Straight line on a plane - necessary information.
In this article we will dwell in detail on one of the primary concepts of geometry - the concept of a straight line on a plane. First, let's define the basic terms and designations. Next, we will discuss the relative position of a line and a point, as well as two lines on a plane, and present the necessary axioms. In conclusion, we will consider ways to define a straight line on a plane and provide graphic illustrations.
Page navigation.
- A straight line on a plane is a concept.
- The relative position of a straight line and a point.
- The relative position of lines on a plane.
- Methods for defining a straight line on a plane.
A straight line on a plane is a concept.
Before giving the concept of a straight line on a plane, you should clearly understand what a plane is. Concept of a plane allows you to get, for example, a flat surface on a table or a wall at home. It should, however, be borne in mind that the dimensions of the table are limited, and the plane extends beyond these boundaries to infinity (as if we had an arbitrarily large table).
If we take a well-sharpened pencil and touch its tip to the surface of the “table”, we will get an image of a point. This is how we get representation of a point on a plane.
Now you can move on to the concept of a straight line on a plane.
Place a sheet of clean paper on the table surface (on a plane). In order to draw a straight line, we need to take a ruler and draw a line with a pencil as far as the size of the ruler and sheet of paper we are using allows us to do. It should be noted that in this way we will only get part of the line. We can only imagine an entire straight line extending into infinity.
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The relative position of a straight line and a point.
We should start with the axiom: on every straight line and in every plane there are points.
Points are usually denoted in capital Latin letters, for example, points A And F. In turn, straight lines are denoted in small Latin letters, for example, straight lines a And d.
Possible two options for the relative position of a line and a point on a plane: either the point lies on the line (in this case it is also said that the line passes through the point), or the point does not lie on the line (it is also said that the point does not belong to the line or the line does not pass through the point).
To indicate that a point belongs to a certain line, the symbol “ ” is used. For example, if the point A lies on a straight line A, then we can write . If the point A does not belong to the line A, then write it down.
The following statement is true: there is only one straight line passing through any two points.
This statement is an axiom and should be accepted as a fact. In addition, this is quite obvious: we mark two points on paper, apply a ruler to them and draw a straight line. A straight line passing through two given points (for example, through points A And IN), can be denoted by these two letters (in our case, the straight line AB or VA).
It should be understood that on a straight line defined on a plane there are infinitely many different points, and all these points lie in the same plane. This statement is established by the axiom: if two points of a line lie in a certain plane, then all points of this line lie in this plane.
The set of all points located between two points given on a line, together with these points, is called straight line segment or simply segment. The points limiting the segment are called the ends of the segment. A segment is denoted by two letters corresponding to the endpoints of the segment. For example, let the points A And IN are the ends of a segment, then this segment can be denoted AB or VA. Please note that this designation for a segment coincides with the designation for a straight line. To avoid confusion, we recommend adding the word “segment” or “straight” to the designation.
To briefly record whether a certain point belongs or does not belong to a certain segment, the same symbols and are used. To show that a certain segment lies or does not lie on a line, use the symbols and, respectively. For example, if the segment AB belongs to the line A, can be briefly written down.
We should also dwell on the case when three different points belong to the same line. In this case, one, and only one point, lies between the other two. This statement is another axiom. Let the points A, IN And WITH lie on the same straight line, and the point IN lies between the points A And WITH. Then we can say that the points A And WITH are on opposite sides of the point IN. It can also be said that the points IN And WITH the points lie on one side A, and the points A And IN lie on one side of the point WITH.
To complete the picture, we note that any point on a line divides this line into two parts - two beam. For this case the axiom is given: arbitrary point ABOUT, belonging to a line, divides this line into two rays, and any two points of one ray lie on the same side of the point ABOUT, and any two points of different rays are on opposite sides of the point ABOUT.
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This publication will help you systematize previously acquired knowledge, as well as prepare for an exam or test and pass it successfully.
2. The condition for three points to be on the same line. Equation of a straight line. The relative position of the points and the line. Bunch of straight lines. Distance from point to line
1. Let three points be given A 1 (X 1 , at 1), A 2 (X 2 , at 2), A 3 (X 3 , at 3), then condition for them to be on the same line:
either ( X 2 – X 1) (at 3 – at 1) – (X 3 – x 1) (at 2 – at 1) = 0.
2. Let two points be given A 1 (X 1 , at 1), A 2 (X 2 , at 2), then alignment of a line passing through these two points:
(X 2 – X 1)(y - y 1) – (x – x 1)(at 2 – at 1) = 0 or ( x – x 1) / (X 2 – X 1) = (y - y 1) / (at 2 – at 1).
3. Let there be a point M (X 1 , at 1) and some straight line L, represented by the equation at = Oh + With. Equation of a line running parallel to a given line L through this point M:
y - y 1 = A(x – x 1).
If straight L given by the equation Oh + Wu + WITH M, is described by the equation A(x – x 1) + IN(y - y 1) = 0.
Equation of a line passing perpendicular to a given line L through this point M:
y - y 1 = –(x – x 1) / A
A(y - y 1) = X 1 – X.
