Let us now consider the axes of symmetry of the sides of the triangle. Recall that the axis of symmetry of a segment is the perpendicular to the segment in its middle.
Any point of such a perpendicular is equally distant from the ends of the segment. Let now be the perpendiculars drawn through the midpoints of the sides BC and AC of triangle ABC (Fig. 220) to these sides, i.e., the symmetry axes of these two sides. The point of their intersection Q is equally distant from the vertices B and C of the triangle, since it lies on the axis of symmetry of side BC, and it is also equally distant from the vertices A and C. Consequently, it is equally distant from all three vertices of the triangle, including vertices A and B. This means that it lies on the axis of symmetry of the third side AB of the triangle. So, the axes of symmetry of the three sides of the triangle intersect at one point. This point is equally distant from the vertices of the triangle. Therefore, if you draw a circle with a radius equal to the distance of this point from the vertices of the triangle, with the center at the found point, then it will pass through all three vertices of the triangle. Such a circle (Fig. 220) is called a circumscribed circle. Conversely, if you imagine a circle passing through the three vertices of a triangle, then its center must be at equal distances from the vertices of the triangle and therefore belongs to each of the axes of symmetry of the sides of the triangle.
Therefore, a triangle has only one circumscribed circle: a given triangle can be circumscribed by a circle, and only one; its center lies at the point of intersection of three perpendiculars raised to the sides of the triangle at their midpoints.
In Fig. 221 shows circles circumscribed around acute, right and obtuse triangles; the center of the circumscribed circle lies in the first case inside the triangle, in the second - in the middle of the hypotenuse of the triangle, in the third - outside the triangle. This follows most simply from the properties of angles supported by an arc of a circle (see paragraph 210).
Since any three points that do not lie on the same line can be considered the vertices of a triangle, it can be argued that a single circle passes through any three points that do not belong to the line. Therefore, two circles have at most two points in common.
You will need
- - properties of symmetrical points;
- - properties of symmetrical figures;
- - ruler;
- - square;
- - compass;
- - pencil;
- - paper;
- - a computer with a graphics editor.
Instructions
Draw a straight line a, which will be the axis of symmetry. If its coordinates are not specified, draw it arbitrarily. Place an arbitrary point A on one side of this line. You need to find a symmetrical point.
Helpful advice
Symmetry properties are used constantly in AutoCAD. To do this, use the Mirror option. To construct an isosceles triangle or isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Reflect them using the specified command and extend the sides to the required size. In the case of a triangle, this will be the point of their intersection, and for a trapezoid, this will be a given value.
You constantly come across symmetry in graphic editors when you use the “flip vertically/horizontally” option. In this case, the axis of symmetry is taken to be a straight line corresponding to one of the vertical or horizontal sides of the picture frame.
Sources:
- how to draw central symmetry
Constructing a cross section of a cone is not such a difficult task. The main thing is to follow a strict sequence of actions. Then this task will be easily accomplished and will not require much labor from you.
You will need
- - paper;
- - pen;
- - circle;
- - ruler.
Instructions
When answering this question, you must first decide what parameters define the section.
Let this be the straight line of intersection of the plane l with the plane and the point O, which is the intersection with its section.
The construction is illustrated in Fig. 1. The first step in constructing a section is through the center of the section of its diameter, extended to l perpendicular to this line. The result is point L. Next, draw a straight line LW through point O, and construct two guide cones lying in the main section O2M and O2C. At the intersection of these guides lie point Q, as well as the already shown point W. These are the first two points of the desired section.
Now draw a perpendicular MS at the base of the cone BB1 and construct generatrices of the perpendicular section O2B and O2B1. In this section, through point O, draw a straight line RG parallel to BB1. Т.R and Т.G are two more points of the desired section. If the cross section of the ball were known, then it could be built already at this stage. However, this is not an ellipse at all, but something elliptical that has symmetry with respect to the segment QW. Therefore, you should build as many section points as possible in order to connect them later with a smooth curve to obtain the most reliable sketch.
Construct an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and construct the corresponding guides O2A and O2N. Through t.O, draw a straight line passing through PQ and WG until it intersects with the newly constructed guides at points P and E. These are two more points of the desired section. Continuing in the same way, you can find as many points as you want.
