creates the reflection from the pre-image to the image, is the line of reflection.
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This is like the mirror itself; here, if this is an image, and this is the pre-image, this is the line of reflection.
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To draw the new image, you would have to draw exactly on the other side.
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So, if this was the mirror, then it would reflect on the other side, the same exact way.
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This would be called the line of reflection.
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And the point of reflection is the point that reflects both images.
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If this is the point of reflection, then it would have to reflect this image on the other side.
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So, here it is the same distance; so if it goes that much that way from the pre-image, then it has to go the same distance to the other side to create the image.
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Here, this is the line of reflection, and this is the point of reflection.
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Now, a symmetry: we have line symmetry and point symmetry; line symmetry is kind of like that line of reflection,
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where it creates two halves: it is the line that makes it symmetric for both sides.
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This equilateral triangle has three lines of symmetry, because you can draw it here to create the two equal halves, this half and this half.
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This is another one, and another; so all three would be lines of symmetry.
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This one is the point of symmetry, because if you go this way, the distance to that point on the image
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will be the same as if you go the opposite way to that point on the image.
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That is point symmetry; and images have point symmetry, or they do not have point symmetry.
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This one does, because no matter where I go...I can go here...then when I go the opposite way,
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it is going to be exactly the same distance away from that point.
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So, the same thing happens to those vertices; I can go this way there, and then this way there--it is exactly the same; that is point symmetry.
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So here, we are going to draw the image over the line of reflection and the point of reflection.
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Here is the line of reflection; here is the point of reflection; I want to reflect this image on this line and on this point.
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To do this, it is best to use their vertices, to reflect the vertices instead of just trying to draw the image (it is not going to be accurate).
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Start with the points: here, if I reflect this point along this line, then it is going to be around here somewhere, around there.
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So, this will be...whenever you create a new image, you are going to call it a "prime"; it looks like an apostrophe: A': it is the same, but it is the prime.
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And then, for C, it is going to go maybe this much; let's call that C'.
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And then, for B, I am going to go maybe this much; so this will be B'.
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And then, just connect them; make your sides; that would be a pre-image, and then this is the image.
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So now, to reflect this on this point of reflection, you are not going straight across, like you did for this line.
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For this one, you are going to go from B; you are not going to go this way; instead, if you want, you can use a ruler;
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and you are going to go directly in the direction of that point, and then you go that same distance from the other side of that point, like that.
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That would be B'; and the same thing happens here, for C; you are going to go directly towards that point of reflection,
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and then keep going to the other side that same distance; there is C';
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and then, for A, go about that much, so it would be right there; so here is A'.
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Then, draw your triangle; there is the reflection along the point of reflection; so here is the first one, and here is the second one.
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Determine how many lines of symmetry each figure has, and identify whether each figure has point symmetry.
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This first one is a circle; now, where can I draw a line that will create symmetry for this image?
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I can draw a line here; I can draw a line here; I can draw a line here.
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A circle has infinite lines of symmetry, because I can draw a line through this circle anywhere I want,
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as long as it is passing through the center; and then, each of those will be a line of symmetry.
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This one, I know, I can cut down this way; and I can cut it down that way, in half--I will have two equal parts.
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Can I cut it this way, diagonally?--no, because if I cut it diagonally, even though it is going to be two equal halves,
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it is not going to be symmetric; it is not going to be exactly the same on both sides;
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only these two work, so this one has two lines of symmetry.
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This one...let's see; can we draw it this way?--no, because these are not the same; they are not the same length.
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It looks like I wouldn't be able to cut it anywhere to make it symmetric, so this has no lines of symmetry--none.
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Now, for point symmetry (I am going to use red for that), can we draw a point within this image so that,
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no matter which way I go, both sides of the point are going to be exactly the same distance to the image, like that and then like that?
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Or like this...is that going to be the same distance as that?
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And this is "yes"; this has point symmetry.
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How about this one--if I draw a point right there in the center, if I go this way, is that the same distance as if I go this way?
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How about if I go there--is that the same distance as from this diagonal?
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It looks like it, so I am going to go here and go here; this one is "yes."
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And then, for this one, let's say I am trying to find as close to the center as possible--somewhere right here.
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If I go this way, is that going to be the same distance as if I go this way?
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No, this is the center; we can tell that this is a lot longer, a lot further away, than to this point; this is "no point symmetry."
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If I go this way, that is not the same distance as from the center to this point; so this is "no"--this does not have point symmetry.
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We are going to graph some reflections: Graph the reflection of the polygon in the line y = x.
