Lesson Objectives

• Educational - repetition, generalization and testing of knowledge on the topic: “Tangent to a circle”; development of basic skills.
• Developing - to develop students' attention, perseverance, perseverance, logical thinking, mathematical speech.
• Educational - through a lesson, to cultivate an attentive attitude towards each other, to instill the ability to listen to comrades, mutual assistance, independence.
• Introduce the concept of a tangent, a point of contact.
• Consider the property of the tangent and its sign and show their application in solving problems in nature and technology.

Lesson objectives

• To form skills in building tangents using a scale ruler, a protractor and a drawing triangle.
• Check students' ability to solve problems.
• Ensure mastery of the basic algorithmic techniques for constructing a tangent to a circle.
• Build skills to apply theoretical knowledge to problem solving.
• To develop the thinking and speech of students.
• Work on the formation of skills to observe, notice patterns, generalize, reason by analogy.
• Cultivate an interest in mathematics.

Lesson Plan

1. The emergence of the concept of tangent.
2. The history of the appearance of the tangent.
3. Geometric definitions.
4. Basic theorems.
5. Construction of a tangent to a circle.
6. Consolidation.

The emergence of the concept of tangent

The concept of a tangent is one of the oldest in mathematics. In geometry, a tangent to a circle is defined as a straight line that has exactly one point of intersection with this circle. The ancients, with the help of a compass and straightedge, were able to draw tangents to a circle, and later to conic sections: ellipses, hyperbolas and parabolas.

The history of the appearance of the tangent

Interest in tangents revived in modern times. Then curves were discovered that were not known to scientists of antiquity. For example, Galileo introduced the cycloid, and Descartes and Fermat built a tangent to it. In the first third of the XVII century. They began to understand that a tangent is a straight line, “most closely adjacent” to a curve in a small neighborhood of a given point. It is easy to imagine a situation where it is impossible to construct a tangent to a curve at a given point (figure).

Geometric definitions

Circle- the locus of points of the plane, equidistant from a given point, called its center.

circle.

Related definitions

• The segment connecting the center of the circle with any point on it (and also the length of this segment) is called radius circles.
• The part of the plane bounded by a circle is called around.
• A line segment that connects two points on a circle is called chord. The chord passing through the center of the circle is called diameter.
• Any two non-coinciding points on the circle divide it into two parts. Each of these parts is called arc circles. The measure of an arc can be the measure of its corresponding central angle. An arc is called a semicircle if the segment connecting its ends is a diameter.
• A line that has exactly one point in common with a circle is called tangent to the circle, and their common point is called the point of contact of the line and the circle.
• A line passing through two points on a circle is called secant.
• A central angle in a circle is a flat angle with a vertex at its center.
• An angle whose vertex lies on a circle and whose sides intersect the circle is called inscribed angle.
• Two circles that have a common center are called concentric.

Tangent line- a straight line passing through a point of the curve and coinciding with it at this point up to the first order.

Tangent to a circle A straight line that has one common point with a circle is called.

A straight line passing through a point of a circle in the same plane perpendicular to the radius drawn to this point, called a tangent. In this case, this point of the circle is called the point of contact.

Where in our case "a" is a straight line which is tangent to the given circle, the point "A" is the point of contact. In this case, a ⊥ OA (the line a is perpendicular to the radius OA).

They say that two circles touch if they have a single common point. This point is called tangent point of circles. Through a tangent point, one can draw a tangent to one of the circles, which is also tangent to the other circle. The tangency of circles is internal and external.

A tangency is called internal if the centers of the circles lie on the same side of the tangent.

A tangency is called external if the centers of the circles lie on opposite sides of the tangent

a is a common tangent to two circles, K is a point of contact.

Basic theorems

Theorem about tangent and secant

If a tangent and a secant are drawn from a point lying outside the circle, then the square of the length of the tangent is equal to the product of the secant and its outer part: MC 2 = MA MB.

Theorem. The radius drawn to the tangent point of the circle is perpendicular to the tangent.

Theorem. If the radius is perpendicular to the line at the point of intersection of the circle, then this line is tangent to this circle.

Proof.