True, the procedure for obtaining them can be slightly simplified using symmetry with respect to QW. To do this, you can draw straight lines SS’ in the plane of the desired section, parallel to RG until they intersect with the surface of the cone. The construction is completed by rounding the constructed polyline from chords. It is enough to construct half of the desired section due to the already mentioned symmetry with respect to QW.
Video on the topic
Tip 3: How to graph a trigonometric function
You need to draw schedule trigonometric functions? Master the algorithm of actions using the example of constructing a sinusoid. To solve the problem, use the research method.
You will need
- - ruler;
- - pencil;
- - knowledge of the basics of trigonometry.
Instructions
Video on the topic
note
If the two semi-axes of a single-strip hyperboloid are equal, then the figure can be obtained by rotating a hyperbola with semi-axes, one of which is the above, and the other, different from the two equal ones, around the imaginary axis.
Helpful advice
When examining this figure relative to the Oxz and Oyz axes, it is clear that its main sections are hyperbolas. And when this spatial figure of rotation is cut by the Oxy plane, its section is an ellipse. The neck ellipse of a single-strip hyperboloid passes through the origin of coordinates, because z=0.
The throat ellipse is described by the equation x²/a² +y²/b²=1, and the other ellipses are composed by the equation x²/a² +y²/b²=1+h²/c².
Sources:
- Ellipsoids, paraboloids, hyperboloids. Rectilinear generators
The shape of a five-pointed star has been widely used by man since ancient times. We consider its shape beautiful because we unconsciously recognize in it the relationships of the golden section, i.e. the beauty of the five-pointed star is justified mathematically. Euclid was the first to describe the construction of a five-pointed star in his Elements. Let's join in with his experience.
You will need
- ruler;
- pencil;
- compass;
- protractor.
Instructions
The construction of a star comes down to the construction and subsequent connection of its vertices to each other sequentially through one. In order to build the correct one, you need to divide the circle into five.
Construct an arbitrary circle using a compass. Mark its center with point O.
Mark point A and use a ruler to draw line segment OA. Now you need to divide the segment OA in half; to do this, from point A, draw an arc of radius OA until it intersects the circle at two points M and N. Construct the segment MN. The point E where MN intersects OA will bisect segment OA.
Restore the perpendicular OD to the radius OA and connect points D and E. Make a notch B on OA from point E with radius ED.
Now, using line segment DB, mark the circle into five equal parts. Label the vertices of the regular pentagon sequentially with numbers from 1 to 5. Connect the dots in the following sequence: 1 with 3, 2 with 4, 3 with 5, 4 with 1, 5 with 2. Here is the regular five-pointed star, into a regular pentagon. This is exactly the way I built it
Points M And M 1 are called symmetrical with respect to a given straight line L, if this line is the perpendicular bisector to the segment MM 1 (Figure 1). Every point is straight L symmetrical to itself. Transformation of a plane, in which each point is mapped to a point symmetrical to it relative to a given line L, called axial symmetry with the L axis and is designated S L :S L (M) = M 1 .
Points M And M 1 are mutually symmetrical with respect to L, That's why S L (M 1 )=M. Consequently, the transformation inverse to axial symmetry is the same axial symmetry: S L -1= S L , S L° S L = E. In other words, the axial symmetry of the plane is involutive transformation.
The image of a given point with axial symmetry can be simply constructed using only one compass. Let L- axis of symmetry, A And B- arbitrary points of this axis (Figure 2). If S L (M) = M 1, then by the property of the points of the perpendicular bisector to the segment we have: AM = AM 1 And BM = BM 1 . So, period M 1 belongs to two circles: a circle with center A radius A.M. and circles with center B radius B.M. (M- given point). Figure F and her image F 1 with axial symmetry are called symmetrical figures relative to a straight line L(Figure 3).
Theorem. Axial symmetry of a plane is movement.
If A And IN- any points of the plane and S L (A) = A 1 , S L (B) = B 1, then we must prove that A 1 B 1 = AB. To do this, we introduce a rectangular coordinate system OXY so that the axis OX coincides with the axis of symmetry. Points A And IN have coordinates A(x 1 ,-y 1 ) And B(x 1 ,-y 2 ) .Points A 1 and IN 1 have coordinates A 1 (x 1 ,y 1 ) And B 1 (x 1 ,y 2 ) (Figure 4 - 8). Using the formula for the distance between two points we find:
From these relations it is clear that AB=A 1 IN 1, which is what needed to be proven.
