When drawing the contours of objects, it is relatively often necessary to build common tangents to two arcs of circles. A common tangent to two circles can be external, if both circles are located on one side of it, and internal, if the circles are located on different sides of the tangent.
Construction of a common external tangent to two circles of radii R and r (Figure 47). From the center of a circle with a larger radius - points O 1 describe a circle with a radius R – r (Figure 47, a). Find the middle of a segment O 2 O 1 – point O 3 and from it draw an auxiliary circle with a radius O 3 O 2 or O 3 O 1. Both drawn circles intersect at points A and IN ... Points O 1 and B connect a straight line and at its intersection with a circle with a radius R define the touch point D (Figure 47, b). From point O 2 parallel straight O 1 D draw a line until it intersects with a circle with a radius r and get the second touch point C ... Straight CD is the required tangent. The second common external tangent to these circles is also constructed (straight line EF ).
Drawing 47
Construction of a common internal tangent to two circles of radii R and r (Figure 48). From the center of any circle, for example: points O 1 , describe a circle with a radius R +r (Figure 48, a). Dividing a segment O 2 O 1 in half, get the point O 3 ... From point O 3 how from the center describe the second auxiliary circle with a radius O 3 O 2 \u003d O 3 ABOUT 1 and mark the points A and IN intersections of construction circles. Connecting straight points A and O 1 (Figure 48, b), at its intersection with a circle of radius R get the touch point D ... Through the center of a circle of radius r draw a straight line parallel to a straight line O 1 D , and at its intersection with a given circle determine the second point of tangency FROM ... Straight CD – internal tangent to the specified circles. The second tangent line is constructed similarly. EF .
Figure 48
3.3 Mates using a circular arc
3.3.1 Conjugation of two straight lines with a circular arc
All tasks for filing with an arc can be reduced to two types. The fillet is carried out either by the specified radius of the fillet arc, or through a point specified on one of the fillet lines. In both cases, it is necessary to build the center of the mating arc.
Conjugation of two intersecting straight lines with an arc of a given radius R c (Figure 49, a). Since the mating arc must touch the given straight lines, its center must be removed from each straight line by an amount equal to the radius R c ... The conjugation is built like this. Draw two straight lines, parallel to the given ones and distant from them by the amount of radius R c and at the intersection of these lines mark the point O – the center of the mating arc. From point ABOUT drop the perpendicular to each of the given lines. Perpendicular bases - points A and B – are the tangency points of the mating arc. This construction of conjugation is valid for two intersecting lines that make up any angle. To mate the sides of a right angle, you can also use the method shown in Figure 49, b.
Figure 49
Conjugation of two intersecting lines, on one of which the tangency point A of the conjugating arc is specified (Figure 50). It is known that the locus of the centers of arcs joining two intersecting lines is the bisector of the angle formed by these lines. Therefore, by constructing the bisector of the angle, from the point of contact A restore the perpendicular to the straight line until it intersects with the bisector and mark the point O – the center of the mating arc. Omitting from point ABOUT perpendicular to another straight line, get the second tangency point B and the radius R c \u003d OA \u003d OB two straight lines are conjugated, on one of which a tangency point was specified.
Conjugation of two parallel straight lines by an arc passing through a given tangency point A (Figure 51). From point A restore the perpendicular to the given straight lines and mark a point at its intersection with the second straight line B ... Section AB halve and get a point ABOUT - the center of the mating arc with the radius.
Picture 50 Picture 51
Lesson objectives
- Educational - repetition, generalization and testing of knowledge on the topic: "Tangent to a circle"; development of basic skills.
- Developing - to develop the attention of students, perseverance, perseverance, logical thinking, mathematical speech.
- Educational - through the lesson to bring up an attentive attitude to each other, instill the ability to listen to comrades, mutual assistance, independence.
- Introduce the concept of a tangent, a point of tangency.
- Consider the property of a tangent and its sign and show their application in solving problems in nature and technology.
Lesson Objectives
- Build skills in building tangents using a scale ruler, protractor and drawing triangle.
- Test the ability of students to solve problems.
- Provide mastery of the basic algorithmic techniques for constructing a tangent to a circle.
- Develop the ability to apply theoretical knowledge to problem solving.
- Develop the thinking and speech of students.
- Work on the formation of the skills to observe, notice patterns, generalize, and reason by analogy.
- Instilling interest in mathematics.
Lesson plan
- The emergence of the concept of tangent.
- The history of the appearance of the tangent.
- Geometric definitions.
- Basic theorems.
- Draws a tangent to a circle.
- Anchoring.
The emergence of the concept of tangent
The concept of a tangent line 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 a ruler, were able to draw tangents to a circle, and later to conical sections: ellipses, hyperbolas and parabolas.