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For this line, we know...we will use red to graph the line of reflection...that the y-intercept is 0; we will plot 0.
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The slope is 1, so that means that those are the point of my line of reflection, so I am going to graph this line.
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We are going to reflect it along that line; so again, to reflect this, do not just reflect the image.
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You want to reflect each of the points, and then you can graph your image.
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The first point: let's do D; D is right here; so then, we can go right here for the D.
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If you are going to draw lines to guide you to find the distance away this way and the distance away that way,
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then make sure it is perpendicular to that line of reflection.
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This is D'; for C, see how I go diagonally that way; it is negative slope; right there, this is C'.
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For this one, this line that I am going draw has a negative slope, so it is as if it is going diagonally 1, 2, 3, 4, and a half;
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here is the half; 1, 2, 3, 4, right here...this is B'.
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And then, for A, it is going to go 1, 2, 3, 1, 2, 3; here is A'; so, my image is right here.
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OK, now, to find the coordinates of my pre-image, this one in black, I know that A is...
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let me do it down here...(-4,2); B is (-4,5); C is (1,5); and D is (1,2).
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Now, for the reflected image, A' became (2,-4); B' became (5,-4); and C' is (5,1); D' is (2,1).
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OK, do you notice something about these coordinates and these coordinates?
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Here, this is (x,y); well, the x in the pre-image became the y in the new image, so it is as if we just flipped it: (y,x).
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When you are reflecting along this line, y = x, you just flip x and y.
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Whenever you reflect along y = x or maybe the y-axis or the x-axis, which we are going to do in the next example,
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it is always going to be something with the coordinates.
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This one, we are going to reflect; we are going to draw; it is going to be a triangle; and then, we are going to reflect over the x-axis and then the y-axis.
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We are going to reflect it twice.
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In the pre-image, A, is (2,0); B is (4,2); so here is A, and here is B; and then, C is (3,-1).
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Now, we are going to reflect over the x-axis; so if this is the mirror, we are going to reflect along this.
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Now, when we reflect along this line right here, this point is on that line.
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This point is on the line of reflection; if it is on the line of reflection, then it doesn't move anywhere; it stays there.
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This is C; so then, this is going to stay there as A'; then, point C is going to reflect to right there, so this is C';
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and then, B is going to reflect two down, so that is going to be 2 this way; this is B'.
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It goes A', B', C'; there is our reflected image, when we reflect along the x-axis.
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I know that it looks kind of confusing, but here, again, if a point is on the line of reflection, then it stays there;
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then the pre-image and the image point is going to stay on that line.
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And then, this one was on the other side; see how this one is on this side and this one is on the other side.
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Well, it just has to go to the opposite side of the line of reflection; so this is that new image.
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And then, we are going to reflect this image along the y-axis, and that is going to be the blue image;
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and then, I am going to draw this new image in red.
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This is the line of reflection, the y-axis; so A' is now going to go here, again, because this is acting as the mirror.
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And this is now called "A double prime," because I reflected it for the second time.
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And then here, 1, 2, 3, 4, 1, 2, 3, 4...here, this is my B double prime, and C...away from this line is 3 spaces, 3 units, so it is 1, 2, 3 units right here: C''.
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It is going to go like that: the black was my pre-image, and the blue was the prime--the first new image.
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And then, this became double-prime; that is the second image.
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It reflected on the x-axis, and then it reflects along the y-axis.
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OK, now, if you were to draw that, that is what happens.
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Let's look at the coordinates now: A is (2,0); B is (4,2); and C is (3,-1).
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For the blue image, A' became (2,0); B' is (4,-2); and C' is (3,1).
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Reflecting on the y-axis, this was a reflection of the blue; the red, the reflection along the y-axis, was a reflection of this one right here.
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A'' is (-2,0); B'' is (-4,-2); and then, C'' is (-3,1).
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If you notice this one right here, see how the y became negative (see the difference right here).
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So, when you reflect along the x-axis, then the y becomes -y.
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This is 0, so it will stay the same; and then, 2 became -2, and -1 became 1.
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When you reflect along the y-axis, then notice how the x changed; the y stayed the same, and the x-coordinates became negative.
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Well, those are good if you want to remember it that way.
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So, if you were given coordinates, ordered pairs to reflect and find the new coordinates,
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if you reflect along the x-axis, if it says to reflect along the x-axis, keep the coordinates the same; just make the y negative.
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And if it says to reflect along the y-axis, the new coordinates would just be the x-coordinates becoming negative.
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If you ever have to reflect along the line y = x, then you have to switch x and y.
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Well, that is it for this lesson; thank you for watching Educator.com.