From a comparison of the orientations of the triangle and its image, we obtain that the axial symmetry of the plane is movement of the second kind.
Axial symmetry maps each line onto a straight line. In particular, each of the lines perpendicular to the axis of symmetry is mapped onto itself by this symmetry.
Theorem. A straight line other than perpendicular to the axis of symmetry and its image at this symmetry intersect on the axis of symmetry or are parallel to it.
Proof. Let a straight line be given, not perpendicular to the axis L symmetry. If m? L=P And S L (m)=m 1, then m 1 ?m And S L (P)=P, That's why Pm1(Figure 9). If m || L, That m 1 || L, since otherwise straight m And m 1 would intersect at a point on a straight line L, which contradicts the condition m ||L(Figure 10).
By virtue of the definition of equal figures, straight lines symmetrical with respect to a straight line L, form with a straight line L equal angles (Figure 9).
Straight L called axis of symmetry of figure F, if with symmetry with the axis L figure F maps to itself: S L (F) =F. They say that the figure F symmetrical about a straight line L.
For example, any straight line containing the center of a circle is the axis of symmetry of this circle. Indeed, let M- arbitrary point on the circle sch with center ABOUT, OL, S L (M)=M 1 . Then S L (O) = O And OM 1 =OM, i.e. M 1 є ь. So, the image of any point on a circle belongs to this circle. Hence, S L (u)=u.
The axes of symmetry of a pair of nonparallel lines are two perpendicular lines containing the bisectors of the angles between these lines. The axis of symmetry of a segment is the straight line containing it, as well as the perpendicular bisector to this segment.
Properties of axial symmetry
- 1. With axial symmetry, the image of a straight line is a straight line, the image of parallel lines is parallel lines
- 3. Axial symmetry preserves the simple relationship of three points.
- 3. With axial symmetry, a segment goes into a segment, a ray into a ray, a half-plane into a half-plane.
- 4. With axial symmetry, an angle transforms into an angle equal to it.
- 5. With axial symmetry with the d axis, every straight line perpendicular to the d axis remains in place.
- 6. With axial symmetry, an orthonormal frame transforms into an orthonormal frame. In this case, point M with coordinates x and y relative to the reference point R goes to point M` with the same coordinates x and y, but relative to the reference point R`.
- 7. The axial symmetry of the plane transforms the right orthonormal frame into the left one and, conversely, the left orthonormal frame into the right one.
- 8. The composition of two axial symmetries of a plane with parallel axes is a parallel translation to a vector perpendicular to the given lines, the length of which is twice the distance between the given lines
People's lives are filled with symmetry. It’s convenient, beautiful, and there’s no need to invent new standards. But what is it really and is it as beautiful in nature as is commonly believed?
Symmetry
Since ancient times, people have sought to organize the world around them. Therefore, some things are considered beautiful, and some are not so much. From an aesthetic point of view, the golden and silver ratios are considered attractive, as well as, of course, symmetry. This term is of Greek origin and literally means “proportionality.” Of course, we are talking not only about coincidence on this basis, but also on some others. In a general sense, symmetry is a property of an object when, as a result of certain formations, the result is equal to the original data. It is found in both living and inanimate nature, as well as in objects made by man.
First of all, the term "symmetry" is used in geometry, but finds application in many scientific fields, and its meaning remains generally unchanged. This phenomenon occurs quite often and is considered interesting, since several of its types, as well as elements, differ. The use of symmetry is also interesting, because it is found not only in nature, but also in patterns on fabric, borders of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely fascinating.
Use of the term in other scientific fields
In the following, symmetry will be considered from the point of view of geometry, but it is worth mentioning that this word is used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon is studied from different angles and under different conditions. For example, the classification depends on what science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged throughout.
Video on the topic
Classification
There are several main types of symmetry, of which three are the most common:
In addition, the following types are also distinguished in geometry; they are much less common, but no less interesting:
- sliding;
- rotational;
- point;
- progressive;
- screw;
- fractal;
- etc.
