Formulate the main property of the location of points on a straight line. Straight line on a plane - information required
A straight line on a plane - the necessary information.
In this article, we will dwell 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 straight lines on the plane, and give the necessary axioms. In conclusion, we will consider ways of defining a straight line on a plane and provide graphic illustrations.
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- A straight line on a plane is a concept.
- Mutual arrangement of a straight line and a point.
- Mutual arrangement of straight lines on a plane.
- Methods for specifying 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, one should clearly understand what the plane is. Plane concept allows you to get, for example, a flat table surface or wall of a house. However, it should be borne in mind that the dimensions of the table are limited, and the plane extends beyond these boundaries to infinity (as if we have an arbitrarily large table).
If you take a well-sharpened pencil and touch it with a rod to the surface of the "table", then we get an image of a point. This is how we get idea of a point on a plane.
Now you can go to the concept of a straight line on a plane.
We put a sheet of clean paper on the surface of the table (on a plane). In order to depict a straight line, we need to take a ruler and draw a line with a pencil as far as the dimensions of the ruler and sheet of paper used allow. It should be noted that in this way we get only a part of the straight line. An entire straight line, stretching to infinity, we can only imagine.
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Mutual arrangement of a straight line and a point.
We should start with the axiom: there are points on every straight line and in every plane.
It is customary to designate points in capital Latin letters, for example, points A and F... In turn, straight lines are denoted in small Latin letters, for example, straight a and d.
Possible two options for the relative position of a straight line and a point on the plane: either a point lies on a straight line (in this case, they also say that a straight line passes through a point), or a point does not lie on a straight line (they also say that a point does not belong to a straight line or a straight line does not pass through a point).
To indicate that a point belongs to a certain straight line, the symbol "" is used. For example, if point A lies on a straight line a, then you can write. If point A does not belong to direct a then write down.
The following statement is true: a single straight line passes through any two points.
This statement is axiomatic and should be accepted as 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 specified points (for example, through points A and V), can be denoted by these two letters (in our case, the straight line AB or VA).
It should be understood that infinitely many different points lie on a straight line defined on a plane, and all these points lie in the same plane. This statement is established by the axiom: if two points of a straight line lie in a certain plane, then all points of this straight line lie in this plane.
The set of all points located between two points given on a straight line, together with these points are called line segment or simply segment... The points that delimit a line are called line ends. The segment is designated by two letters corresponding to the points of the ends of the segment. For example, let the points A and V are the ends of the segment, then this segment can be denoted AB or VA... Please note that this designation of a line segment coincides with the designation of a straight line. To avoid confusion, we recommend adding the word "segment" or "straight" to the designation.
To briefly record the belonging and non-belonging of a point to a certain segment, all the same symbols and are used. To show that a certain segment lies or does not lie on a straight line, use symbols and, respectively. For example, if the segment AB belongs to direct a, can be briefly written.
It should also dwell on the case when three different points belong to the same straight line. In this case, one and only one point lies between the other two. This statement is another axiom. Let the points A, V and WITH lie on one straight line, and the point V lies between the points A and WITH... Then we can say that the points A and WITH are on opposite sides of the point V... You can also say that the points V and WITH lie on one side then points A and the points A and V lie on one side of the point WITH.
For the sake of completeness, note that any point on a straight line divides this straight line into two parts - two ray... For this case, an axiom is given: an arbitrary point O belonging to a straight line divides this straight line into two rays, and any two points of one ray lie on the same side of the point O, and any two points of different rays are on opposite sides of the point O.
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This publication will help to systematize the previously acquired knowledge, as well as prepare for an exam or test and pass them successfully.
2. The condition for finding three points on one straight line. Equation of a straight line. Mutual arrangement of points and a straight line. A bunch of straight lines. Distance from point to line
1. Let there be given three points A 1 (NS 1 , at 1), A 2 (NS 2 , at 2), A 3 (NS 3 , at 3), then the condition for finding them on one straight line:
either ( NS 2 – NS 1) (at 3 – at 1) – (NS 3 – x 1) (at 2 – at 1) = 0.