Tangent history
Interest in tangents revived in modern times. Then curves were discovered that were unknown to ancient scientists. For example, Galileo introduced a 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 line 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 line 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 of its points (as well as the length of this segment) is called radiuscircles.
- The part of the plane bounded by a circle is called around.
- A segment connecting two points of a circle is called it chord... The chord passing through the center of the circle is called diameter.
- Any two non-coincident points of the circle divide it into two parts. Each of these parts is called arccircles. The measure of the arc can be the measure of the corresponding central angle. An arc is called a semicircle if the segment connecting its ends is a diameter.
- A straight line that has exactly one common point with a circle is called tangentto the circle, and their common point is called the tangent point of the line and the circle.
- A straight line passing through two points of a circle is called secant.
- The central angle in a circle is called a flat angle with a vertex at its center.
- An angle whose vertex lies on a circle and the sides intersect this circle is called inscribed angle.
- Two circles with 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 the circle is called a straight line that has one point in common with a circle.
A straight line passing through a point of a circle in the same plane perpendicular to the radius drawn to this point, called tangent... In this case, this point of the circle is called the tangency point.
Where in our case "a" is a straight line which is tangent to a given circle, point "A" is a point of tangency. In this case, a⊥OA (straight line a is perpendicular to the radius OA).
They say that two circles touchif they have a single point in common. This point is called point of tangency of the circles... Through the point of tangency, you can draw a tangent to one of the circles, which is at the same time tangent to the other circle. The touch of the circles is internal and external.
The tangency is called internal if the centers of the circles lie on one side of the tangent.
The tangent is called external if the centers of the circles lie on opposite sides of the tangent
a - common tangent to two circles, K - tangent point.
Basic theorems
Theoremabout 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 \u003d 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.
Evidence.
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 straight line. Suppose that OA is not perpendicular to the tangent, but there is a straight line OS perpendicular to the tangent. The length of the OS includes the length of the radius and also some segment of the BC, which is certainly greater than the radius. Thus, one can prove for any straight line. We conclude that the radius, the radius drawn to the point of tangency, is the shortest distance to the tangent from point O, i.e. OS is perpendicular to the tangent line. 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 this line have one more common point B with a circle. Triangle AOB is rectangular and its two sides are equal as radii, which cannot be. Thus, we find that this line has no more common points with the circle except 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.
Evidence.
The proof is very simple. Using the previous theorem, we claim that OB is perpendicular to AB, and OS is AC. The right-angled triangles ABO and ASO are equal in leg and hypotenuse (OB \u003d OS - radii, AO - common). Therefore, their legs AB \u003d AC and the angles OAC and OAB are also equal.
Theorem. The value of the angle formed by the tangent and the chord having a common point on the circle is equal to half the angular value of the arc between its sides.
Evidence.
Consider the angle NAB formed by the tangent and the chord. Let's draw the diameter of the speaker. The tangent line is perpendicular to the diameter drawn to the point of tangency, therefore, ∠CAN \u003d 90 о. Knowing the theorem, we see that the angle alpha (a) is equal to half and half the angular value of the BC arc or half of the BOC angle. ∠NAB \u003d 90 о -a, hence we obtain ∠NAB \u003d 1/2 (180 о -∠BOC) \u003d 1 / 2∠АВ or \u003d half the angular value of the arc BA. ch.d.
Theorem. If a tangent and a secant are drawn from a point to the circle, then the square of the segment of the tangent from this point to the point of tangency is equal to the product of the lengths of the secant segments from this point to the points of its intersection with the circle.
Evidence.
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 value of the arc AC, but also the angle ABC is equal to half the angular value 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 BMA have a common angle at the vertex M, we state the similarity of these triangles in two angles (the second feature). From the similarity we have: MA / MB \u003d MC / MA, whence we get MA 2 \u003d MV * MS
Drawing tangents to a circle
Now let's try to figure it out and find out what needs to be done to build a tangent to the 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, which is bounded by this circle. In this case, it is not possible to construct a tangent line through this circle.
In the second case, the point is on a circle, and we can build a tangent line by drawing a perpendicular line to the radius, which is drawn to a known point.
Third, let’s assume that the point is outside the circle, which is bounded by a circle. In this case, before drawing the tangent, you need 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 should be equal to the radius and the segment that lies on the circle on the opposite side.
Here we see that the point on the circle is the midpoint of the segment, which 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, centered at the ends of the line, which is equal to twice the radius. Now we can draw a straight line through any intersection point of these circles and a given point. Such a straight line is the median perpendicular to the radius of the circle that was drawn at the beginning. Thus, we can see that this line is perpendicular to the circle and from this it follows that it is tangent to the circle.