If straight L given by the equation Oh + Wu + WITH= 0, then a line parallel to it passing through the point M(X 1 , at 1), described by the equation A (y - y 1) – IN(x – x 1) = 0.
4. Let two points be given A 1 (X 1 , at 1), A 2 (X 2 , at 2) and the straight line given by the equation Oh + Wu + C = 0. The relative position of the points relative to this line:
1) points A 1 , A 2 lie on one side of a given line if the expressions ( Oh 1 + Wu 1 + WITH) And ( Oh 2 + Wu 2 + WITH) have the same signs;
2) points A 1 ,A 2 lie on opposite sides of a given line if the expressions ( Oh 1 + Wu 1 + WITH) And ( Oh 2 + Wu 2 + WITH) have different signs;
3) one or both points A 1 , A 2 lie on a given line if one or both expressions, respectively ( Oh 1 + + Wu 1 + WITH) And ( Oh 2 + Wu 2 + WITH) take zero value.
5. Central beam is a set of lines passing through one point M (X 1 , at 1), called beam center. Each of the straight lines of the pencil is described by the pencil equation y - y 1 = To(x – x 1) (beam parameter To each line has its own).
All straight lines of the beam can be represented by the equation: l(y–y 1) = m(x – x 1), where l, m– arbitrary numbers that are not equal to zero at the same time.
If two straight beams L 1 and L 2 respectively have the form ( A 1 X + IN 1 at+ WITH 1) = 0 and ( A 2 X+ IN 2 at+ WITH 2) = 0, then the beam equation: m 1 (A 1 X + IN 1 at + WITH 1) + m 2 (A 2 X + IN 2 at + WITH 2) = 0. If straight L 1 and L 2 intersecting, then the beam is central; if the lines are parallel, then the beam is parallel.
6. Let the points be given M(X 1 ,at 1) and the straight line given by the equation Ax + Wu + C = 0. Distance dfrom this points M to a straight line:
- 1. Basic concepts. Coordinate systems. Straight lines and their relative positions
- 2. The condition for three points to be on the same line. Equation of a straight line. The relative position of the points and the line. Bundle of straight lines. Distance from point to line
The article talks about the concept of a straight line on a plane. Let's look at the basic terms and their designations. Let's work with the relative position of a line and a point and two lines on a plane. Let's talk about axioms. Finally, we will discuss methods and methods for defining a straight line on a plane.
Straight line on a plane - concept
First you need to have a clear understanding of what a plane is. Any surface of something can be classified as a plane, only it differs from objects in its boundlessness. If we imagine that the plane is a table, then in our case it will not have boundaries, but will be infinitely huge.
If you touch the table with a pencil, a mark will remain, which can be called a “dot”. Thus, we get an idea of a point on the plane.
Let's consider the concept of a straight line on a plane. If you draw a straight line on a sheet, it will appear on it with a limited length. We did not get the entire straight line, but only part of it, since in fact it does not have an end, just like a plane. Therefore, the depiction of lines and planes in the notebook is formal.
We have an axiom:
Definition 1
Points can be marked on each straight line and in each plane.
Points are designated in both large and small Latin letters. For example, A and D or a and d.
For a point and a line, only two possible locations are known: a point on a line, in other words, that the line passes through it, or a point not on a line, that is, the line does not pass through it.
To indicate whether a point belongs to a plane or a point to a line, use the sign “∈”. If the condition is given that the point A lies on the line a, then it has the following form of writing A ∈ a. In the case when point A does not belong, then another entry A ∉ a.
Fair judgment:
Definition 2
Through any two points located in any plane, there is a single straight line that passes through them.
This statement is considered an akisoma, and therefore does not require proof. If you consider this yourself, you can see that with two existing points there is only one option for connecting them. If we have two given points A and B, then the line passing through them can be called by these letters, for example, line A B. Consider the figure below.
A straight line located on a plane has a large number of points. This is where the axiom comes from:
Definition 3
If two points of a line lie in a plane, then all other points of this line belong to the plane.
The set of points located between two given points is called a straight segment. It has a beginning and an end. A two-letter designation has been introduced.
If it is given that points A and P are the ends of a segment, then its designation will take the form P A or A P. Since the designations of a segment and a line coincide, it is recommended to add or finish the words “segment”, “straight line”.
A shorthand notation for membership involves the use of the signs ∈ and ∉. In order to fix the location of a segment relative to a given line, use ⊂. If the condition states that the segment A P belongs to the line b, then the entry will look like this: A P ⊂ b.
The case where three points simultaneously belong to one line occurs. This is true when one point lies between two others. This statement is considered to be an axiom. If points A, B, C are given, which belong to the same line, and point B lies between A and C, it follows that all given points lie on the same line, since they lie on both sides of point B.
A point divides a line into two parts, called rays. We have an axiom:
Definition 4
Any point O located on a straight line divides it into two rays, with any two points of one ray lying on one side of the ray relative to point O, and others on the other side of the ray.
The arrangement of straight lines on a plane can take the form of two states.
Definition 5
coincide.