To prove these theorems, we need to remember what a perpendicular from a point to a line is. This is the shortest distance from this point to this line. Let us assume that OA is not perpendicular to the tangent, but there is a straight line OC perpendicular to the tangent. The length of the OS includes the length of the radius and a certain segment BC, which is certainly greater than the radius. Thus, one can prove for any line. We conclude that the radius, the radius drawn to the point of contact, is the shortest distance to the tangent from the point O, i.e. OS is perpendicular to the tangent. In the proof of the converse theorem, we will proceed from the fact that the tangent has only one common point with the circle. Let the given line have one more common point B with the circle. Triangle AOB is right-angled and its two sides are equal as radii, which cannot be. Thus, we obtain that the given line has no more points in common with the circle except for the point A, i.e. is tangent.

Theorem. The segments of the tangents drawn from one point to the circle are equal, and the straight line connecting this point with the center of the circle divides the angle between the tangents into hits.

Proof.

The proof is very simple. Using the previous theorem, we assert that OB is perpendicular to AB, and OS is perpendicular to AC. Right-angled triangles ABO and ACO are equal in leg and hypotenuse (OB = OS - radii, AO - total). Therefore, their legs AB = AC and the angles OAC and OAB are also equal.

Theorem. The value of the angle formed by a tangent and a chord having a common point on a circle is equal to half the angular value of the arc enclosed between its sides.

Proof.

Consider the angle NAB formed by the tangent and the chord. Draw the diameter AC. The tangent is perpendicular to the diameter drawn to the point of contact, therefore, ∠CAN=90 o. Knowing the theorem, we see that the angle alpha (a) is equal to half the angular magnitude of the arc BC or half the angle BOC. ∠NAB=90 o -a, hence we get ∠NAB=1/2(180 o -∠BOC)=1/2∠AOB or = half the angular value of the arc BA. h.t.d.

Theorem. If a tangent and a secant are drawn from a point to a circle, then the square of the segment of the tangent from the given point to the point of tangency is equal to the product of the lengths of the segments of the secant from the given point to the points of its intersection with the circle.

Proof.

In the figure, this theorem looks like this: MA 2 \u003d MV * MS. Let's prove it. According to the previous theorem, the angle MAC is equal to half the angular size of the arc AC, but also the angle ABC is equal to half the angular size of the arc AC, according to the theorem, therefore, these angles are equal to each other. Taking into account the fact that triangles AMC and VMA have a common angle at the vertex M, we state the similarity of these triangles in two angles (the second sign). From the similarity we have: MA / MB = MC / MA, from which we get MA 2 \u003d MB * MC

Construction of tangents to a circle

And now let's try to figure it out and find out what needs to be done to build a tangent to a circle.

In this case, as a rule, a circle and a point are given in the problem. And you and I need to build a tangent to the circle so that this tangent passes through a given point.

In the event that we do not know the location of the point, then let's consider the cases of the possible location of the points.

First, the point can be inside a circle that is bounded by the given circle. In this case, it is not possible to construct a tangent through this circle.

In the second case, the point is on a circle, and we can build a tangent by drawing a perpendicular line to the radius, which is drawn to the point known to us.

Thirdly, let's assume that the point is outside the circle, which is bounded by a circle. In this case, before constructing a tangent, it is necessary to find a point on the circle through which the tangent must pass.

With the first case, I hope you understand everything, but to solve the second option, we need to build a segment on the straight line on which the radius lies. This segment must be equal to the radius and the segment that lies on the circle, on the opposite side.

Here we see that a point on a circle is the midpoint of a segment that is equal to twice the radius. The next step is to draw two circles. The radii of these circles will be equal to twice the radius of the original circle, with centers at the ends of the segment, which is equal to twice the radius. Now we can draw a straight line through any point of intersection of these circles and a given point. Such a straight line is the median perpendicular to the radius of the circle, which was drawn at the beginning. Thus, we see that this line is perpendicular to the circle, and from this it follows that it is tangent to the circle.

In the third option, we have a point lying outside the circle, which is bounded by a circle. In this case, we first construct a segment that will connect the center of the provided circle and the given point. And then we find its middle. But for this you need to build a perpendicular bisector. And you already know how to build it. Then we need to draw a circle, or at least part of it. Now we see that the point of intersection of the given circle and the newly constructed one is the point through which the tangent passes. It also passes through the point that was specified by the condition of the problem. And finally, through the two points you already know, you can draw a tangent line.