2. Let two points be given A 1 (NS 1 , at 1), A 2 (NS 2 , at 2), then y alignment of a straight line passing through these two points:
(NS 2 – NS 1)(y - y 1) – (x - x 1)(at 2 – at 1) = 0 or ( x - x 1) / (NS 2 – NS 1) = (y - y 1) / (at 2 – at 1).
3. Let there be a point M (NS 1 , at 1) and some straight line L represented by the equation at = Oh + with. Equation of a straight line passing parallel to a given straight line L through this point M:
y - y 1 = a(x - x 1).
If straight L given by the equation Oh + Woo + WITH M, is described by the equation A(x - x 1) + V(y - y 1) = 0.
Equation of a straight line passing perpendicular to a given straight line L through this point M:
y - y 1 = –(x - x 1) / a
a(y - y 1) = NS 1 – NS.
If straight L given by the equation Oh + Woo + WITH= 0, then a straight line parallel to it passing through the point M(NS 1 , at 1) is described by the equation A (y - y 1) – V(x - x 1) = 0.
4. Let two points be given A 1 (NS 1 , at 1), A 2 (NS 2 , at 2) and the straight line given by the equation Oh + Woo + C = 0. The relative position of points relative to this straight line:
1) points A 1 , A 2 lie on one side of this straight line if expressions ( Oh 1 + Woo 1 + WITH) and ( Oh 2 + Woo 2 + WITH) have the same signs;
2) points A 1 ,A 2 lie on opposite sides of this straight line if expressions ( Oh 1 + Woo 1 + WITH) and ( Oh 2 + Woo 2 + WITH) have different signs;
3) one or both points A 1 , A 2 lie on this line if one or both expressions, respectively ( Oh 1 + + Woo 1 + WITH) and ( Oh 2 + Woo 2 + WITH) take zero.
5. Central beam Is a set of straight lines passing through one point M (NS 1 , at 1) called center of the beam... Each of the straight lines of the beam is described by the beam equation y - y 1 = To(x - x 1) (beam parameter To for each line 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 not equal to zero at the same time.
If two straight beams L 1 and L 2 respectively have the form ( A 1 NS + V 1 at+ WITH 1) = 0 and ( A 2 NS+ V 2 at+ WITH 2) = 0, then the beam equation: m 1 (A 1 NS + V 1 at + WITH 1) + m 2 (A 2 NS + V 2 at + WITH 2) = 0. If the straight lines L 1 and L 2 intersecting, then the bundle is central, if the straight lines are parallel, then the bundle is parallel.
6. Let a point be given M(NS 1 ,at 1) and the straight line given by the equation Ax + Wu + C = 0. Distance dfrom this points M to straight:
- 1. Basic concepts. Coordinate systems. Straight lines and their relative position
- 2. The condition for finding three points on one straight line. Equation of a straight line. Mutual arrangement of points and a straight line. A bunch of straight lines. Distance from point to line
A segment is a part of a straight line, which consists of all the points of this straight line lying between its two given points. These points are called line ends. The segment is indicated by the indication of its ends.
In Figure 7, b, the segment AB is part of the 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, therefore it does not belong to the 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 straight line, one and only one lies between the other two.
The following axiom expresses the basic property of measuring line 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 any point C is taken on the segment MK, then the length of the segment MK is equal to the sum of the lengths of the segments MC 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 straight 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 We check the fulfillment of this condition:. Conclusion: point P lies between points O and M.
Point B belongs to the segment PM if it lies between points P and M, that is, 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 line segments on a plane so that each of them intersects exactly three others?
Solution. 6 segments can be arranged so (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. Suppose that such an arrangement of the seven line segments is possible. Let's number the segments and compose 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 that is also set. 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 there are only characters. On the other hand, the table is filled symmetrically with respect to the diagonal:
if in cell C: j) is in the cell too. This means that the total number of characters must be even. We got a contradiction.
Here we have used proof by contradiction.
5. Ray.
A semi-straight or ray is a part of a straight line, which consists of all points of this straight line, lying on one side of its 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 straight line with a common starting point are called complementary.