This opportunity arises when straight lines have common points. Based on the axiom written above, we have that a straight line passes through two points and only one. This means that when 2 straight lines pass through given 2 points, they coincide.
Definition 6
Two straight lines on a plane can cross.
This case shows that there is one common point, which is called the intersection of lines. The intersection is designated by the sign ∩. If there is a notation form a ∩ b = M, then it follows that the given lines a and b intersect at the point M.
When straight lines intersect, we deal with the resulting angle. The section where straight lines intersect on a plane to form an angle of 90 degrees, that is, a right angle, is subject to separate consideration. Then the lines are called perpendicular. The form of writing two perpendicular lines is as follows: a ⊥ b, which means that line a is perpendicular to line b.
Definition 7
Two straight lines on a plane can be parallel.
Only if two given lines have no common intersections, and therefore no points, are they parallel. A notation is used that can be written for a given parallelism of lines a and b: a ∥ b.
A straight line on a plane is considered together with vectors. Particular importance is attached to zero vectors that lie on a given line or on any of the parallel lines; they are called direction vectors of a line. Consider the figure below.
Non-zero vectors located on lines perpendicular to a given one are otherwise called normal line vectors. There is a detailed description in the article of the normal vector of a line on a plane. Consider the figure below.
If there are 3 lines on a plane, their location can be very different. There are several options for their location: the intersection of all, parallelism, or the presence of different intersection points. The figure shows the perpendicular intersection of two lines relative to one.
To do this, we present the necessary factors that prove their relative position:
- if two lines are parallel to a third, then they are all parallel;
- if two lines are perpendicular to a third, then these two lines are parallel;
- If on a plane a straight line intersects one parallel line, then it will also intersect another.
Let's look at this in the pictures.
A straight line on a plane can be specified in several ways. It all depends on the conditions of the problem and on what its solution will be based. This knowledge can help for the practical arrangement of straight lines.
Definition 8
The straight line is defined using the specified two points located in the plane.
From the considered axiom it follows that through two points it is possible to draw a straight line and, moreover, only one single one. When a rectangular coordinate system specifies the coordinates of two divergent points, then it is possible to fix the equation of a straight line passing through the two given points. Consider a drawing where we have a line passing through two points. 
Definition 9
A straight line can be defined through a point and a line to which it is parallel.
This method exists because through a point it is possible to draw a straight line parallel to a given one, and only one. The proof is already known from a school course in geometry.
If a line is given relative to a Cartesian coordinate system, then it is possible to construct an equation for a line passing through a given point parallel to a given line. Let's consider the principle of defining a straight line on a plane.
Definition 10
The straight line is specified through the specified point and the direction vector.
When a straight line is specified in a rectangular coordinate system, it is possible to compose canonical and parametric equations on the plane. Let us consider in the figure the location of the straight line in the presence of a direction vector.
The fourth point in specifying a straight line makes sense when the point through which it should be drawn and the straight line perpendicular to it are indicated. From the axiom we have:
Definition 11
Through a given point located on a plane, only one straight line will pass, perpendicular to the given one.
And the last point related to specifying a line on a plane is given the specified point through which the line passes, and in the presence of a normal vector of the line. Given the known coordinates of a point located on a given line and the coordinates of the normal vector, it is possible to write down the general equation of the line.
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On straight line a (Fig. 7, o) points A, B and C are taken. Point B lies between points A and C. We can also say that points A and C lie on opposite sides of point B. Points A and B lie along one side from point C, they are not separated by point C. Points B and C lie on the same side of point A.A segment is a part of a line that consists of all points of this line lying between two given points. These points are called the ends of the segment. A segment is indicated by indicating its ends.
In Figure 7, b, segment AB is part of straight line a. Point M lies between points A and B, and therefore belongs to the segment AB; Point K does not lie between points A and B, and therefore does not belong to segment AB.
The axiom (main property) of the location of points on a straight line is formulated as follows:
Of the three points on a line, one and only one lies between the other two.
The following axiom expresses the basic property of measuring segments.
Each segment has a certain length greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its points.
This means that if we take any point C on the segment MK, then the length of the segment MK is equal to the sum of the lengths of the segments MS and SK (Fig. 7, c).
The length of the segment MK is also called the distance between points M and K.
Example 1. Three points O, P and M are given on a line. It is known that . Does point P lie between O and M? Can point B belong to the segment PM if ? Explain the answer.
Solution. Point P lies between points O and M, if Let's check whether this condition is met: . Conclusion: point P lies between points O and M.
Point B belongs to the segment RM if it lies between points P and M, i.e. Let's check: , and by condition . Conclusion: point B does not belong to the segment PM.
Example 2. Is it possible to arrange 6, 7 and 8 segments on a plane so that each of them intersects exactly three others?
Solution. 6 segments can be arranged this way (Fig. 8, o). 8 segments can also be arranged in this way (Fig. 8, b). 7 segments cannot be arranged like this.
Let us prove the last statement. Let us assume that such an arrangement of seven segments is possible. Let's number the segments and make such a table in a cell at the intersection of a row and a column, put “+” if the segment intersects with the j-th one, and “-” if it does not intersect. If then we also put Let’s count in two ways how many characters are in the table.