And finally, in order to prove that the straight line we have constructed is a tangent, you need to pay attention to the angle that was formed by the radius of the circle and the segment known by the condition and connecting the point of intersection of the circles with the point given by the condition of the problem. Now we see that the resulting angle rests on a semicircle. And from this it follows that this angle is right. Therefore, the radius will be perpendicular to the newly constructed line, and this line is the tangent.

Construction of a tangent.

The construction of tangents is one of those problems that led to the birth of differential calculus. The first published work relating to differential calculus, written by Leibniz, was entitled "A new method of maxima and minima, as well as tangents, for which neither fractional nor irrational quantities are an obstacle, and a special kind of calculus for this."

Geometric knowledge of the ancient Egyptians.

If we do not take into account the very modest contribution of the ancient inhabitants of the valley between the Tigris and Euphrates and Asia Minor, then geometry originated in ancient Egypt before 1700 BC. During the tropical rainy season, the Nile replenished its water supply and flooded. Water covered patches of cultivated land, and for tax purposes it was necessary to establish how much land was lost. Surveyors used a tightly stretched rope as a measuring tool. Another incentive for the accumulation of geometric knowledge by the Egyptians was their activities such as the construction of pyramids and fine arts.

The level of geometric knowledge can be judged from ancient manuscripts, which are specifically devoted to mathematics and are something like textbooks, or rather, problem books, where solutions to various practical problems are given.

The oldest mathematical manuscript of the Egyptians was copied by a certain student between 1800 - 1600. BC. from an older text. The papyrus was found by the Russian Egyptologist Vladimir Semenovich Golenishchev. It is stored in Moscow - in the Museum fine arts named after A.S. Pushkin, and is called the Moscow papyrus.

Another mathematical papyrus, written two or three hundred years later than Moscow, is kept in London. It is called: "Instruction on how to achieve knowledge of all dark things, all the secrets that hide things in themselves ... According to the old monuments, the scribe Ahmes wrote this." and bought this papyrus in Egypt. The Papyrus of Ahmes gives the solution of 84 problems for various calculations that may be needed in practice.

Transects, tangents - all this could be heard hundreds of times in geometry lessons. But graduation from school is over, years pass, and all this knowledge is forgotten. What should be remembered?

Essence

The term "tangent to a circle" is probably familiar to everyone. But it is unlikely that everyone will be able to quickly formulate its definition. Meanwhile, a tangent is such a straight line lying in the same plane with a circle that intersects it at only one point. There may be a huge variety of them, but they all have the same properties, which will be discussed below. As you might guess, the point of contact is the place where the circle and the line intersect. In every specific case she is one, but if there are more of them, then it will be a secant.

History of discovery and study

The concept of tangent appeared in antiquity. The construction of these straight lines, first to a circle, and then to ellipses, parabolas and hyperbolas with the help of a ruler and a compass, was carried out even at the initial stages of the development of geometry. Of course, history has not preserved the name of the discoverer, but it is obvious that even at that time people were quite aware of the properties of a tangent to a circle.

In modern times, interest in this phenomenon flared up again - a new round of studying this concept began, combined with the discovery of new curves. So, Galileo introduced the concept of a cycloid, and Fermat and Descartes built a tangent to it. As for circles, it seems that there are no secrets left for the ancients in this area.

Properties

The radius drawn to the point of intersection will be

the main, but not the only property that a tangent to a circle has. Another important feature includes already two straight lines. So, through one point lying outside the circle, two tangents can be drawn, while their segments will be equal. There is another theorem on this topic, but it is rarely passed within the framework of the standard school course, although it is extremely convenient for solving some problems. It sounds like this. From one point located outside the circle, a tangent and a secant are drawn to it. Segments AB, AC and AD are formed. A is the intersection of lines, B is the point of contact, C and D are the intersections. In this case, the following equality will be valid: the length of the tangent to the circle, squared, will be equal to the product of segments AC and AD.