Semi-straight are denoted by lowercase Latin letters. You can designate a half-line with two letters: an initial and some other letter corresponding to a point belonging to the half-line. In this case, the starting point is put in the first place. For example, in Figure 9, a, beams AB and AC are shown, which are additional, in Figure 9, b, beams MA, MB and beam c are shown.
The following axiom reflects the main property of postponing line segments.
On any half-line from its starting point, you can postpone a segment of a given length, and only one.
Example. You are given two points A and B. How many lines can you draw through points A and B? How many rays exist on line AB with origin at point A, at point B? Mark two points on line A B, different from A and B. Do they belong to segment AB?
Solution. 1) According to the axiom, you can always draw a straight line through points A and B, and only one.
2) On the straight line AB with the origin at point A, there are two rays, which are called additional. Similarly for point B.
3) The answer depends on the location of the marked points. Let's consider the 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 a segment, and the other does not; in cases d) and e) the points M and N do not belong to the segment AB.
6. Circumference. Circle.
A circle is a shape that consists of all points on the plane that are at a given distance from a given point. This point is called the center of the circle.
The distance from the points of the circle to its center is called the radius of the circle. Any line segment connecting a point of a circle with its center is also called a radius.
A segment connecting two points of a circle is called a chord. The chord passing through the center is called the diameter.
Figure 11, a shows a circle centered 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 that are at a distance not more 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 border of the circle is a circle with the same center and radius (Fig. 11, b).
Example. What is the largest number of different parts that do not have common points, except for their boundaries, the 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 one more axiom of geometry.
The straight line splits the plane into two half-planes.
In Figure 13, the straight line a splits the plane into two half-planes so that each point of the plane that does not belong to the straight line o lies in one of them. This partition has the following property: if the ends of some segment belong to one half-plane, then the segment does not intersect with a straight line; if the ends of the segment belong to different half-planes, then the segment intersects with a straight line. In Figure 13, the points lie in one of the half-planes into which Line a splits the plane. Therefore, the segment AB does not intersect with the straight line a. Points C and D lie in different half-planes. Therefore, the segment CD intersects the line a.
8. Angle. The degree measure of the 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 the corner are additional half-lines, then the angle is called unfolded.
An angle is indicated either by indicating its vertex, or by indicating its sides, or by indicating three points; vertices and two points on the sides of the corner. The word “corner is sometimes replaced by the symbol Z.
The angle in Figure 14 can be denoted in three ways:
They say that a ray c passes between the sides of an angle if it emanates from its vertex and crosses some segment with ends on the sides of the angle.
In Figure 15, ray c passes between the sides of the angle, as it intersects segment AB.
In the case of a flat corner, any ray emanating from its vertex and other than its sides passes between the sides of the corner.
Angles are measured in degrees. If you take an extended angle and divide it by 180 equal angles, then the degree measure of each of these angles is called a degree.
The basic properties of measuring angles are expressed in the following axiom:
Each angle has a certain degree measure, greater than zero. The flattened angle is 180 °. The degree measure of the 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 the ray c passes between the sides of the angle, then the angle is equal to the sum of the angles
The degree measure of the 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 the deposition of corners.
From any half-line to a given half-plane, you can postpone 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 splits the plane into two half-planes. Figure 16 shows how, using a protractor, to set aside an angle with a given degree measure of 60 ° from the half-line a to the upper half-plane.
If two corners are set aside from a given half-line into one half-plane, then the side of the smaller angle, different from this half-line, passes between the sides of the larger angle.
Let the angles, plotted from the given half-line and in 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 a ray that emanates from its vertex, passes between its sides and divides the angle in half. In Figure 18, ray OM is the bisector of the angle AOB.
In geometry, there is the concept of a flat angle. A plane angle is the part of a plane bounded by two different rays emanating from one point. These rays are called the sides of the angle. There are two planar corners with these sides. They are called complementary. In Figure 19, one of the flat corners 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 the plane angle contains a half-plane, then its degree measure is 360 ° - a, where a is the degree measure of the additional plane angle.
