 Hi, I'm Zor. Welcome to Unizor Education. I will continue talking about certain introductory concepts of geometry. Today's topic is symmetry. Now, during my previous lecture about congruent geometrical figures, I was talking about certain transformations which help to identify or to prove that certain geometrical objects are congruent. And one of these transformations was called a reflection, reflection relative to the axis. Another was a rotation. Now, let's consider the word symmetry. It's very much, it's very closely related to these two transformations. Rotation around certain point and reflection around certain axis. We will consider the term symmetry in two different meanings. One is a characteristic of an object to be symmetrical. And another symmetry would be a transformation which I was just talking about. So let me address first the transformation. Rotation around certain point by certain angle is very simple. As you know, this is just rotation of this segment which connects the center of rotation with our point into a different location, a prime. So point A after rotation by this angle alpha will take the position of a prime. So this is a rotation. And now the question is, what kind of a geometrical object has a property of transforming into itself after this particular rotation? Well, consider, for instance, a circle. And the center of rotation will be the center of the circle. Now, if you will rotate any point on the curve, which is the circle, by any angle, it will also fall onto the point of a circle. So basically, regardless of the value of the angle, alpha of rotation, the circle transforms into itself after this particular rotation. And that's what actually makes a circle a figure which is symmetrical relative to rotation around its center by any angle. Now, if you will take a different figure, let's say you have a square. Now, this is the center of the square. Now, if you will rotate a square around this particular center of rotation by 90 degree, only 90 degree, then this particular segment will be converted into this position, this into this, this into this, et cetera. So square will be converted into itself. So rotation by 90 degree is transforming a circle into its square into itself. So that's why we're calling a square a geometrical figure which is symmetrical relative to rotation by 90 degree around its center, where the diagonals are crossing each other. By the way, if it's symmetrical relative to this particular rotation by 90 degree, it's also symmetrical by 180 and 270 and every multiple of 90. So these are examples of symmetrical figures. Now, I would also like to talk about the symmetry relative to the axis. Now, the definition of this is the following. If you have a line which we call nexus of symmetry and a point anywhere outside of this line, then you draw a perpendicular and extend it to the same length. So this length is equal to this length. So this will be a reflection relative to this particular axis of reflection. So if we can transform a point, we can transform obviously any kind of a geometrical figure. And reflection will convert one into another. Now, there are certain geometrical figures which are transformed onto themselves during this operation of reflection. And here is an example. Let's have again a circle with this line crossing its center. Now, it can be proven that if you will reflect relative to this axis, our circle, it will turn into itself. This side will overlap with this and this one will overlap with that. So basically, a circle is a symmetrical figure relative to a reflection of the line which is crossing its center. Now, if it's not crossing the center, then this, for instance, line, then the circle is not symmetrical relative to this because it will be converted into this, not to itself. So it's very important that the line which is the axis of reflection is crossing the center. Now, other examples of symmetrical in this particular sense, figures, for instance, you have a socialist triangle and you have the line which is basically its altitude from the vertex down to the base. Now, it can be proven that this is exactly the axis of symmetry because every segment of this type will be transformed into this one. So the whole triangle will turn upon itself. Now, so we actually have two different kinds of symmetry. The symmetry which is related to rotation by some angle and the symmetry which is related to reflection. There is only one little detail about rotation. Sometimes you might hear that this particular figure is centrally symmetrical. Well, centrally symmetrical basically means it's symmetrical relative to rotation by 180 degree. And let's say you have the same square. Now, square is centrally symmetrical relative to its center because if you will turn by 180 degree, the whole thing will turn upon itself. And the definition of this is very much equivalent to the following definition. If you have a center of rotation, then any point to convert it into centrally symmetrical would be connected to the center and expanded by the same length. So you will have this point. Now, obviously, A prime can be obtained from A not by this process of connecting by a segment and then expanding the segment, but by rotation of this segment by 180 degree because 180 degree makes the whole line straight. So basically, when you're talking about centrally symmetrical figures, it means either the definition of the central symmetry as this one, which means you are connecting any point with the center and then extended by the same length. Or you turn the whole figure by 180 degree equivalent. So the centrally symmetrical and reflected, basically, kind of symmetry, reflection kind of symmetry are most thoroughly studied in geometry. Now, there are certain figures much more complex than I could just draw, which do have this property of being symmetrical. Well, for instance, let me, for instance, this is a regular hexagon. Well, it has lots of different symmetries in it. Well, number one, from the rotational standpoint, obviously, now this is 60 degrees. So it's symmetrical relative to rotation around the center of the hexagon if you turn it by 60 degrees. Also, obviously, it's symmetrical by 120, 180, 240, and 360 degree because they're all multiples of 60. Now, another symmetry can be observed is related to a reflection relative to, let's say, this particular line. It's symmetrical in this way. So this might be an axis of symmetry. Or you can have this as an axis of symmetry. It will also be symmetrical. So there are many different axes of symmetry as far as the reflection is concerned. Well, actually, at least one, two, three, and one, two, three, at least six, three axis of symmetry which contain the vertexes and three axis of symmetry which are perpendicular to the lines. So these are all axis of symmetry. There are six of them. And also, it's centrally symmetrical because, obviously, 180 degrees is multiple of 60, and its rotation converts it into the same figure after rotating by 60 degree around the center. So it has many different axes and one center but many different angles of symmetry, reflection or rotation. OK, so what's interesting about symmetrical figures? Well, obviously, the most important thing is if two figures are symmetrical to each other, then they are congruent because symmetry is one of those non-deforming transformations. And so if you want to prove that two different geometrical objects are congruent, if you see that they are symmetrical, if you can prove that they are symmetrical, that's sufficient, basically. OK, so that would be it for lecture about the symmetry. It's just an introduction. The real geometrical theorems and different logical concepts will be introduced in some other later lectures. But I would like actually to make sure that the concept of congruency and symmetry are more familiar to you and you can use them. Thanks very much.