There is an important consequence of the above. For each point of the circle, you can build a tangent, but only one. The proof of this is quite simple: theoretically dropping a perpendicular from the radius onto it, we find out that the formed triangle cannot exist. And this means that the tangent is unique.

Building

Among other tasks in geometry, there is a special category, as a rule, not

favored by pupils and students. To solve tasks from this category, you only need a compass and a ruler. These are building tasks. There are also methods for constructing a tangent.

So, given a circle and a point lying outside its boundaries. And it is necessary to draw a tangent through them. How to do it? First of all, you need to draw a segment between the center of the circle O and given point. Then, using a compass, divide it in half. To do this, you need to set the radius - a little more than half the distance between the center of the original circle and the given point. After that, you need to build two intersecting arcs. Moreover, the radius of the compass does not need to be changed, and the center of each part of the circle will be the initial point and O, respectively. The intersections of the arcs must be connected, which will divide the segment in half. Set a radius on the compass equal to this distance. Next, with the center at the intersection point, draw another circle. Both the initial point and O will lie on it. In this case, there will be two more intersections with the circle given in the problem. They will be the touch points for the initially given point.

It was the construction of tangents to the circle that led to the birth

differential calculus. The first work on this topic was published by the famous German mathematician Leibniz. He provided for the possibility of finding maxima, minima and tangents, regardless of fractional and irrational values. Well, now it is used for many other calculations as well.

In addition, the tangent to the circle is related to the geometric meaning of the tangent. That is where its name comes from. Translated from Latin, tangens means "tangent". Thus, this concept is connected not only with geometry and differential calculus, but also with trigonometry.

Two circles

A tangent does not always affect only one figure. If a huge number of straight lines can be drawn to one circle, then why not vice versa? Can. But the task in this case is seriously complicated, because the tangent to two circles may not pass through any points, and the relative position of all these figures can be very

different.

Types and varieties

When it comes to two circles and one or more straight lines, even if it is known that these are tangents, it does not immediately become clear how all these figures are located in relation to each other. Based on this, there are several varieties. So, circles can have one or two common points or not have them at all. In the first case, they will intersect, and in the second, they will touch. And here there are two varieties. If one circle is, as it were, embedded in the second, then the touch is called internal, if not, then external. You can understand the relative position of the figures not only based on the drawing, but also having information about the sum of their radii and the distance between their centers. If these two quantities are equal, then the circles touch. If the first is greater, they intersect, and if less, then they do not have common points.

Same with straight lines. For any two circles that do not have common points, one can

build four tangents. Two of them will intersect between the figures, they are called internal. A couple of others are external.

If we are talking about circles that have one common point, then the task is greatly simplified. The point is that for any relative position in this case they will have only one tangent. And it will pass through the point of their intersection. So the construction of the difficulty will not cause.

If the figures have two points of intersection, then a straight line can be constructed for them, tangent to the circle, both one and the second, but only the outer one. The solution to this problem is similar to what will be discussed below.

Problem solving

Both internal and external tangents to two circles are not so simple in construction, although this problem can be solved. The fact is that an auxiliary figure is used for this, so think of this method yourself

quite problematic. So, given two circles with different radii and centers O1 and O2. For them, you need to build two pairs of tangents.

First of all, near the center of the larger circle, you need to build an auxiliary one. In this case, the difference between the radii of the two initial figures must be established on the compass. Tangents to the auxiliary circle are built from the center of the smaller circle. After that, from O1 and O2, perpendiculars are drawn to these lines until they intersect with the original figures. As follows from the main property of the tangent, the desired points on both circles are found. The problem is solved, at least, its first part.

In order to construct the internal tangents, one has to solve practically

a similar task. Again, an auxiliary figure is needed, but this time its radius will be equal to the sum of the original ones. Tangents are constructed to it from the center of one of the given circles. The further course of the solution can be understood from the previous example.

Tangent to a circle or even two or more is not such a difficult task. Of course, mathematicians have long ceased to solve such problems manually and trust calculations special programs. But do not think that now it is not necessary to be able to do it yourself, because in order to correctly formulate a task for a computer, you need to do and understand a lot. Unfortunately, there are fears that after the final transition to the test form of knowledge control, construction tasks will cause more and more difficulties for students.