Example. Beam a passes between the sides of an angle equal to 120 °. Find angles if their degree measures are 4: 2.
Solution. Ray a passes between the sides of the angle, which means, according to the basic property of measuring angles (see item 8)
Since the degree measures are related as 4: 2, then
9. Adjacent and vertical corners.
Two corners are called adjacent if they have one side in common, and the other sides of these corners are additional half-lines. In Figure 20, the corners are adjacent.
The sum of adjacent angles is 180 °.
Theorem 1.3 implies the following properties:
1) if two angles are equal, then the angles adjacent to them 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 corners are called vertical if the sides of one corner are complementary half-straight sides of the other. In Figure 21, and the corners are vertical.
The vertical angles are equal.
Obviously, two intersecting straight lines form adjacent and vertical angles. Adjacent angles complement each other up to 180 °. The angular measure of the smaller of them is called the angle between straight lines.
Example. In Figure 21, b, the angle is 30. ° What are the angles AOK and
Solution. The angles COD and AOK are vertical, therefore, by Theorem 1.4, they are equal, that is, the angle TYUK adjacent to the angle SOD means, by Theorem 1.3
10. Central and inscribed corners.
The central angle in a circle is a flat angle with a vertex at its center. The part of a circle located inside a flat angle is called a circular arc corresponding to that central angle. The degree measure of an arc of a circle is the degree measure of the corresponding central angle.
In Figure 22, the angle AOB is the central angle of the circle, its vertex O is the center of this circle, and the sides OA and OB intersect the circle. The arc AB is part of a circle inside the central corner.
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.
The angle, the vertex of which lies on the circle, and the 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 of the circle, is equal to half the angle between the radii drawn to 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 corner (Fig. 23, b); the center of the circle lies outside the inscribed angle (Fig. 23, c).
Theorem 1. 5 implies the following corollary: all angles inscribed in a circle, the sides of which pass through two given points of the circle, and the vertices lie on one side of the straight line connecting these points, are equal; inscribed angles, the sides of which pass through the ends of the diameter of the circle, are straight.
In Figure 24, the sides of the inscribed angle ABC pass through the ends of the diameter AC, therefore
Example. Points A, B and C lie on a circle with center O. Find the angle AOC if
Solution. Angle ABC, inscribed in a circle, rests on the arc AC, and the central angle of this circle (Fig. 25). , hence, by Theorem 1.5, and since the angle AOC is central, its degree measure is equal to the degree measure of the arc AC, i.e.
11. Parallel lines.
Two straight lines on a plane are called parallel if they do not intersect.
Figure 26 shows how, using a square and a ruler, draw a straight line 6 through a given point B, parallel to a given straight line a.
To denote the parallelism of straight lines, the symbol II is used. The entry reads: "Line a is parallel to line b".
The parallelism axiom expresses the main property of parallel lines.
Through a point that does not lie on a given straight line, at most one straight line parallel to the given one can be drawn on the plane.
Two straight lines, parallel to the third, are parallel to each other.
In Figure 27, straight lines a and b are parallel to straight line c. Theorem 1. 6 states that.
You can prove that through a point that does not belong to a straight line, you can draw a straight line parallel to the given one. In Figure 28, a straight line a is drawn through a point A, which does not belong to b, parallel to a straight line b.
Comparing this statement and the axiom of parallels, they come to an important conclusion: on a plane through a point that does not lie on a given straight line, it is possible to draw a straight line parallel to it, and only one line.
The axiom of parallelism in Euclid's book "Beginnings was called" the fifth postulate. Ancient geometers tried to prove the uniqueness of the parallel. These unsuccessful attempts continued for over 2000 years, until the 19th century.
The great Russian mathematician NI Lobachevsky and, independently of him, the Hungarian mathematician J. Boyai showed that, assuming the possibility of drawing 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 parallel axiom, is Thales's theorem. Thales of Miletus was an ancient Greek mathematician who lived in 625-547. BC NS.
