Seconds, 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 each case, it is one, but if there are more, then it will be a secant.
History of discovery and study
The concept of a 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, butit is obvious that even at that time, people were quite aware of the properties of the tangent to the 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 the 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 perpendicular to the line. This is
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 covered in 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 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.
From the above there is an important consequence. 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 formedtriangle cannot exist. And this means that the tangent is the only one.
Building
Among other problems in geometry, there is a special category, as a rule, not
loved 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 a 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.
Interesting
It was the construction of tangents to the circle that led to the birth of
differential calculus. The first work on this topic waspublished 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.
Besides, 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
Not always a tangent affects only one shape. 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 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. understand mutu althe location of the figures is possible 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 one is larger, they intersect, and if it is smaller, then they do not have common points.
The same with straight lines. For any two circles that do not have common points, you can
construct 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 fact is that for any mutual arrangement 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 easy to construct, 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 largercircles need to be built auxiliary. 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. Problem solved, at least the first part of it.
In order to construct internal tangents, you will have to solve practically
a similar task. Again, you will need an auxiliary figure, 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 the 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, construction tasks will cause more and more difficulties for students.
As for finding common tangents for more circles, it's not always possible, even if they lie in the same plane. But in some cases you can find such a straight line.
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 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 applications in engineering, physics, construction and many other areas.