As for finding common tangents for more circles, this is not always possible, even if they lie in the same plane. But in some cases it is possible to find such a line.

Real life examples

A common tangent to two circles is often encountered in practice, although this is not always noticeable. Conveyors, block systems, pulley transmission belts, thread tension in sewing machine, and even just a bicycle chain - all these are examples from life. So do not think that geometric problems remain only in theory: in engineering, physics, construction and many other areas, they find practical application.

points $x\_0\backslash in\; \backslash mathbb(R)$, and is differentiable in it: $f\; \backslash in\; \backslash mathcal(D)(x\_0)$. Tangent to the graph of a function $f$ at the point $x\_0$ is called the graph of a linear function, given by the equation $y\; =\; f(x\_0)\; +\; f"(x\_0)(x-x\_0),\backslash quad\; x\backslash in\; \backslash mathbb(R)$.

If the function $f$ has at the point $x\_0$ infinite derivative $f"(x\_0)\; =\; \backslash pm\backslash infty,$ then the tangent line at this point is the vertical line given by the equation $x\; =\; x\_0.$

Comment

It follows directly from the definition that the graph of the tangent line passes through the point $(x\_0,f(x\_0))$. Corner $\backslash alpha$ between the tangent to the curve and the x-axis satisfies the equation

$\backslash operatorname(tg)\backslash ,\backslash alpha\; =\; f"(x\_0)=k,$

Where $\backslash operatorname(tg)$ stands for tangent, and $\backslash operatorname\; (k)$- tangent slope coefficient. Derivative at a point $x\_0$ is equal to angular coefficient tangent to the graph of the function $y\; =\; f(x)$ at this point.

Tangent as the limiting position of a secant

Let $f\backslash colon\; U(x\_0)\; \backslash to\; \backslash R$ And $x\_1\backslash in\; U(x\_0).$ Then a straight line passing through the points $(x\_0,f(x\_0))$ And $(x\_1,f(x\_1))$ given by the equation

$y\; =\; f(x\_0)\; +\; \backslash frac(f(x\_1)\; -\; f(x\_0))(x\_1\; -\; x\_0)(x-x\_0).$

This line passes through the point $(x\_0,f(x\_0))$ for anyone $x\_1\backslash in\; U(x\_0),$ and its angle of inclination $\backslash alpha(x\_1)$ satisfies the equation

$\backslash operatorname(tg)\backslash ,\backslash alpha(x\_1)\; =\; \backslash frac(f(x\_1)\; -\; f(x\_0))(x\_1\; -\; x\_0).$

Due to the existence of the derivative of the function $f$ at the point $x\_0,$ passing to the limit at $x\_1\backslash to\; x\_0,$ we get that there is a limit

$\backslash lim\backslash limits\_(x\_1\; \backslash to\; x\_0)\; \backslash operatorname(tg)\backslash ,\backslash alpha(x\_1)\; =\; f"(x\_0),$

and due to the continuity of the arc tangent and the limiting angle

$\backslash alpha\; =\; \backslash operatorname(arctg)\backslash ,f"(x\_0).$

A line passing through a point $(x\_0,f(x\_0))$ and having a limit angle of inclination that satisfies $\backslash operatorname(tg)\backslash ,\backslash alpha\; =\; f"(x\_0),$ is given by the tangent equation:

$y\; \backslash u003d\; f\; (x\_0)\; +\; f\; "(x\_0)\; (x-x\_0).$

Tangent to circle

A straight line that has one common point with a circle and lies in the same plane with it is called a tangent to the circle.

Properties

1. The tangent to the circle is perpendicular to the radius drawn to the point of contact.
2. The segments of tangents to the circle drawn from one point are equal and make equal angles with the line passing through this point and the center of the circle.
3. The length of the segment of the tangent drawn to a circle of unit radius, taken between the point of tangency and the point of intersection of the tangent with the ray drawn from the center of the circle, is the tangent of the angle between this ray and the direction from the center of the circle to the point of tangency. "Tangens" from lat. tangents- "tangent".