If parallel straight lines intersecting the sides of an angle cut off equal segments on one side of it, then they cut off equal segments on its other side (Thales' theorem).
Let the points of intersection of parallel straight lines on one of the sides of the corner and lie between (Fig. 29). Let the corresponding points of intersection of these lines with the other side of the corner. 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 straight lines, three of which are parallel.
Example 2. An arbitrary segment of the AC is divided into 6 equal parts.
Solution. Let's draw a segment of the AC. Let us draw from point A a ray AM that does not lie on the line AC. On the ray AM from point A, we successively set aside 6 equal segments (Fig. 31). The ends of the segments will be labeled.Connect the point with a segment with point C and through the points we will draw straight lines parallel to the straight line. The intersection points of these lines with the segment AC will divide it into 6 equal parts (by Theorem 1.7).
12. Signs of parallelism of straight lines.
Let AB and CD be two lines. Let AC be the third line intersecting lines AB and CD (Fig. 32, c). Direct AC in relation to direct AB and CD is called secant. The right angles formed by these right angles are often viewed in pairs. The pairs of angles have received special names. So, if points B and D lie in the same half-plane relative to the straight line AC, then the angles BAC and DCA are called internal one-sided (Fig. 32, c). If points B and D lie in different half-planes relative to the straight line AC, then the angles BAC and DCA are called internal crosswise (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 the inner cross lying angles are equal or the sum of the inner one-sided angles is 180 °, then the straight lines are parallel.
In Figure 32, c, four pairs of corners are numbered. Theorem 1.8 states that if or then the lines c and b are parallel. Theorem 1.8 also states that if or, then lines a and b are parallel.
Theorems 1.6 and 1.8 are criteria for parallelism of lines. The converse theorem to Theorem 1.8 is also true.
If two parallel straight lines are intersected by a third straight line, then the inner cross lying angles are equal, and the sum of the inner one-sided angles is 180 °.
Example. One of the inner one-sided corners, formed at the intersection of two parallel straight lines of the third straight line, is 4 times larger than the other. What are these angles equal to?
Solution. By Theorem 1.9, the sum of interior one-sided angles for two parallel lines and a secant is 180 °. Let us denote these angles by the letters a and P, then a is known that a is 4 times more, which means then So,
13. Perpendicular straight lines.
Two straight lines are called perpendicular if they intersect at right angles (fig. 33).
The perpendicularity of straight lines is written using the symbol. The entry reads: "Line a is perpendicular to line b".
Perpendicular to a given straight line is a segment of a straight line perpendicular to a given one, having an endpoint of their intersection. This end of the line is called the base of the perpendicular.
In Figure 34, the perpendicular AB is drawn from point A to line a. Point B is the base of the perpendicular.
Through each point of the straight line, you can draw a straight line perpendicular to it, and only one.
From any point not lying on the straight line, you can drop a perpendicular to this straight line, and only one.
The length of a perpendicular dropped from a given point onto a straight line is called the distance from a point to a straight line.
The distance between parallel straight lines is the distance from any point of one straight line to another straight line.
Let BA be a perpendicular dropped from a point on a straight line a, and C - any point of a 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 oblique. The segment AC is called an oblique projection.
A straight line passing through the middle of a segment perpendicular to it is called the midpoint perpendicular.
In Figure 36, straight line a is perpendicular to segment AB and passes through point C - the middle of segment AB, that is, a is the midpoint perpendicular.
Example. Equal segments AD and CB, enclosed between parallel lines AC and BD, intersect at point O. Prove that.
Solution. Let's draw from points A to C perpendiculars to line BD (fig. 37). AK = CM as the distance between parallel straight lines, ZAKD and DSLYAV are rectangular, they
are equal in hypotenuse and leg (see T. 1.25), which means isosceles (T. 1.19), which means from the equality of triangles AKT) and CTAB it follows that, and then, i.e. A. AOS is isosceles , which means
14. Tangent to the circle. Tangency of circles.
A straight line passing through a point on a circle perpendicular to the radius drawn to this point is called a tangent line. In this case, this point of the circle is called the tangency point. In Figure 38, straight line a is drawn through point A of the circle perpendicular to the radius OA. Line c is tangent to the circle. Point A is the touch point. We can also say that the circle touches the straight line a at point A.
