Everything possible cases the relative position of a straight line and a plane in space :

The straight line lies on the plane if all points of the straight line belong to the plane.

Comment ... For a straight line to lie on a plane, it is necessary and sufficient that any two points of this straight line belong to this plane.

A straight line intersects a plane if the line and the plane have the only common point

A straight line is parallel to a plane, if a straight line and a plane do not have common points... (they do not intersect

Statement 1 ... Suppose the straight line a and the plane α are parallel, and the plane β passes through the straight line a. Then two cases are possible:

But then the point P turns out to be the intersection point of the straight line a and the plane α, and we get a contradiction with the fact that the line a and the plane α are parallel. This contradiction completes the proof of Statement 1.

Statement 2 (sign of parallelism of a straight line and a plane) ... If straight a, not lying in the plane α, parallel to some straight line b lying in the plane α, then the straight line a and the plane α are parallel.

Proof. Let us prove the "contradictory" criterion for parallelism of a straight line and a plane. Suppose the straight line a intersects the plane α at some point P. Draw the plane β through parallel lines a and b.

Point P lies on a straight line a and belongs to the plane β. But on the assumption the point P also belongs to the plane α, hence the point P lies on a straight line b, along which the planes α and β intersect. However, direct a and b are parallel by condition and cannot have common points.

The resulting contradiction completes the proof of the parallelism criterion for a straight line and a plane.

Theorems

• If a straight line intersecting a plane is perpendicular to two straight lines lying in this plane and passing through the point of intersection of this line and the plane, then it is perpendicular to the plane.
• If a plane is perpendicular to one of two parallel straight lines, then it is perpendicular to the other.
• If two lines are perpendicular to the same plane, then they are parallel.
• If a straight line lying in a plane is perpendicular to the inclined projection, then it is perpendicular to the most inclined one.
• If a straight line that does not lie in a given plane is parallel to some straight line located in this plane, then it is parallel to this plane.
• If a straight line is parallel to a plane, then it is parallel to some straight line on this plane.
• If a line and a plane are perpendicular to the same line, then they are parallel.
• All points of a straight line parallel to a plane are equally distant from this plane.

A straight line and a plane are called parallel if they have no common points. If a straight line not lying in a given plane is parallel to some straight line lying in this

1. If a plane passes through a given straight line, parallel to another plane, and intersects this plane, then the line of intersection of the planes is parallel to this straight line.

2. If one of two parallel lines is parallel to a given plane, and the other line has a common point with the plane, then this line lies in this plane. plane, then it is parallel to the plane itself.

Cases of relative position of a straight line and a plane: a) the straight line lies in the plane;

b) a straight line and a plane have only one common point; c) a straight line and a plane have no common point.

2. Determination of the natural size of a line segment in general position by the method of a right-angled triangle.

The natural value (n.v.) of the segment AB of a straight line in general position is the hypotenuse of the right-angled triangle ABK. In this triangle, the leg AK is parallel to the plane of projections π1 and is equal to the horizontal projection of the segment A "B". Leg BK is equal to the difference between the distances of points A and B from the plane π1.

In the general case, to determine the natural value of a straight line segment, it is necessary to construct the hypotenuse of a right-angled triangle, one leg of which is the horizontal (frontal) projection of the segment, and the other leg is a segment equal in magnitude to the algebraic difference of coordinates Z (Y) of the extreme points of the segment.

From a right-angled triangle, find the angle α - the angle of inclination of the straight line to the horizontal plane of the projections.

To determine the angle of inclination of a straight line to the frontal plane of the projections, it is necessary to perform similar constructions on the frontal projection of the segment.

3. The main lines of the plane (horizontal, frontal).

The horizontal plane of P is a straight line that lies in this plane and is parallel to the horizontal plane. The horizontal as a straight line parallel to the horizontal plane has a frontal projection ѓ parallel to the x-axis.

The frontal of the P plane is a straight line that lies in this plane and is parallel to the frontal plane.

The frontal is a straight line parallel to the frontal plane, and its horizontal projection φ is parallel to the x-axis.