Variations and Generalizations

One-sided semi-tangents

• If there is a right derivative That right semitangent to the graph of the function $f$ at the point $x\_0$ called beam

$y\; =\; f(x\_0)\; +\; f"\_+(x\_0)(x\; -\; x\_0),\backslash quad\; x\; \backslash geqslant\; x\_0.$

• If there is a left derivative That left semitangent to the graph of the function $f$ at the point $x\_0$ called beam

$y\; =\; f(x\_0)\; +\; f"\_-(x\_0)(x\; -\; x\_0),\backslash quad\; x\; \backslash leqslant\; x\_0.$

• If there is an infinite right derivative $f"\_+(x\_0)\; =\; +\backslash infty\backslash ;\; (-\backslash infty),$ $f$ at the point $x\_0$ called beam

$x\; =\; x\_0,\; \backslash ;\; y\; \backslash geqslant\; f(x\_0)\backslash ;\; (y\backslash leqslant\; f(x\_0)).$

• If there is an infinite left derivative $f"\_-(x\_0)\; =\; +\backslash infty\backslash ;\; (-\backslash infty),$ then the right semitangent to the graph of the function $f$ at the point $x\_0$ called beam

$x\; =\; x\_0,\; \backslash ;\; y\; \backslash leqslant\; f(x\_0)\backslash ;\; (y\backslash geqslant\; f(x\_0)).$

see also

• Normal, binormal

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Literature

• Toponogov V. A. Differential geometry of curves and surfaces. - Fizmatkniga, 2012. - ISBN 9785891552135.
• // Encyclopedic Dictionary of Brockhaus and Efron: in 86 volumes (82 volumes and 4 additional). - St. Petersburg. , 1890-1907.

An excerpt characterizing the tangent line

- In places! - shouted a young officer at the soldiers gathered around Pierre. This young officer, apparently, performed his position for the first or second time, and therefore treated both the soldiers and the commander with particular distinctness and uniformity.
The erratic firing of cannons and rifles intensified throughout the field, especially to the left, where Bagration's flashes were, but because of the smoke of shots from the place where Pierre was, it was almost impossible to see anything. Moreover, observations of how, as it were, a family (separated from all others) circle of people who were on the battery, absorbed all the attention of Pierre. His first unconsciously joyful excitement, produced by the sight and sounds of the battlefield, was now replaced, especially after the sight of this lonely soldier lying in the meadow, by another feeling. Sitting now on the slope of the ditch, he watched the faces around him.
By ten o'clock, twenty people had already been carried away from the battery; two guns were broken, more and more shells hit the battery and flew, buzzing and whistling, long-range bullets. But the people who were on the battery did not seem to notice this; cheerful conversation and jokes were heard from all sides.
- Chinenko! - the soldier shouted at the approaching, whistling grenade. - Not here! To the infantry! - another added with a laugh, noticing that the grenade flew over and hit the ranks of the cover.
- What, friend? - laughed another soldier at the crouching peasant under the flying cannonball.
Several soldiers gathered at the rampart, looking at what was happening ahead.
“And they took off the chain, you see, they went back,” they said, pointing over the shaft.
“Look at your business,” the old non-commissioned officer shouted at them. - They went back, which means there is work back. - And the non-commissioned officer, taking one of the soldiers by the shoulder, pushed him with his knee. Laughter was heard.
- Roll on to the fifth gun! shouted from one side.
“Together, more amicably, in burlatski,” the cheerful cries of those who changed the gun were heard.
“Ay, I almost knocked off our master’s hat,” the red-faced joker laughed at Pierre, showing his teeth. “Oh, clumsy,” he added reproachfully to the ball that had fallen into the wheel and leg of a man.
- Well, you foxes! another laughed at the squirming militiamen who were entering the battery for the wounded.
- Al is not tasty porridge? Ah, crows, swayed! - they shouted at the militia, who hesitated in front of a soldier with a severed leg.
“Something like that, little one,” the peasants mimicked. - They don't like passion.
Pierre noticed how after each shot that hit, after each loss, a general revival flared up more and more.
As from an advancing thundercloud, more and more often, brighter and brighter flashed on the faces of all these people (as if in repulse to what was happening) lightning bolts of hidden, flaring fire.
Pierre did not look ahead on the battlefield and was not interested in knowing what was happening there: he was completely absorbed in contemplating this, more and more burning fire, which in the same way (he felt) flared up in his soul.
At ten o'clock the infantry soldiers, who were ahead of the battery in the bushes and along the Kamenka River, retreated. From the battery it was visible how they ran back past it, carrying the wounded on their guns. Some general with his retinue entered the mound and, after talking with the colonel, looking angrily at Pierre, went down again, ordering the infantry cover, which was standing behind the battery, to lie down so as to be less exposed to shots. Following this, in the ranks of the infantry, to the right of the battery, a drum was heard, shouts of command, and from the battery it was clear how the ranks of the infantry moved forward.
Pierre looked over the shaft. One face in particular caught his eye. It was an officer who, with a pale young face, was walking backwards, carrying a lowered sword, and looking around uneasily.
The ranks of infantry soldiers disappeared into the smoke, their long-drawn cry and frequent firing of guns were heard. A few minutes later, crowds of wounded and stretchers passed from there. Shells began to hit the battery even more often. Several people lay uncleaned. Near the cannons, the soldiers moved busier and more lively. No one paid any attention to Pierre anymore. Once or twice he was angrily shouted at for being on the road. The senior officer, with a frown on his face, moved with large, quick steps from one gun to another. The young officer, flushed even more, commanded the soldiers even more diligently. Soldiers fired, turned, loaded and did their job with intense panache. They bounced along the way, as if on springs.

