Geometry Reflections

Welcome back to Educator.com.0000

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

Remember: a reflection is a type of transformation whose image and pre-image mirror each other.0007

It is a congruence transformation, meaning that, when you have the pre-image,0014

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.0020

And again, reflection is like a mirror; think of them as reflecting each other.0029

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.0038

This is like the mirror itself; here, if this is an image, and this is the pre-image, this is the line of reflection.0053

To draw the new image, you would have to draw exactly on the other side.0067

So, if this was the mirror, then it would reflect on the other side, the same exact way.0072

This would be called the line of reflection.0078

And the point of reflection is the point that reflects both images.0083

If this is the point of reflection, then it would have to reflect this image on the other side.0091

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.0099

Here, this is the line of reflection, and this is the point of reflection.0114

Now, a symmetry: we have line symmetry and point symmetry; line symmetry is kind of like that line of reflection,0122

where it creates two halves: it is the line that makes it symmetric for both sides.0134

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.0145

This is another one, and another; so all three would be lines of symmetry.0162

This one is the point of symmetry, because if you go this way, the distance to that point on the image0170

will be the same as if you go the opposite way to that point on the image.0186

That is point symmetry; and images have point symmetry, or they do not have point symmetry.0192

This one does, because no matter where I go...I can go here...then when I go the opposite way,0199

it is going to be exactly the same distance away from that point.0206

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.0210

So here, we are going to draw the image over the line of reflection and the point of reflection.0227

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.0232

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).0244

Start with the points: here, if I reflect this point along this line, then it is going to be around here somewhere, around there.0256

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.0269

And then, for C, it is going to go maybe this much; let's call that C'.0282

And then, for B, I am going to go maybe this much; so this will be B'.0294

And then, just connect them; make your sides; that would be a pre-image, and then this is the image.0306

So now, to reflect this on this point of reflection, you are not going straight across, like you did for this line.0323

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;0335

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.0343

That would be B'; and the same thing happens here, for C; you are going to go directly towards that point of reflection,0355

and then keep going to the other side that same distance; there is C';0370

and then, for A, go about that much, so it would be right there; so here is A'.0380

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.0396

Determine how many lines of symmetry each figure has, and identify whether each figure has point symmetry.0422

This first one is a circle; now, where can I draw a line that will create symmetry for this image?0429

I can draw a line here; I can draw a line here; I can draw a line here.0439

A circle has infinite lines of symmetry, because I can draw a line through this circle anywhere I want,0450

as long as it is passing through the center; and then, each of those will be a line of symmetry.0480

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.0489

Can I cut it this way, diagonally?--no, because if I cut it diagonally, even though it is going to be two equal halves,0503

it is not going to be symmetric; it is not going to be exactly the same on both sides;0510

only these two work, so this one has two lines of symmetry.0515

This one...let's see; can we draw it this way?--no, because these are not the same; they are not the same length.0529

It looks like I wouldn't be able to cut it anywhere to make it symmetric, so this has no lines of symmetry--none.0542

Now, for point symmetry (I am going to use red for that), can we draw a point within this image so that,0559

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?0571

Or like this...is that going to be the same distance as that?0585

And this is "yes"; this has point symmetry.0591

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?0597

How about if I go there--is that the same distance as from this diagonal?0610

It looks like it, so I am going to go here and go here; this one is "yes."0618

And then, for this one, let's say I am trying to find as close to the center as possible--somewhere right here.0626

If I go this way, is that going to be the same distance as if I go this way?0638

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."0648

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.0662

We are going to graph some reflections: Graph the reflection of the polygon in the line y = x.0679

For this line, we know...we will use red to graph the line of reflection...that the y-intercept is 0; we will plot 0.0686

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.0699

We are going to reflect it along that line; so again, to reflect this, do not just reflect the image.0720

You want to reflect each of the points, and then you can graph your image.0729

The first point: let's do D; D is right here; so then, we can go right here for the D.0735

If you are going to draw lines to guide you to find the distance away this way and the distance away that way,0752

then make sure it is perpendicular to that line of reflection.0758

This is D'; for C, see how I go diagonally that way; it is negative slope; right there, this is C'.0765

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;0785

here is the half; 1, 2, 3, 4, right here...this is B'.0806

And then, for A, it is going to go 1, 2, 3, 1, 2, 3; here is A'; so, my image is right here.0815

OK, now, to find the coordinates of my pre-image, this one in black, I know that A is...0838

let me do it down here...(-4,2); B is (-4,5); C is (1,5); and D is (1,2).0850

Now, for the reflected image, A' became (2,-4); B' became (5,-4); and C' is (5,1); D' is (2,1).0879

OK, do you notice something about these coordinates and these coordinates?0909

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).0916

When you are reflecting along this line, y = x, you just flip x and y.0938

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,0948

it is always going to be something with the coordinates.0958

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.0969

We are going to reflect it twice.0978

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).0984

Now, we are going to reflect over the x-axis; so if this is the mirror, we are going to reflect along this.1010

Now, when we reflect along this line right here, this point is on that line.1021

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.1029

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';1036

and then, B is going to reflect two down, so that is going to be 2 this way; this is B'.1052

It goes A', B', C'; there is our reflected image, when we reflect along the x-axis.1062

I know that it looks kind of confusing, but here, again, if a point is on the line of reflection, then it stays there;1078

then the pre-image and the image point is going to stay on that line.1089

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.1094

Well, it just has to go to the opposite side of the line of reflection; so this is that new image.1099

And then, we are going to reflect this image along the y-axis, and that is going to be the blue image;1108

and then, I am going to draw this new image in red.1118

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.1124

And this is now called "A double prime," because I reflected it for the second time.1136

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''.1145

It is going to go like that: the black was my pre-image, and the blue was the prime--the first new image.1171

And then, this became double-prime; that is the second image.1191

It reflected on the x-axis, and then it reflects along the y-axis.1195

OK, now, if you were to draw that, that is what happens.1201

Let's look at the coordinates now: A is (2,0); B is (4,2); and C is (3,-1).1207

For the blue image, A' became (2,0); B' is (4,-2); and C' is (3,1).1229

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.1255

A'' is (-2,0); B'' is (-4,-2); and then, C'' is (-3,1).1275

If you notice this one right here, see how the y became negative (see the difference right here).1295

So, when you reflect along the x-axis, then the y becomes -y.1314

This is 0, so it will stay the same; and then, 2 became -2, and -1 became 1.1327

When you reflect along the y-axis, then notice how the x changed; the y stayed the same, and the x-coordinates became negative.1333

Well, those are good if you want to remember it that way.1355

So, if you were given coordinates, ordered pairs to reflect and find the new coordinates,1361

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.1370

And if it says to reflect along the y-axis, the new coordinates would just be the x-coordinates becoming negative.1377

If you ever have to reflect along the line y = x, then you have to switch x and y.1387

Well, that is it for this lesson; thank you for watching Educator.com.1394