4. Mutual position of straight lines in space. Determination of visibility by competing points. Two straight lines in space can have a different arrangement: A) intersect (lie in the same plane). A special case of intersection - at a right angle; B) can be parallel (lie in the same plane); C) coincide - a special case of parallelism; D) intersect (lie in different planes and not intersect).

The points at which the projections on P1 coincide are called competing with respect to the plane P1, and the points at which the projections on P2 coincide are called competing in relation to the plane P2.

Points K and L are competing with respect to the plane P1, since on the plane P1 points K and L are projected into one point: K1 = L1.

Point K is higher than point L, because K2 is higher than point L2, therefore K1 is visible on P1.

Theorem

If a straight line that does not belong to a plane is parallel to some straight line in this plane, then it is parallel to the plane itself.

Proof

Let α be a plane, a a straight line not lying in it, and a1 a straight line in the α plane parallel to a. Let us draw the plane α1 through lines a and a1. The planes α and α1 intersect along the straight line a1. If the straight line a intersected the plane α, then the intersection point would belong to the straight line a1. But this is impossible, since the lines a and a1 are parallel. Consequently, the straight line a does not intersect the plane α, and therefore is parallel to the plane α. The theorem is proved.

18. PLANES

If two parallel planes intersect a third, then the straight lines intersect parallel(fig. 333).

Indeed, according to the definition Parallel straight lines are straight lines that lie in the same plane and do not intersect. Our lines lie in the same plane - the secant plane. They do not intersect, because the parallel planes containing them do not intersect.

Hence, the lines are parallel, as required.

Properties

§ If the plane α is parallel to each of two intersecting lines lying in the other plane β, then these planes are parallel

§ If two parallel planes are crossed by a third, then the lines of their intersection are parallel

§ Through a point outside the given plane, you can draw a plane parallel to the given one, and moreover, only one

§ Sections of parallel straight lines bounded by two parallel planes are equal

§ Two angles with respectively parallel and equally directed sides are equal and lie in parallel planes

19.

If two straight lines lie in the same plane, the angle between them is easy to measure - for example, using a protractor. How to measure angle between line and plane?

Let the straight line intersect the plane, and not at a right angle, but at some other angle. Such a straight line is called oblique.

Let us drop the perpendicular from any point inclined to our plane. Connect the base of the perpendicular to the intersection of the inclined plane and the plane. We got inclined projection.

The angle between a straight line and a plane is the angle between a straight line and its projection onto a given plane.

Pay attention - we choose an acute angle as the angle between the line and the plane.

If the line is parallel to the plane, then the angle between the line and the plane is zero.

If the line is perpendicular to the plane, its projection onto the plane will be a point. Obviously, in this case, the angle between the straight line and the plane is 90 °.

A straight line is perpendicular to a plane if it is perpendicular to any straight line lying in this plane.

This is the definition. But how do you work with him? How to check that a given line is perpendicular to all lines in the plane? After all, there are infinitely many of them.

In practice, it is applied sign of perpendicularity of a line and a plane:

A straight line is perpendicular to a plane if it is perpendicular to two intersecting straight lines lying in this plane.

21 Dihedral- a spatial geometric figure formed by two half-planes emanating from one straight line, as well as a part of the space bounded by these half-planes.

Two planes are called perpendicular if the dihedral angle between them is 90 degrees.

§ If a plane passes through a straight line perpendicular to another plane, then these planes are perpendicular.

§ If from a point belonging to one of the two perpendicular planes, draw a perpendicular to the other plane, then this perpendicular lies completely in the first plane.

§ If, in one of the two perpendicular planes, draw a perpendicular to their line of intersection, then this perpendicular will be perpendicular to the second plane.

Two intersecting planes form four dihedral angles with a common edge: the pairs of vertical angles are equal, and the sum of two adjacent angles is 180 °. If one of the four corners is straight, then the other three are also equal and straight. Two planes are called perpendicular if the angle between them is a straight line.

Theorem. If a plane passes through a straight line perpendicular to another plane, then these planes are perpendicular.

Let and be two planes such that passes through the straight line AB, perpendicular to and intersecting with it at point A (Fig. 49). Let us prove that _ | _. The planes and intersect along some straight line AC, and AB _ | _ AC, since AB _ | _. Let us draw line AD in the plane, perpendicular to line AC.