Definition. A tangent to a circle is a straight line in the plane that has exactly one common point with the circle.

Here are a couple of examples:

Circle with center O touches a straight line l at the point A From anywhere M Exactly two tangents can be drawn outside the circle difference between tangent l, secant BC and direct m, which has no common points with the circle

This could be the end, but practice shows that it is not enough just to memorize the definition - you need to learn to see the tangents in the drawings, know their properties and, in addition, how to practice using these properties when solving real problems. We will deal with all this today.

Basic properties of tangents

In order to solve any problems, you need to know four key properties. Two of them are described in any reference book / textbook, but the last two are somehow forgotten about, but in vain.

1. Segments of tangents drawn from one point are equal

A little higher, we already talked about two tangents drawn from one point M. So:

The segments of the tangents to the circle, drawn from one point, are equal.

Segments AM And BM equal

2. The tangent is perpendicular to the radius drawn to the point of contact

Let's look at the picture above again. Let's draw the radii OA And OB, after which we find that the angles OAM And OBM- straight.

The radius drawn to the tangent point is perpendicular to the tangent.

This fact can be used without proof in any problem:

The radii drawn to the tangent point are perpendicular to the tangents

By the way, note: if you draw a segment OM, then we get two equal triangles: OAM And OBM.

3. Relationship between tangent and secant

But this is a more serious fact, and most schoolchildren do not know it. Consider a tangent and a secant that pass through the same common point M. Naturally, the secant will give us two segments: inside the circle (segment BC- it is also called a chord) and outside (it is called that - the outer part MC).

The product of the entire secant by its outer part is equal to the square of the tangent segment

Relationship between secant and tangent

4. Angle between tangent and chord

An even more advanced fact that is often used to solve complex problems. I highly recommend taking it on board.

The angle between a tangent and a chord is equal to the inscribed angle based on this chord.

Where does the dot come from B? In real problems, it usually "pops up" somewhere in the condition. Therefore, it is important to learn to recognize this configuration in the drawings.

Sometimes it still applies :)

Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.

Collection and use of personal information

Personal information refers to data that can be used to identify or contact a specific person.

You may be asked to provide your personal information at any time when you contact us.

The following are some examples of the types of personal information we may collect and how we may use such information.

What personal information we collect:

• When you submit an application on the site, we may collect various information, including your name, phone number, email address, etc.

How we use your personal information:

• The personal information we collect allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
• From time to time, we may use your personal information to send you important notices and messages.
• We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
• If you enter a prize draw, contest or similar incentive, we may use the information you provide to administer such programs.

Disclosure to third parties

We do not disclose information received from you to third parties.

Exceptions:

• If necessary - in accordance with the law, judicial order, in legal proceedings, and / or based on public requests or requests from government agencies on the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest reasons.
• In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.

Protection of personal information

We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.

Maintaining your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.