They say that two circles having a common point touch at this point if they have a common tangent line at that point. The tangency of the circles is called internal if the centers of the circles lie on one side of their common tangent. The tangency of the circles is called external if the centers of the circles lie on opposite sides of their common
tangent. In Figure 39, c, the tangency of the circles is internal, and in Figure 39, b - external.
Example 1. Construct a circle of a given radius tangent to a given straight line at a given point.
Solution. The tangent to the circle is perpendicular to the radius drawn to the tangent point. Therefore, the center of the desired circle lies on the perpendicular to the given straight line passing through the given point, and is located from this point at a distance equal to the radius. The problem has two solutions - two circles symmetric to each other with respect to a given straight line (Fig. 40).
Example 2. Two circles with a diameter of 4 and 8 cm touch externally. What is the distance between the centers of these circles?
Solution. The radii of the 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 one straight line, and three segments that connect these points in pairs. The points are called the vertices of the triangle, and the line segments are called the sides. The triangle is indicated by its vertices. Instead of the word "triangle, the symbol D. is used.
Figure 42 shows a triangle ABC; A, B, C - the vertices of this triangle; A B, BC and AC are its sides.
The angle of the triangle ABC at the vertex A is the angle formed by the half-lines AB and AC. The angles of the triangle at the vertices B to C. are also determined.
If a straight line that does not pass through any of the vertices of the triangle intersects one of its sides, then it intersects only one of the other two sides.
The height of a triangle dropped from a given vertex is called a perpendicular drawn from this vertex to a straight line containing the opposite side of the triangle. In Figure 43, c, the 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 point D - lies on the continuation of the BC side.
The bisector of a triangle is the segment of the bisector of the angle of the triangle that connects the 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 middle line of a triangle is the segment that connects the midpoints of its two sides.
The middle line of the triangle, which connects the midpoints of these two sides, is parallel to and equal to half of the third side.
Let DE be the midline of triangle ABC (Fig. 46).
The theorem states that.
The triangle inequality is the property of the 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 more than the sum of the distances from them to the third point.
Let 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 one straight line (Fig. 47, a), one of them, for example B, lies between two others, in this case whence it follows that each of the three distances is not 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 asserts 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. For the sides of triangle ABC, the following inequalities must be satisfied:
In case a) inequality (2) does not hold, which means that such an arrangement of points cannot be; in case b) the inequalities hold, that is, 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 line segments are said to be 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 and are called equal if
This is briefly expressed in words: triangles are equal if they have the corresponding sides and the corresponding angles are equal.
Let us formulate the main property of the existence of equal triangles (the axiom of the existence of a triangle equal to a given one):
Whatever the triangle, there is an equal triangle in a given location relative to a given half-line.
There are three criteria for equality of triangles:
If two sides and the angle between them of one triangle are equal respectively to the two sides and the angle between them of another triangle, then such triangles are equal (sign of equality of triangles on two sides and the angle between them).
If the side and the angles adjacent to it of one triangle are respectively equal to the side and the angles adjacent to it of the other triangle, then such triangles are equal (a sign of equality of triangles along the side and the angles adjacent to it).
If three sides of one triangle are equal, respectively, to three sides of another triangle, then such triangles are equal (sign of equality of triangles on three sides).
Example. Points B and D lie in different half-planes relative to the straight line AC (Fig. 49). It is known that Prove that
Solution. by condition, and since these angles are obtained by subtracting from equal angles BCD and DAB equal angles BC A and DAC. In addition, the speaker side is common in the indicated triangles. These triangles are equal in side and angles adjacent to it.
17. Isosceles triangle.
A triangle is called isosceles if its two sides are equal. These equal sides are called sides, and the third side is called the base of the triangle.
In a triangle means ABC is isosceles with base AC.
In an isosceles triangle, the angles at the base are equal.