Then the angle BAD is the linear angle of the dihedral angle formed by and. But< ВАD - 90° (ибо AB _|_ ), а тогда, по определению, _|_ . Теорема доказана.

22. A polyhedron is a body whose surface consists of a finite number of flat polygons.

1.any of the polygons that make up a polyhedron, you can reach any of them by going to the adjacent to it, and from this, in turn, to the adjacent to it, etc.

These polygons are called facets, their sides are ribs, and their tops are peaks polyhedron. The simplest examples of polyhedra are convex polyhedra, that is, the boundary of a bounded subset of Euclidean space, which is the intersection of a finite number of half-spaces.

The above definition of a polyhedron acquires a different meaning depending on how to define a polygon, for which the following two options are possible:

§ Plane closed polygonal lines (even if they are self-intersecting);

§ Parts of the plane bounded by polygonal lines.

In the first case, we get the concept of a stellated polyhedron. In the second, a polyhedron is a surface made up of polygonal pieces. If this surface does not intersect itself, then it is the full surface of some geometric body, which is also called a polyhedron. This gives rise to the third definition of a polyhedron as the geometric body itself

Straight prism

The prism is called straight if its lateral edges are perpendicular to the bases.
The prism is called oblique if its lateral edges are not perpendicular to the bases.
A straight prism has rectangles.

The prism is called correct if its bases are regular polygons.
The area of ​​the lateral surface of the prism called the sum of the areas of the side faces.
Full prism surface equal to the sum of the lateral surface and the areas of the bases

Prism elements:
Points - called vertices
The segments are called side ribs.
Polygons and - are called bases. Also, the planes themselves are called bases and

24. Parallelepiped(from the Greek παράλλος - parallel and the Greek επιπεδον - plane) - a prism, the base of which is a parallelogram, or (equivalently) a polyhedron, which has six faces and each of them is a parallelogram.

§ The parallelepiped is symmetrical about the midpoint of its diagonal.

§ Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is halved by it; in particular, all the diagonals of the parallelepiped meet at one point and are bisected by it.

§ Opposing faces of the box are parallel and equal.

§ The square of the length of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.

Surface area of ​​a rectangular parallelepiped is equal to twice the sum of the areas of the three faces of this parallelepiped:

1.	S= 2(S a+S b+S c)= 2(ab+bc+ac)

25 .Pyramid and its elements

Consider a plane, a polygon lying in it and a point S not lying in it. Connect S to all the vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called side ribs. The polygon is called the base and point S is called the top of the pyramid. Depending on the number n, the pyramid is called triangular (n = 3), quadrangular (n = 4), ptyagonal (n = 5), and so on. An alternative name for the triangular pyramid is tetrahedron... The height of the pyramid is called the perpendicular, lowered from its top to the plane of the base.

A pyramid is called regular if a regular polygon, and the base of the pyramid's height (the base of the perpendicular) is its center.

The program is designed to calculate the lateral surface area of ​​a regular pyramid.
The pyramid is a polyhedron with a base in the form of a polygon, and the rest of the faces are triangles with a common vertex.

Formula for calculating the lateral surface area of ​​a regular pyramid:

where p is the perimeter of the base (polygon ABCDE),
a - apothem (OS);

Apothem is the height of the side face of a regular pyramid, which is drawn from its top.

To find the lateral surface area of ​​a regular pyramid, enter the values ​​for the perimeter of the pyramid and apothem, then click the CALCULATE button. The program will determine the lateral surface area of ​​the regular pyramid, the value of which can be placed on the clipboard.

Truncated pyramid

A truncated pyramid is a part of a complete pyramid, enclosed between the base and a section parallel to it.
The section is called the upper base of the truncated pyramid, and the base of the full pyramid is bottom base truncated pyramid. (The bases are similar.) The side faces of the truncated pyramid are trapeziums. In a truncated pyramid 3 n ribs, 2 n peaks, n+ 2 faces, n(n- 3) diagonals. The distance between the upper and lower bases is the height of the truncated pyramid (a segment cut off from the height of the full pyramid).
The total surface area of ​​the truncated pyramid is equal to the sum of the areas of its faces.
Truncated pyramid volume ( S and s- the area of ​​the bases, H- height)

Body of rotation is called a body formed as a result of rotation of a line around a straight line.