In biology, all species are called slightly differently, although in essence they may be the same. Division into certain groups occurs on the basis of the presence or absence, as well as the quantity of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.
Basic elements
The phenomenon has certain features, one of which is necessarily present. The so-called basic elements include planes, centers and axes of symmetry. It is in accordance with their presence, absence and quantity that the type is determined.
The center of symmetry is the point inside a figure or crystal at which the lines connecting in pairs all sides parallel to each other converge. Of course, it does not always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since it does not exist. According to the definition, it is obvious that the center of symmetry is that through which a figure can be reflected onto itself. An example would be, for example, a circle and a point in its middle. This element is usually designated as C.
The plane of symmetry, of course, is imaginary, but it is precisely it that divides the figure into two parts equal to each other. It can pass through one or more sides, be parallel to it, or divide them. For the same figure, several planes can exist at once. These elements are usually designated as P.
But perhaps the most common is what is called “axis of symmetry”. This is a common phenomenon that can be seen both in geometry and in nature. And it is worthy of separate consideration.
Axles
Often the element in relation to which a figure can be called symmetrical is 
a straight line or segment appears. In any case, we are not talking about a point or a plane. Then the axes of symmetry of the figures are considered. There can be a lot of them, and they can be located in any way: dividing the sides or being parallel to them, as well as intersecting corners or not doing so. Axes of symmetry are usually designated as L.
Examples include isosceles and equilateral triangles. In the first case, there will be a vertical axis of symmetry, on both sides of which there are equal faces, and in the second, the lines will intersect each angle and coincide with all bisectors, medians and altitudes. Ordinary triangles do not have this.
By the way, the totality of all the above elements in crystallography and stereometry is called the degree of symmetry. This indicator depends on the number of axes, planes and centers.
Examples in geometry
Conventionally, we can divide the entire set of objects of study by mathematicians into figures that have an axis of symmetry and those that do not. All regular polygons, circles, ovals, as well as some special cases automatically fall into the first category, while the rest fall into the second group.
As in the case when we talked about the axis of symmetry of a triangle, this element does not always exist for a quadrilateral. For a square, rectangle, rhombus or parallelogram it is, but for an irregular figure, accordingly, it is not. For a circle, the axis of symmetry is the set of straight lines that pass through its center.
In addition, it is interesting to consider three-dimensional figures from this point of view. In addition to all regular polygons and the ball, some cones, as well as pyramids, parallelograms and some others, will have at least one axis of symmetry. Each case must be considered separately.
Examples in nature
Mirror symmetry in life is called bilateral, it is most common
often. Any person and many animals are an example of this. The axial one is called radial and is found much less frequently, as a rule, in the plant world. And yet they exist. For example, it is worth thinking about how many axes of symmetry a star has, and does it have any at all? Of course, we are talking about marine life, and not about the subject of study by astronomers. And the correct answer would be: it depends on the number of rays of the star, for example five, if it is five-pointed.
In addition, radial symmetry is observed in many flowers: daisies, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.
Arrhythmia
This term, first of all, reminds most of medicine and cardiology, but it initially has a slightly different meaning. In this case, the synonym will be “asymmetry”, that is, the absence or violation of regularity in one form or another. It can be found as an accident, and sometimes it can become a wonderful technique, for example in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous Leaning Tower of Pisa is slightly tilted, and although it is not the only one, it is the most famous example. It is known that this happened by accident, but this has its own charm.
In addition, it is obvious that the faces and bodies of people and animals are not completely symmetrical either. There have even been studies that show that “correct” faces are judged to be lifeless or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore are extremely interesting.
What is an axis of symmetry? This is a set of points that form a straight line, which is the basis of symmetry, that is, if a certain distance is set aside from a straight line on one side, then it will be reflected in the other direction in the same size. The axis can be anything - a point, a straight line, a plane, and so on. But it’s better to talk about this with clear examples.
Symmetry
In order to understand what an axis of symmetry is, you need to delve into the very definition of symmetry. This is the correspondence of a certain fragment of the body relative to any axis, when its structure is unchanged, and the properties and shape of such an object remain the same relative to its transformations. We can say that symmetry is the property of bodies to display. When a fragment cannot have such a correspondence, this is called asymmetry or arrhythmia.