In the third version, we have a point lying outside the aisles of a circle, which is bounded by a circle. In this case, we first draw a line segment that connects the center of the provided circle and the specified point. And then we find the middle. But for this it is necessary to build a middle perpendicular. And you already know how to build it. Then we need to draw a circle or at least a 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 problem statement. And finally, you can draw a tangent line through the two points you know.
And finally, 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 formed corner rests on a semicircle. And from this it follows that this angle is straight. Therefore, the radius will be perpendicular to the newly constructed line, and this line is the tangent line.
Constructing a tangent line.
The construction of tangents is one of the problems that led to the birth of differential calculus. The first published work related to differential calculus and penned by Leibniz was called "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 you do not take into account the very modest contribution of the ancient inhabitants of the valley between the Tigris and the 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 overflowed. Water covered tracts of cultivated land, and for tax purposes it was necessary to establish how much land was lost. The surveyors used a tightly stretched rope as a measuring instrument. Another stimulus for the accumulation of geometric knowledge by the Egyptians was such types of their activities as the construction of pyramids and the visual 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 rewritten by a student between 1800 and 1600. BC. from an older text. The papyrus was found by the Russian Egyptologist Vladimir Semenovich Golenishchev. It is kept in Moscow - at the 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: “Instructions on how to attain knowledge of all dark things, all the secrets that things hide in themselves ... According to the old monuments, the scribe Ahmes wrote this.” and bought this papyrus in Egypt. In the Ahmes papyrus, 84 problems are given for various calculations that may be needed in practice.
Secants, 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?
The essence
The term "tangent to a circle" is probably familiar to everyone. But hardly everyone will be able to quickly formulate its definition. Meanwhile, a tangent line is called a straight line lying in the same plane with a circle, which intersects it only at one point. There can 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 each case, it is one, but if there are more of them, then it will already be a secant.
History of discovery and study
The concept of a tangent line dates back to ancient times. 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 at the initial stages of the development of geometry. Of course, history did not preserve the name of the discoverer, but it is obvious that even at that time people were quite aware of the properties of the tangent to a circle.
In modern times, interest in this phenomenon flared up again - a new round of studying this concept began in combination 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 the circles, it seems that there were no secrets left for the ancients in this area.
Properties
The radius drawn to the intersection point will be This
the main, but not the only property that the tangent to the circle has. Another important feature already includes two straight lines. So, through one point lying outside the circle, you can draw two tangents, while their segments will be equal. There is one more theorem on this topic, but it is rarely passed within the framework of a 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 - intersection of lines, B - point of tangency, C and D - intersections. In this case, the following equality will be true: the length of the tangent to the circle, squared, will be equal to the product of the segments AC and AD.
There is an important consequence from the above. For each point of the circle, you can draw a tangent, but only one. The proof of this is quite simple: theoretically, dropping the perpendicular from the radius onto it, we find out that the formed triangle cannot exist. And this means that the tangent is the only one.
Building
Among other problems in geometry, there is a special category, usually not
loved by pupils and students. To solve tasks from this category, you only need a compass and a ruler. These are building tasks. They also exist for the construction of a tangent line.
So, given a circle and a point lying outside its boundaries. And you need to draw a tangent through them. How can this be done? First of all, you need to draw a segment between the center of the circle O and a given point. Then, using a compass, you should divide it in half. To do this, you need to set a radius - just over half the distance between the center of the original circle and this point. Then 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 need to be connected, which will split the line in half. Set the radius on the compass equal to this distance. Then, with the center at the intersection point, build 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 points of tangency for the originally specified 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. It 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. It is from this that its name comes from. Translated from Latin tangens - "tangent". Thus, this concept is associated not only with geometry and differential calculus, but also with trigonometry.
Two circles
The tangent does not always affect only one figure. If a huge set of straight lines can be drawn to one circle, then why not the other way around? 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 we are talking about two circles and one or more straight lines, even if it is known that these are tangents, it is not immediately clear how all these figures are located in relation to each other. Based on this, several varieties are distinguished. 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 two varieties are distinguished. If one circle is, as it were, embedded in the second, then the touch is called internal, if not, then external. It is possible to understand the relative position of 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 values \u200b\u200bare equal, then the circles touch. If the first is more, they intersect, and if it is less, then they have no common points.
It's the same with straight lines. For any two circles that have no common points, you can
build four tangents. Two of them will intersect between the shapes, 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 fact 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 will not cause difficulty.
If the figures have two intersection points, then a straight line can be constructed for them, tangent to the circle of both one and the second, but only the outer one. The solution to this problem is similar to what will be discussed below.
Solving problems
Both the internal and external tangent 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, you need to build an auxiliary circle near the center of the larger circle. In this case, the difference between the radii of the two original figures must be established on the compass. Tangents to the auxiliary circle are constructed 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 line, the required points on both circles are found. The problem is solved, at least its first part.