If two angles in a triangle are equal, then it is isosceles (the opposite of Theorem T. 1.18).
In an isosceles triangle, the median drawn to the base is the bisector and the height.
You can also prove that in an isosceles triangle, the height drawn to the base is the bisector and the median. Similarly, the bisector of an isosceles triangle, drawn from the apex opposite the base, is the median and height.
A triangle in which all sides are equal is called equilateral.
Example. In triangle ADB, the angle D is 90 °. On the continuation of the side AD there is a segment (point D lies between points A and C) (Fig. 51). Prove that triangle ABC is isosceles.
The outer corner of a triangle is equal to the sum of two inner angles that are not adjacent to it.
From Theorem 1.22 it follows that the outer angle of a triangle is greater than any inner angle not adjacent to it.
Example. In a triangle
The bisector AD of this triangle cuts off from it Find the corners of this triangle.
Solution. since AD is the bisector of angle A (see subsection as the outer angle by the sum of angles theorem
19. Rectangular triangle. Pythagorean theorem.
A triangle is called rectangular if it has a right angle. Since the sum of the angles of a triangle is 180 °, then a right-angled triangle has only one right angle. The other two corners of the right-angled triangle are sharp, and they complement each other up to 90 °. The side of a right-angled triangle opposite the right angle is called the hypotenuse, the other two sides are called legs. A ABC, shown in Figure 54, rectangular, straight, hypotenuse, CB and BA - legs.
For right-angled triangles, you can formulate your own equality criteria.
If the hypotenuse and acute angle of one right-angled triangle are respectively equal to the hypotenuse and acute angle of another triangle, then such triangles are equal (a sign of equality for the hypotenuse and acute angle).
If the leg and the opposite corner of one right-angled triangle are respectively equal to the leg and the opposite corner of the other triangle, then such triangles are equal (a sign of equality in the leg and the opposite corner).
If the hypotenuse and leg of one right-angled triangle are respectively equal to the hypotenuse and leg of the other triangle, then such triangles are equal (a sign of equality for the hypotenuse and leg).
In a right-angled triangle with an angle of 30 °, the leg opposite to the atom angle is half the hypotenuse bus.
In the triangle ABC, shown in the figure is a straight line, So, in this triangle.
In a right-angled triangle, the Pythagorean theorem is valid, named after the ancient Greek scientist Pythagoras, who lived in the 6th century. BC NS.
In a right-angled 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-angled 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-angled triangle any of the legs is less than the hypotenuse.
From the Pythagorean theorem it follows that if a perpendicular and an oblique line from one point are drawn, then the oblique is greater than the perpendicular; equal oblique have equal projections; of the two oblique, the larger is the one with the larger projection.
In Figure 57, from point O to straight line a, a perpendicular OA and oblique OB, OS and OD are drawn, while 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 40 and 30 cm long are drawn on one side of its center. Find the distance between these chords.
Solution. Let's draw the 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); ОМ 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.,
Triangles OCM and O AN are rectangular, in them. ON and ОМ are found by the Pythagorean theorem.
20. Circles inscribed in a triangle and circumscribed around 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 intersection point of the perpendiculars to the sides of the triangle.
In Figure 61, a circle is described around a triangle ABC. The center of this circle O is the intersection point of the mid-perpendiculars ОМ, ON and OJT, drawn respectively to the sides AB, BC and C A.
A circle is called 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 point of intersection of the bisectors AO, BO and CO of the corresponding angles of the triangle.
Example. In a right-angled triangle, the legs are 12 and 16 cm. Calculate the radii: 1) the inscribed circle; 2) the circumcircle.
Solution. 1) Let a 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 doubled hypotenuse and the diameter of the circle inscribed in the triangle (use the definition of the tangent to the circle and the equality of right-angled triangles AOM and AOK, MOC and LOC along the hypotenuse and leg).
Thus, whence, by the Pythagorean theorem, i.e.
2) The center of a circle circumscribed about a right-angled triangle coincides with the middle of the hypotenuse, whence the radius of the circumscribed circle is cm (Fig. 63, b).
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