A straight circular cylinder is inscribed in a ball if the circles of its bases lie on the sphere. The bases of the cylinder are small circles of the ball, the center of the ball coincides with the middle of the cylinder axis. [ 2 ]

A straight circular cylinder is inscribed in a ball if the circles of its bases lie on the sphere. Obviously, the center of the ball lies not in the middle of the cylinder axis. [ 3 ]

The volume of any cylinder is equal to the product of the base area by the height:

1.

V=π r 2 h

The total surface area of ​​the cylinder is equal to the sum of the lateral surface of the cylinder and twice the area of ​​the base of the cylinder.

Formula for calculating the total surface area of ​​a cylinder:

27. A round cone can be obtained by rotating a rectangular triangle around one of its legs, therefore a round cone is also called a cone of revolution. See also Round Cone Volume

Total surface area of ​​a round cone is equal to the sum of the areas of the lateral surface of the cone and its base. The base of the cone is a circle and its area is calculated using the formula for the area of ​​a circle:

2.	S=π r l+π r 2 = π r(r+l)

28. Frustum it will turn out if a section is drawn in a cone parallel to the base. The body bounded by this section, the base and the lateral surface of the cone is called a truncated cone. See also Truncated Cone Volume

Total surface area of ​​a truncated cone is equal to the sum of the areas of the lateral surface of the truncated cone and its bases. The bases of the truncated cone are circles and their area is calculated using the formula for the area of ​​a circle: S= π (r 1 2 + (r 1 + r 2)l+ r 2 2)

29. A ball is a geometric body bounded by a surface, all points of which are at an equal distance from the center. This distance is called the radius of the ball.

Sphere(Greek σφαῖρα - ball) - a closed surface, the locus of points in space, equidistant from a given point, called the center of the sphere. A sphere is a special case of an ellipsoid in which all three axes (semiaxes, radii) are equal. The sphere is the surface of the ball.

The area of ​​the spherical surface of the spherical segment (spherical sector) and the spherical layer depends only on their height and the radius of the ball and is equal to the circumference of the great circle of the ball, multiplied by the height

Ball volume is equal to the volume of the pyramid, the base of which has the same area as the surface of the ball, and the height is the radius of the ball

The volume of the sphere is one and a half times less than the volume of the cylinder described around it.

Ball elements

Ball Segment A cutting plane splits a ball into two ball segments. H- segment height, 0< H < 2 R, r- the radius of the segment base, Ball segment volume Spherical surface area of ​​a spherical segment
Spherical layer A spherical layer is the part of a ball enclosed between two parallel sections. Distance ( H) between the sections is called layer height, and the sections themselves are the bases of the layer... Spherical surface area ( volume) of the spherical layer can be found as the difference between the areas of the spherical surfaces (volumes) of the spherical segments.

 1. Multiplication of a vector by a number(fig. 56).

Product of vector A by the number λ is called a vector V whose modulus is equal to the product of the modulus of the vector A per module number λ :

The direction does not change if λ > 0 ; reverses if λ < 0 ... If λ = −1, then the vector

Called the vector opposite to the vector A, and denoted

 2. Vector addition... To find the sum of two vectors A and V vector

Then the sum will be a vector, the beginning of which coincides with the beginning of the first, and the end - with the end of the second. This rule for adding vectors is called the "triangle rule" (Fig. 57). it is necessary to depict the vector-terms so that the beginning of the second vector coincides with the end of the first.

It is easy to prove that for vectors “the sum does not change from the change of places of the terms”.
Let us indicate one more rule for vector addition - the "parallelogram rule". If we combine the beginnings of the vector addends and build a parallelogram on them, then the sum will be a vector that coincides with the diagonal of this parallelogram (Fig. 58).

It is clear that addition according to the "parallelogram rule" leads to the same result as according to the "triangle rule".
The "triangle rule" is easy to generalize (for the case of several terms). To find the sum of vectors

It is necessary to combine the beginning of the second vector with the end of the first, the beginning of the third - with the end of the second, etc. Then the beginning of the vector WITH matches the beginning of the first, and the end WITH- with the end of the latter (Fig. 59).