On the one hand, there are 3 of them in each line, so the total number of characters is . On the other hand, the table is filled symmetrically about the diagonal:
if in cell C: j) stands then in cell too. This means that the total number of characters must be even. We got a contradiction.
Here we used the proof by contradiction.
5. Beam.
A half-line or ray is a part of a line that consists of all points of this line lying on one side of a given point. This point is called the starting point of the half-line or the beginning of the ray. Different half-lines of the same line with a common starting point are called complementary.
Half-straight lines are designated by lowercase Latin letters. You can denote a half-line by two letters: an initial letter and some other letter corresponding to a point belonging to the half-line. In this case, the starting point is placed in the first place. For example, in Figure 9, a, rays AB and AC are shown, which are additional; in Figure 9, b, rays MA, MB and ray c are shown.
The following axiom reflects the main property of laying down segments.
On any half-line from its starting point, you can plot a segment of a given length, and only one.
Example. Given two points A and B. How many lines can be drawn through points A and B? How many rays are there on the straight line AB starting at point A and point B? Mark two points on line A B that are different from A and B. Do they belong to segment AB?
Solution. 1) According to the axiom, it is always possible to draw a straight line through points A and B, and only one.
2) On the line AB with the beginning at point A, there are two rays, which are called additional. Likewise for point B.
3) The answer depends on the location of the marked points. Let's consider possible cases (Fig. 10). It is clear that in case a) the points belong to the segment AB; in cases b), c) one point
belongs to the segment, but the other does not; in cases d) and e) points M and N do not belong to the segment AB.
6. Circle. Circle.
A circle is a figure that consists of all points of the plane located at a given distance from a given point. This point is called the center of the circle.
The distance from the points of a circle to its center is called the radius of the circle. A radius is also called any segment connecting a point on a circle to its center.
A segment connecting two points on a circle is called a chord. The chord passing through the center is called the diameter.
Figure 11, a shows a circle with a center at point O. Segment OA is the radius of this circle, BD is the chord of the circle, CM is the diameter of the circle.
A circle is a figure that consists of all points of the plane located at a distance not greater than a given one from a given point. This point is called the center of the circle, and this distance is called the radius of the circle. The boundary of a circle is a circle with the same center and radius (Fig. 11, b).
Example. What is the greatest number of different parts that have no common points except their boundaries that a plane can be divided into: a) a straight line and a circle; b) two circles; c) three circles?
Solution. Let us depict in the figure the cases of mutual arrangement of figures corresponding to the condition. Let's write down the answer: a) four parts (Fig. 12, o); b) four parts (Fig. 12, b); c) eight parts (Fig. 12, c).
7. Half-plane.
Let us formulate another axiom of geometry.
A straight line divides a plane into two half-planes.
In Figure 13, line a divides the plane into two half-planes so that every point of the plane that does not belong to line o lies in one of them. This partition has the following property: if the ends of any segment belong to the same half-plane, then the segment does not intersect with a line; if the ends of a segment belong to different half-planes, then the segment intersects the line. In Figure 13, the points lie in one of the half-planes into which Line a divides the plane. Therefore, segment AB does not intersect line a. Points C and D lie in different half-planes. Therefore, segment CD intersects line a.
8. Angle. Degree measure of angle.
An angle is a figure that consists of a point - the vertex of the angle and two different half-lines emanating from this point - the sides of the angle (Fig. 14). If the sides of an angle are complementary half-lines, then the angle is called a developed angle.
An angle is indicated either by indicating its vertex, or by indicating its sides, or by indicating three points; vertex and two points on the sides of the angle. The word "angle" is sometimes replaced by the symbol Z.
The angle in Figure 14 can be designated in three ways:
A ray c is said to pass between the sides of an angle if it comes from its vertex and intersects some segment with ends on the sides of the angle.
In Figure 15, ray c passes between the sides of the angle since it intersects segment AB.
In the case of a straight angle, any ray emanating from its vertex and different from its sides passes between the sides of the angle.
Angles are measured in degrees. If you take a straight angle and divide it into 180 equal angles, then the degree measure of each of these angles is called a degree.
The basic properties of angle measurement are expressed in the following axiom:
Each angle has a certain degree measure greater than zero. The rotated angle is 180°. The degree measure of an angle is equal to the sum of the degree measures of the angles into which it is divided by any ray passing between its sides.
This means that if a ray c passes between the sides of an angle, then the angle is equal to the sum of the angles
The degree measure of an angle is found using a protractor.
An angle equal to 90° is called a right angle. An angle less than 90° is called an acute angle. An angle greater than 90° and less than 180° is called obtuse.
Let us formulate the main property of setting aside corners.
From any half-line, into a given half-plane, you can put an angle with a given degree measure less than 180°, and only one.
Consider the half-line a. Let us extend it beyond the starting point A. The resulting straight line divides the plane into two half-planes. Figure 16 shows how, using a protractor, to plot an angle with a given degree measure of 60° from a half-line to the upper half-plane.
