Secant, tangent - 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 his definition. Meanwhile, a tangent is a line lying on the same plane with a circle that intersects it at only one point. There may be a great many 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 line intersect. In each case, it is one, but if there are more, then it will already be secant.
History of discovery and study
The concept of tangent appeared in antiquity. The construction of these lines first to the circle, and then to the ellipses, parabolas and hyperbolas using a ruler and 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 completely aware of the properties of a tangent to a circle.
In modern times, interest in this phenomenon flared up again - a new round of study of this concept began, combined with the discovery of new curves. So, Galileo introduced the concept of cycloid, and Fermat and Descartes built a tangent to it. As for the circles, it seems that for the ancients there were no secrets left in this area.
The properties
The radius drawn to the intersection will be perpendicular to the line. it
the main, but not the only property, which has a tangent to a circle. Another important feature includes two straight lines. So, through one point lying outside the circle, two tangents can be drawn, and their segments will be equal. There is one more theorem on this topic, however, it is rarely passed within the framework of a standard school course, although it is extremely convenient for solving some problems. It sounds as follows. From one point located outside the circle, a tangent and secant to it are drawn. The segments AB, AC and AD are formed. A is the intersection of lines, B is the point of tangency, C and D are the 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.
From the foregoing, there is an important consequence. For each point of the circle, one can construct 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 geometry problems, there is a special category, as a rule, not
enjoyed by pupils and students. To solve tasks from this category, you need only a compass and a ruler. These are building tasks. There are also on the construction of a tangent.
So, given a circle and a point lying outside its borders. And you need to draw a tangent through them. How to do this? First of all, you need to draw a line between the center of the circle O and a given point. Then, with the help of a compass, it should be divided in half. To do this, you must specify a radius - a little more than half the distance between the center of the original circle and this 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. Define a radius equal to this distance on the compass. 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 tangent 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 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.
In addition, a tangent to a circle is associated with 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
Not always a tangent will 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 can not go 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 they 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 at all. In the first case, they will intersect, and in the second, touch. And here there are two varieties. If one circle is enclosed in the second, then the contact is called internal, if not, then external. You can understand the relative position of the figures not only on the basis of 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 are tangent. If the first is greater, they intersect, and if less, then they do not have common points.
So it is with straight lines. For any two circles that do not have common points, we 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 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 intersection points, then for them a straight line can be constructed, 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 later.
Problem solving
Both the internal and the 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, two circles with different radii and centers O1 and O2 are given. 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. In this case, the difference between the radii of the two original figures should be established on the compass. From the center of the smaller circle, tangents to the auxiliary are constructed. After that, perpendiculars to these straight lines are drawn from O1 and O2 to the intersection 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 build the internal tangents, you have to solve practically
a similar task. Again, an auxiliary figure will be needed, but this time its radius will be equal to the sum of the original ones. The tangents from the center of one of these circles are constructed for it. The further course of the solution can be understood from the previous example.
A 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 computing to special programs. But do not think that now it is not necessary to be able to do it yourself, because for the correct formulation of the task for the computer you need to do a lot and understand. 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, you can find such a straight line.
Life examples
A common tangent to two circles is often found in practice, although this is not always noticeable. Conveyors, block systems, transmission belts of pulleys, thread tension in a sewing machine, and even just a bicycle chain are all 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.