In order to build internal tangents, you will have to solve practically
a similar task. You will need a secondary shape again, but this time its radius will be equal to the sum of the original ones. Tangents are drawn to it from the center of one of these 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 entrust calculations to 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, building tasks will cause more and more difficulties for students.
As for finding common tangents for a large number of circles, this is not always possible, even if they lie in the same plane. But in some cases you can find such a straight line.
Examples from life
A common tangent line to two circles is often found in practice, although this is not always noticeable. Conveyors, block systems, pulley transfer belts, thread tension in a 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: they find practical application in engineering, physics, construction and many other fields.
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Geometric constructions
Drawing tangents to circles
Consider the problem underlying the solution of other problems of drawing tangent lines to circles.
Let from the pointAND (Fig. 1) it is necessary to draw tangents to the circle centered at the pointABOUT.
For accurate construction of tangents, it is necessary to determine the points of tangency of the lines to the circle. For this pointAND should be connected with a stitchABOUT and split the segmentOA in half. From the middle of this segment - pointsFROM, as from the center, describe a circle, the diameter of which should be equal to the segmentOA... PointsTO1 andTO2 intersection of circles centered at a pointFROM and centered at the pointABOUT are the points of tangency of the linesAK1 andAK2 to a given circle.
The correctness of the solution to the problem is confirmed by the fact that the radius of the circle drawn to the point of tangency is perpendicular to the tangent to the circle. CornersOK1 AND andOK2 AND are straight because they rely on the diameterJSC circle centered at a pointFROM.
Figure: 1.
When constructing tangents to two circles, tangents are distinguishedinternal andexternal... If the centers of the given circles are located on one side of the tangent, then it is considered external, and if the centers of the circles are on opposite sides of the tangent, they are internal.
ABOUT1 andABOUT2 R1 andR2 ... It is required to draw external tangents to the given circles.
For an accurate construction, it is necessary to determine the points of tangency of lines and given circles. If the radii of the circles with centersABOUT1 andABOUT2 start successively decreasing by the same value, then you can get a series of concentric circles of smaller diameters. In this case, in each case of decreasing the radius, the tangents to the smaller circles will be parallel to the desired one. After reducing both radii to the size of the smaller radiusR2 circle with centerABOUT2 will turn to a point, and the circle with the centerABOUT1 will transform into a concentric circle with a radiusR3 equal to the difference between the radiiR1 andR2 .
Using the method described earlier, from the pointABOUT2 draw the outer tangents to the circle with the radiusR3 , connect the dotsABOUT1 andABOUT2 , divided by a dotFROM sectionABOUT1 ABOUT2 in half and draw with a radiusCO1 an arc whose intersection with a given circle defines the tangency points of the linesABOUT2 TO1 andABOUT2 TO2 .
PointAND1 andAND2 tangency of the sought lines with the larger circle is located on the continuation of the linesABOUT1 TO1 andABOUT1 TO2 ... PointsIN1 andIN2 tangent lines with a smaller circle are on perpendiculars with a baseABOUT2 respectively to auxiliary tangentsABOUT2 TO1 andABOUT2 TO2 ... With the points of contact, you can draw the desired straight linesAND1 IN1 andAND2 IN2 .
Figure: 2.
Let two circles be given with centers at the pointsABOUT1 andABOUT2 (Fig. 2) having radii respectivelyR1 andR2 ... It is required to draw inner tangents to the given circles.
To determine the points of tangency of lines with circles, we use reasoning similar to those given in solving the previous problem. If you decrease the radiusR2 to zero, then the circle with the centerABOUT2 turn to the point. However, in this case, to preserve the parallelism of the auxiliary tangents with the required radiusR1 should be increased by sizeR2 and draw a circle with a radiusR3 equal to the sum of the radiiR1 andR2 .
From pointABOUT2 draw tangents to a circle with a radiusR3 , for which we connect the pointsABOUT1 andABOUT2 , divided by a dotFROM sectionABOUT1 ABOUT2 in half and draw an arc of a circle centered at the pointFROM and radiusCO1 ... Intersect an arc with a circle with a radiusR3 will determine the position of the pointsTO1 andTO2 tangency of auxiliary linesABOUT2 TO1 andABOUT2 TO2 .
PointAND1 andAND2 R1 is at the intersection of this circle with the segmentABOUT1 TO1 andABOUT1 TO2 ... To define pointsIN 1 andAT 2 tangency of the sought lines with a circle of radiusR2 follows from the pointО2 restore perpendiculars to auxiliary linesO2K1 andO2K2 before crossing the specified circle. Having the points of tangency of the desired straight lines and given circles, we draw straight linesA1B1 andA2B2.
Figure: 3.