If two angles from a given half-line are put into one half-plane, then the side of the smaller angle, different from the given half-line, passes between the sides of the larger angle.
Let the angles plotted from a given half-line a be one half-plane, and let the angle be less than the angle . Theorem 1. 2 states that ray b passes between the sides of the angle (ac) (Fig. 17).
The bisector of an angle is the ray that emanates from its vertex, passes between its sides and divides the angle in half. In Figure 18, the ray OM is the bisector of the angle AOB.
In geometry there is the concept of a plane angle. A plane angle is a part of a plane bounded by two different rays emanating from one point. These rays are called sides of the angle. There are two plane angles with given sides. They are called additional. In Figure 19, one of the plane angles with sides a and b is shaded.
If a plane angle is part of a half-plane, then its degree measure is the degree measure of an ordinary angle with the same sides. If a plane angle contains a half-plane, then its degree measure is 360° - a, where a is the degree measure of an additional plane angle.
Example. Ray a passes between the sides of an angle equal to 120°. Find the angles if their degree measures are in the ratio 4:2.
Solution. Ray a passes between the sides of the angle, which means, according to the main property of measuring angles (see paragraph 8)
Since degree measures have a ratio of 4:2, then
9. Adjacent and vertical angles.
Two angles are called adjacent if they have one side in common, and the other sides of these angles are complementary half-lines. In Figure 20 the angles are adjacent.
The sum of adjacent angles is 180°.
The following properties follow from Theorem 1.3:
1) if two angles are equal, then their adjacent angles are equal;
2) an angle adjacent to a right angle is a right angle;
3) an angle adjacent to an acute one is obtuse, and an angle adjacent to an obtuse one is acute.
Two angles are called vertical if the sides of one angle are complementary half-lines of the sides of the other. In Figure 21, the corners are vertical.
Vertical angles are equal.
Obviously, two intersecting lines form adjacent and vertical angles. Adjacent angles are complementary to each other up to 180°. The angular measure of the smaller of them is called the angle between the lines.
Example. In Figure 21, b, the angle is 30.° What are the angles AOC and
Solution. Angles COD and AOK are vertical, therefore, by Theorem 1.4 they are equal, i.e. Angle TYUK adjacent to angle SOD means, by Theorem 1.3
10. Central and inscribed angles.
A central angle in a circle is a plane angle with a vertex at its center. The part of the circle located inside a plane angle is called the circular arc corresponding to this central angle. The degree measure of an arc of a circle is the degree measure of the corresponding central angle.
In Figure 22, angle AOB is the central angle of a circle, its vertex O is the center of the circle, and sides OA and OB intersect the circle. Arc AB is the part of the circle located inside the central angle.
The degree measure of the arc AB in Figure 22 is equal to the degree measure of the angle AOB. The degree measure of the arc AB is designated AB.
An angle whose vertex lies on a circle and whose sides intersect this circle is called inscribed in the circle. Figure 23 shows inscribed angles.
An angle inscribed in a circle, the sides of which pass through two given points on the circle, is equal to half the angle between the radii drawn at these points, or complements this half to 180°.
When proving Theorem 1.5, it is necessary to consider three different cases, which are shown in Figure 23: one of the sides of the inscribed angle passes through the center of the circle (Figure 23, c); the center of the circle lies inside the inscribed angle (Fig. 23, b); the center of the circle lies outside the inscribed angle (Fig. 23, c).
A corollary follows from Theorem 1.5: all angles inscribed in a circle, the sides of which pass through two given points of the circle, and the vertices lie on the same side of the straight line connecting these points, are equal; inscribed angles whose sides pass through the ends of the diameter of a circle are right angles.
In Figure 24, the sides of the inscribed angle ABC pass through the ends of the diameter AC, so
Example. Points A at B and C lie on a circle with center O. Find angle AOC if
Solution. Angle ABC, inscribed in a circle, rests on arc AC, and the central angle of this circle (Fig. 25). , which means, by Theorem 1.5, and since the angle AOS is central, its degree measure is equal to the degree measure of the arc AC, i.e.
11. Parallel lines.
Two lines in a plane are called parallel if they do not intersect.
Figure 26 shows how, using a square and a ruler, draw line 6 through a given point B, parallel to a given straight line a.
To indicate the parallelism of lines, the symbol II is used. The entry reads: “Line a is parallel to line b.”
The axiom of parallelism expresses the basic property of parallel lines.
Through a point not lying on a given line, it is possible to draw on the plane at most one straight line parallel to the given one.
Two lines parallel to a third are parallel to each other.
In Figure 27, lines a and b are parallel to line c. Theorem 1.6 states that .
It can be proven that through a point that does not belong to a line, one can draw a line parallel to the given one. In Figure 28, through point A, which does not belong to b, a line a is drawn parallel to line b.
Comparing this statement and the axiom of parallels, we come to an important conclusion: on a plane, through a point not lying on a given line, you can draw a line parallel to it, and only one.
The axiom of parallelism in Euclid’s book “Elements” was called the “fifth postulate.” Ancient geometers tried to prove the uniqueness of the parallel. These unsuccessful attempts continued for more than 2000 years, until the 19th century.