 3. Subtraction of vectors... The operation of subtraction is reduced to the two previous operations: the difference of two vectors is the sum of the first with the vector opposite to the second:

You can also formulate the "triangle rule" for the subtraction of vectors: it is necessary to match the origins of the vectors A and V, then their difference will be the vector

Drawn from the end of the vector V towards the end of the vector A(fig. 60).

In the future, we will talk about the vector of movement of the material point, that is, the vector connecting the initial and final positions of the point. Agree that the introduced rules of action on vectors are quite obvious for displacement vectors.

 4. Dot product of vectors... The result of the dot product of two vectors A and V is the number c equal to the product of the moduli of the vectors and the cosine of the angle α between

The dot product operation of vectors is very widely used in physics. In the future, we will often have to deal with such an operation.

1.Formulate the definition of intersecting lines. Formulate and prove a theorem expressing the criterion of intersecting lines. 2 / Prove that if two

straight lines are parallel to the third line, then they are parallel. 3. Draw a section of the parallelepiped ABCDA1B1C1D1 with a plane passing through points A, C and M, where M is the midpoint of the edge AlDl.

Which of the figures is not the main figure in space? 1) point; 2) segment; 3) straight; 4) plane.

2. Straighta andb interbreeding. How is the straight lineb relative to the plane α, if the straight line a ϵ α?

1) crosses; 2) is parallel; 3) lies in the plane; 4) crosses.

3. Determine which statement is true:

1) The perpendicular is longer than the oblique.

2) If two slopes are not equal, then the larger slope has a smaller projection.

3) A straight line is perpendicular to the plane if it is perpendicular to the two sides of the triangle lying in this plane.

4) The angle between the parallel straight line and the plane is 90º.

4. The distance between two parallel planes is 8 cm. A segment of a straight line, the length of which is 17 cm, is located between them so that its ends belong to the planes. Find the projection of this line segment to each of the planes.

1) 15 cm; 2) 9 cm; 3) 25 cm) 4) 12 cm.

5. A perpendicular TE equal to 6 dm is drawn to the MCRT plane. Calculate the distance from point E to the top of the rhombus K, if MK = 8 dm, the angle M of the rhombus is 60º.

1) 10 dm; 2) 14 dm; 3) 8 dm; 4) 12 dm.

6. The hypotenuse of a right-angled triangle is 12 cm. Outside the plane of the triangle, a point is given at a distance of 10 cm from each vertex of the triangle. Find the distance from the point to the plane of the triangle.

1) 4 cm; 2) 16 cm; 3) 8 cm; 4) 10 cm.

7. From some point a perpendicular and an inclined line are drawn to this plane, the angle between which is 60º. Find an inclined projection on a given plane if the perpendicular is 5 cm.

1) 5√3 cm; 2) 10 cm; 3) 5 cm; 4) 10√3 cm.

8. Find lateral surface a regular triangular pyramid, if the side of the base is 2 cm, and all dihedral angles at the base are equal to 30º.

1) 2 cm2; 2) 2√3 cm2; 3) √3 cm2; 4) 3√2 cm2.

9. Find the surface area of ​​a rectangular parallelepiped by its three dimensions, equal to 3 cm, 4 cm, 5 cm.

1) 94 cm2; 2) 47 cm2; 3) 20 cm2; 4) 54 cm2.

plane.

b) if one of the two parallel lines intersects this plane, then the other line also intersects this plane.

c) if two lines are parallel to the third line, then they intersect

d) if the line and the plane have no common points, then the line lies in the plane

e) a straight line and a plane are called crossing if they have no common points

plane; b) if one of two parallel straight lines intersects this plane, then the other straight line also intersects this plane; c) if two straight lines are parallel to the third straight line, then they intersect; d) if the straight line and the plane have no common points, then the straight line lies in planes), a straight line and a plane are called crossing if they have no common points.
2. The straight line c, parallel to the straight line a, intersects the plane β. Line b is parallel to line a, then:

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