Some figures do not have symmetry, which is why they are called irregular or asymmetrical. These include various trapezoids (except isosceles), triangles (except isosceles and equilateral) and others.
Types of symmetry
We will also discuss some types of symmetry in order to fully explore this concept. They are divided like this:
History of symmetry
The very concept of symmetry is often the starting point in the theories and hypotheses of scientists of ancient times, who were confident in the mathematical harmony of the universe, as well as in the manifestation of the divine principle. The ancient Greeks firmly believed that the Universe was symmetrical, because symmetry is magnificent. Man has long used the idea of symmetry in his knowledge of the picture of the universe.
In the 5th century BC, Pythagoras considered the sphere to be the most perfect form and thought that the Earth was shaped like a sphere and moved in the same way. He also believed that the Earth moved in the form of some kind of “central fire”, around which 6 planets (known at that time), the Moon, the Sun and all other stars were supposed to revolve.
And the philosopher Plato considered polyhedra to be the personification of the four natural elements:
- tetrahedron is fire, since its vertex is directed upward;
- cube - earth, since it is the most stable body;
- octahedron - air, no explanation;
- icosahedron - water, since the body does not have rough geometric shapes, angles, and so on;
- The image of the entire Universe was the dodecahedron.
Because of all these theories, regular polyhedra are called Platonic solids.
The architects of Ancient Greece used symmetry. All their buildings were symmetrical, as evidenced by images of the ancient temple of Zeus at Olympia.
The Dutch artist M.C. Escher also used symmetry in his paintings. In particular, a mosaic of two birds flying towards them became the basis of the painting “Day and Night”.
Also, our art critics did not neglect the rules of symmetry, as can be seen in the example of Vasnetsov’s painting “Bogatyrs”.
What can we say, symmetry has been a key concept for all artists for many centuries, but in the 20th century its meaning was also appreciated by all workers in the exact sciences. Accurate evidence is provided by physical and cosmological theories, for example, the theory of relativity, string theory, and absolutely all quantum mechanics. From the times of Ancient Babylon and ending with the advanced discoveries of modern science, the ways of studying symmetry and the discovery of its basic laws are traced.
Symmetry of geometric shapes and bodies
Let's take a closer look at geometric bodies. For example, the axis of symmetry of a parabola is a straight line passing through its vertex and cutting the given body in half. This figure has one single axis.
But with geometric figures the situation is different. The axis of symmetry of a rectangle is also straight, but there are several of them. You can draw the axis parallel to the width segments, or you can draw it parallel to the length segments. But it's not that simple. Here the straight line has no axes of symmetry, since its end is not defined. Only central symmetry could exist, but, accordingly, there will not be such.
You should also know that some bodies have many axes of symmetry. This is not difficult to guess. There is no need to even talk about how many axes of symmetry a circle has. Any straight line passing through the center of a circle is such, and there are an infinite number of these straight lines.
Some quadrilaterals may have two axes of symmetry. But the second ones must be perpendicular. This happens in the case of a rhombus and a rectangle. In the first, the axes of symmetry are diagonals, and in the second, the middle lines. Only a square has many such axes.
Symmetry in nature
Nature amazes with many examples of symmetry. Even our human body is symmetrical. Two eyes, two ears, a nose and a mouth are located symmetrically relative to the central axis of the face. The arms, legs and the whole body in general are arranged symmetrically to an axis passing through the middle of our body.
And how many examples surround us all the time! These are flowers, leaves, petals, vegetables and fruits, animals and even honeycombs of bees that have a pronounced geometric shape and symmetry. All of nature is arranged in an orderly manner, everything has its place, which once again confirms the perfection of the laws of nature, in which symmetry is the main condition.
Conclusion
We are constantly surrounded by some phenomena and objects, for example, a rainbow, a drop, flowers, petals, and so on. Their symmetry is obvious; to some extent it is due to gravity. Often in nature, the concept of “symmetry” is understood as the regular change of day and night, seasons, and so on.
Similar properties are observed wherever there is order and equality. Also, the laws of nature themselves - astronomical, chemical, biological and even genetic - are subject to certain principles of symmetry, since they are perfectly systematic, which means that balance has an all-encompassing scale. Consequently, axial symmetry is one of the fundamental laws of the universe as a whole.