The great Russian mathematician N. I. Lobachevsky and, independently of him, the Hungarian mathematician J. Bolyai showed that, by accepting the assumption that it is possible to draw through a point several straight lines parallel to a given one, it is possible to construct another, equally “correct” non-Euclidean geometry. This is how Lobachevsky's geometry was born.
An example of a theorem that uses the concept of parallelism, and its proof is based on the axiom of parallels, is Thales' theorem. Thales of Miletus - ancient Greek mathematician who lived in 625-547. BC e.
If parallel lines intersecting the sides of an angle cut off equal segments on one side, then they cut off equal segments on the other side (Thales’ theorem).
Let the points of intersection of parallel lines with one of the sides of the angle lie between (Fig. 29). Let the corresponding points of intersection of these lines be with the other side of the angle. Theorem 1.7 states that if then
Example 1. Can seven lines intersect at eight points?
Solution. They can. For example, Figure 30 shows seven such lines, three of which are parallel.
Example 2. Divide an arbitrary segment AC into 6 equal parts.
Solution. Let's draw a line segment AC. Let us draw a ray AM from point A that does not lie on the straight line AC. On the ray AM from point A we will sequentially lay out 6 equal segments (Fig. 31). Let's give designations to the ends of the segments. Connect the point with a segment to point C and draw straight lines parallel to the straight line through the points. The points of intersection of these lines with the segment AC will divide it into 6 equal parts (according to Theorem 1.7).
12. Signs of parallel lines.
Let AB and CD be two straight lines. Let AC be the third straight line intersecting straight lines AB and CD (Fig. 32, c). The straight line AC in relation to the straight lines AB and CD is called a secant. The angles formed by these right angles are often considered in pairs. Pairs of angles received special names. So, if points B and D lie in the same half-plane relative to straight line AC, then angles BAC and DCA are called internal one-sided (Fig. 32, c). If points B and D lie in different half-planes relative to straight line AC, then angles BAC and DCA are called internal cross-lying ones (Fig. 32, b).
The secant AC forms with straight lines AB and CD two pairs of internal one-sided two pairs of internal cross-lying angles Fig. 32, c).
If internal crosswise angles are equal or the sum of internal one-sided angles is equal to 180°, then the lines are parallel.
In Figure 32, c, four pairs of angles are indicated by numbers. Theorem 1.8 states that if or then the lines c and b are parallel. Theorem 1.8 also states that if or , then the lines a and b are parallel.
Theorems 1.6 and 1.8 are tests for the parallelism of lines. The converse of Theorem 1.8 is also true.
If two parallel lines are intersected by a third line, then the intersecting internal angles are equal, and the sum of the internal one-sided angles is 180°.
Example. One of the interior one-sided angles formed by the intersection of two parallel lines with a third line is 4 times larger than the other. What are these angles equal to?
Solution. By Theorem 1.9, the sum of internal one-sided angles for two parallel lines and a transversal is equal to 180°. Let us denote these angles by the letters a and P, then a is known that a is 4 times greater, which means then So,
13. Perpendicular lines.
Two lines are called perpendicular if they intersect at right angles (Fig. 33).
The perpendicularity of lines is written using the symbol. The entry reads: “Line a is perpendicular to line b.”
A perpendicular to a given line is a segment of a line perpendicular to a given line, whose end is their intersection point. This end of the segment is called the base of the perpendicular.
In Figure 34, a perpendicular A B is drawn from point A to straight line a. Point B is the base of the perpendicular.
Through each point of a line you can draw a line perpendicular to it, and only one.
From any point not lying on a given line, you can drop a perpendicular to this line, and only one.
The length of a perpendicular drawn from a given point to a straight line is called the distance from the point to the straight line.
The distance between parallel lines is the distance from any point on one line to another line.
Let BA be a perpendicular dropped from a point to straight line a, and C be any point of straight line c, different from A. The segment BC is called inclined, drawn from point B to straight line a (Fig. 35). Point C is called the base of the inclined point. The segment AC is called an oblique projection.
A straight line passing through the middle of a segment perpendicular to it is called a perpendicular bisector.
In Figure 36, straight line a is perpendicular to segment AB and passes through point C - the middle of segment AB, i.e. a is the perpendicular bisector.
Example. Equal segments AD and CB, enclosed between parallel lines AC and BD, intersect at point O. Prove that.
Solution. Let us draw perpendiculars to straight line BD from points A to C (Fig. 37). AK=CM as the distance between parallel lines, ZAKD and DSLYAV are rectangular, they
equal in hypotenuse and leg (see T. 1.25), and therefore isosceles (T. 1.19), and therefore, From the equality of triangles ACT) and CTAB it follows that , and then, i.e. A. AOS isosceles , which means
14. Tangent to a circle. Touching circles.
A straight line passing through a point on a circle perpendicular to the radius drawn to this point is called a tangent. In this case, this point on the circle is called the point of tangency. In Figure 38, straight line a is drawn through point A of the circle perpendicular to the radius OA. The line c is tangent to the circle. Point A is the point of contact. We can also say that the circle touches line a at point A.
Two circles having a common point are said to touch at that point if they have a common tangent at that point. The tangency of circles is called internal if the centers of the circles lie on the same side of their common tangent. The tangency of circles is called external if the centers of the circles lie on opposite sides of their common
tangent. In Figure 39, c, the contact of the circles is internal, and in Figure 39, b - external.
Example 1. Construct a circle of given radius tangent to a given line at a given point.
Solution. A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Therefore, the center of the desired circle lies on the perpendicular to the given line passing through the given point, and is located from the given point at a distance equal to the radius. The problem has two solutions - two circles, symmetrical to each other relative to a given straight line (Fig. 40).
Example 2. Two circles with diameters of 4 and 8 cm touch externally. What is the distance between the centers of these circles?
Solution. The radii of circles OA and O, A are perpendicular to the common tangent passing through point A (Fig. 41). Therefore see
15. Triangles.
A triangle is a figure that consists of three points that do not lie on the same line and three segments connecting these points in pairs. The points are called the vertices of the triangle, and the segments are its sides. A triangle is indicated by its vertices. Instead of the word “triangle” the symbol D is used.
Figure 42 shows triangle ABC; A, B, C are the vertices of this triangle; A B, BC and AC are its sides.
The angle of triangle ABC at vertex A is the angle formed by half lines A B and AC. The angles of the triangle at vertices B to C are also determined.
If a line that does not pass through any of the vertices of a triangle intersects one of its sides, then it intersects only one of the other two sides.
The altitude of a triangle dropped from a given vertex is the perpendicular drawn from this vertex to the line containing the opposite side of the triangle. In Figure 43, c, segment AD is the height of the acute-angled A. ABC, and in Figure 43, b, the base of the height of the obtuse-angled segment - point D - lies on the continuation of side BC.
The bisector of a triangle is the bisector segment of an angle of a triangle connecting a vertex to a point on the opposite side. In Figure 44, segment AD is the bisector of triangle ABC.
The median of a triangle drawn from a given vertex is the segment connecting this vertex with the middle
the opposite side of the triangle. In Figure 45, segment AD is the median of the triangle
The midline of a triangle is the segment connecting the midpoints of its two sides.
The middle line of a triangle connecting the midpoints of two given sides is parallel to the third side and equal to half of it.
Let DE be the midline of triangle ABC (Fig. 46).
The theorem states that .
The triangle inequality is the property of distances between three points, which is expressed by the following theorem:
Whatever the three points, the distance between any two of these points is not greater than the sum of the distances from them to the third point.
Let there be three given points. The relative position of these points can be different: a) two points out of three or all three coincide, in this case the statement of the theorem is obvious; b) the points are different and lie on the same line (Fig. 47, a), one of them, for example B, lies between the other two, in this case it follows that each of the three distances is no more than the sum of the other two; c) the points do not lie
on one straight line (Fig. 47, b), then Theorem 1.14 states that .
In case c) three points A, B, C are the vertices of the triangle. Therefore, in any triangle, each side is less than the sum of the other two sides.
Example 1. Is there a triangle ABC with sides: a) ; b)
Solution. The sides of triangle ABC must satisfy the following inequalities:
In case a) inequality (2) is not satisfied, which means that such an arrangement of points cannot exist; in case b) the inequalities are satisfied, i.e. the triangle exists.
Example 2. Find the distance between points A and separated by an obstacle.
Solution. To find the distance, we hang the basis CD and draw straight lines BC and AD (Fig. 48). Find point M - the middle of CD. We also carry out MPAD. It follows that PN is the middle line, i.e.
By measuring PN, it is not difficult to find AB.
16. Equality of triangles.
Two segments are called equal if they have the same length. Two angles are said to be equal if they have the same angular measure in degrees.
Triangles ABC are said to be congruent if
This is briefly expressed in words: triangles are congruent if their corresponding sides and corresponding angles are equal.
Let us formulate the basic property of the existence of equal triangles (the axiom of the existence of a triangle equal to a given one):
Whatever a triangle is, there is an equal triangle in a given location relative to a given half-line.
Three conditions for equality of triangles are valid:
If two sides and the angle between them of one triangle are equal, respectively, to two sides and the angle between them of another triangle, then such triangles are congruent (a sign that triangles are equal along two sides and the angle between them).
If the side and adjacent angles of one triangle are equal, respectively, to the side and adjacent angles of another triangle, then such triangles are congruent (a sign of equality of triangles along a side and adjacent angles).
If three sides of one triangle are equal, respectively, to three sides of another triangle, then such triangles are congruent (a sign that triangles are equal on three sides).
Example. Points B and D lie in different half-planes relative to straight line AC (Fig. 49). It is known that Prove that
Solution. according to the condition, and since these angles are obtained by subtracting equal angles BC A and DAC from equal angles BCD and DAB. In addition, in these triangles the side AC is common. These triangles are equal in side and adjacent angles
17. Isosceles triangle.
A triangle is called isosceles if its two sides are equal. These equal sides are called the sides, and the third side is called the base of the triangle.
In a triangle it means ABC is isosceles with base AC.
In an isosceles triangle, the base angles are equal.
If two angles in a triangle are equal, then it is isosceles (the converse of Theorem T. 1.18).
In an isosceles triangle, the median drawn to the base is the bisector and the altitude.
It can also be proven that in an isosceles triangle the altitude drawn to the base is the bisector and the median. Similarly, the bisector of an isosceles triangle drawn from the vertex opposite the base is the median and altitude.
A triangle in which all sides are equal is called equilateral.
Example. In triangle ADB, angle D is 90°. A segment is laid out on the continuation of side AD (point D lies between points A and C) (Fig. 51). Prove that triangle ABC is isosceles.
An exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it.
From Theorem 1.22 it follows that the exterior angle of a triangle is greater than any interior angle not adjacent to it.
Example. In a triangle
The bisector AD of this triangle cuts off from it Find the angles of this triangle.
Solution. since AD is the bisector of angle A (see paragraph as an external angle according to the theorem on the sum of angles
19. Right triangle. Pythagorean theorem.
A triangle is called right-angled if it has a right angle. Since the sum of the angles of a triangle is 180°, a right triangle has only one right angle. The other two angles of a right triangle are acute, and they complement each other up to 90°. The side of a right triangle opposite the right angle is called the hypotenuse, the other two sides are called legs. A ABC, shown in Figure 54, is rectangular, straight, hypotenuse, NE and BA - legs.
For right triangles, you can formulate your own signs of equality.
If the hypotenuse and acute angle of one right triangle are respectively equal to the hypotenuse and acute angle of another triangle, then such triangles are congruent (a sign of equality by hypotenuse and acute angle).
If the leg and the opposite angle of one right triangle are respectively equal to the leg and the opposite angle of another triangle, then such triangles are congruent (a sign of equality along the leg and the opposite angle).
If the hypotenuse and leg of one right triangle are respectively equal to the hypotenuse and leg of another triangle, then such triangles are congruent (a sign of equality by hypotenuse and leg).
In a right triangle with an angle of 30°, the leg opposite the atom is equal to half the hypotenuse.
In triangle ABC, shown in the drawing of a straight line, So, in this triangle .
In a right triangle, the Pythagorean theorem is valid, named after the ancient Greek scientist Pythagoras, who lived in the 6th century. BC e.
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs (Pythagorean theorem).
Let ABC be a given right triangle with right angle C, legs a and b and hypotenuse c (Fig. 56). The theorem states that
From the Pythagorean theorem it follows that in a right triangle any of the legs is less than the hypotenuse.
From the Pythagorean theorem it follows that if there is a perpendicular and an inclined line to a straight line from one point, then the inclined one is greater than the perpendicular; equal obliques have equal projections; Of the two inclined, the one with the larger projection is greater.
In Figure 57, from point O to straight line a, a perpendicular OA and inclined OB, OS and OD are drawn, and based on the above: a)
The perimeter of the KDMA rectangle is 18 cm
Example 3. In a circle with a radius of 25 cm, two parallel chords of length 40 and 30 cm are drawn on one side of its center. Find the distance between these chords.
Solution. Let's draw a radius OK perpendicular to the chords AB and CD, connect the center of the circle O with points C, A, D and B (Fig. 60). Triangles COD and AOB are isosceles, since (as radii); OM and ON are the heights of these triangles. By Theorem 1.20, each of the heights is simultaneously the median of the corresponding triangle, i.e., and
Triangles OSM and O AN are rectangular, in them . ON and OM we will find using the Pythagorean theorem.
20. Circles inscribed in a triangle and circumscribed about the triangle.
A circle is called circumscribed about a triangle if it passes through all its vertices.
The center of a circle circumscribed about a triangle is the point of intersection of the perpendicular bisectors to the sides of the triangle.
In Figure 61, a circle is circumscribed about triangle ABC. The center of this circle O is the intersection point of the bisectoral perpendiculars OM, ON and OJT, drawn respectively to the sides AB, BC and CA of A.
A circle is said to be inscribed in a triangle if it touches all its sides.
The center of a circle inscribed in a triangle is the intersection point of its bisectors.
In Figure 62, the circle is inscribed in triangle ABC. The center of this circle O is the intersection point of the bisectors AO, BO and CO of the corresponding angles of the triangle.
Example. In a right triangle, the legs are equal to 12 and 16 cm. Calculate the radii of: 1) the circle inscribed in it; 2) circumscribed circle.
Solution. 1) Let triangle ABC be given, in which is the center of the inscribed circle (Fig. 63, a). The perimeter of triangle ABC is equal to the sum of the double hypotenuse and the diameter of the circle inscribed in the triangle (use the definition of a tangent to a circle and the equality of right triangles AOM and AOC, MOC and LOC by hypotenuse and leg).
Thus, from where, according to the Pythagorean theorem, i.e.
2) The center of the circumscribed circle around a right triangle coincides with the middle of the hypotenuse, where the radius of the circumscribed circle is cm (Fig. 63, b).
