Crossing straight lines. Examples of problems with and without solutions

Stereometry

Independent work N 1

Option 1

1. Draw a straight line a and dots A,B And C, not belonging to this line. Make the necessary notes.

2. Draw plane b, points E,F belonging to her, and period G, which does not belong to her. Make the necessary notes.

3. Draw a straight line a, lying in plane a. Make the necessary entry.

4. Draw two intersecting planes a and b. Make the necessary entry.

Option 2

1. Draw two intersecting at a point O straight a And b and dots A,B,C, and point A belongs to the line a, B belongs to the line b, dot C does not belong to the given lines.

2. Draw the plane g and points not belonging to it K,L and the point belonging to it M. Make the necessary notes.

3. Draw a straight line b, intersecting the plane b at the point O. Make the necessary entry.

4. Draw three intersecting lines a planes a, b and g. Make the necessary entry.

Independent work N 2

Option 1

1) The angles at the base of an isosceles triangle are equal.

2) A single straight line passes through two points in space.

3) Vertical angles are equal.

4) A parallelogram is a quadrilateral whose opposite sides pairwise parallel.

2. Define mutual arrangement planes a and b, if a triangle lies in them ABC. Justify your answer.

3. How many planes can pass through three points?

4. Find greatest number lines passing through different pairs of four points.

Option 2

1. From the following sentences, indicate axioms, definitions, theorems:

1) If two planes have common point, then they intersect in a straight line.

2) Middle line of a triangle is the segment connecting the midpoints of its two sides.

3) For straight lines and planes in space, the axioms of planimetry are satisfied.

4) The diagonals of a parallelogram are divided in half by the point of intersection.

2. Determine the relative position of two planes b and g if they contain points B And C. Justify your answer.

3. Find the greatest number of lines passing through different pairs of 5 points.

4. Find the greatest number of planes passing through different triplets of four points.

2. Corollaries from the axioms of stereometry

Option 1

1. In the plane of two intersecting lines a And b point given C, not belonging to these lines. Straight c, lying in a given plane, passes through the point C c relative to these straight lines?

2. Given three points that do not belong to the same line. Prove that all lines intersecting two of the three segments connecting these points lie in the same plane.

3. The plane is given by a straight line c and a point not belonging to it C a, different from the given line and not passing through this point.

4. A plane is defined by two intersecting at a point O straight a And b. Draw a straight line c, which intersects these lines and does not lie in the given plane.

Option 2

1. Direct d, lying in the plane of the triangle ABC, crosses his side AB. What could be the relative position of the lines? d And B.C.?

2. Two parallel lines are drawn in plane a a And b. Prove that all lines intersecting these lines lie in the same plane.

3. A plane is defined by two intersecting at a point O straight m And n. Construct a straight line in this plane k, different from the given lines and not passing through the point O.

4. The plane is defined by three points D,E,F, not belonging to the same line. Draw a straight line a, which intersects the sides DE And DF triangle DEF and does not lie in this plane.

3. Spatial figures

Option 1

1. Draw a pentagonal prism and divide it into tetrahedrons.

2. Determine the number of vertices, edges and faces: a) cube; b) 7-gonal prism; V) n-coal pyramid.

3. Determine the type of prism if it has: a) 10 vertices; b) 21 ribs; c) 5 faces.

4. How can the faces of a 4-gonal prism be colored so that the adjacent (having a common edge) faces are colored in different colors? Which smallest number Will you need flowers?

Option 2

1. Draw a pentagonal pyramid and divide it into tetrahedrons.

2. Determine the number of vertices, edges and faces: a) rectangular parallelepiped; b) 6-sided pyramid; V) n- carbon prism.

3. Determine the type of pyramid if it has: a) 5 vertices; b) 14 ribs; c) 9 faces.

4. How can the faces of an octahedron be colored so that neighboring (sharing a common edge) faces are painted in different colors. What is the smallest number of colors needed?

4. Modeling polyhedra

Option 1

1. Draw several nets of the cube.

2. Draw a figure consisting of four equal equilateral triangles, which is not a net of a regular tetrahedron.

3. Draw the development of a regular quadrangular pyramid and color it in such a way that when gluing adjacent faces have different colors. What is the smallest number of flowers you need to take?

4. Draw a development of a rectangular parallelepiped and color it in such a way that when gluing adjacent faces have different colors. What is the smallest number of flowers you need to take?

Option 2

1. Draw several nets of a regular tetrahedron.

2. Draw a figure consisting of six squares that is not a net of a cube.

3. Draw a cube development and color it in such a way that when gluing adjacent faces have different colors. What is the smallest number of flowers you need to take?

4. Draw the development of a regular 6-gonal pyramid and color it in such a way that when gluing adjacent faces have different colors. What is the smallest number of flowers you need to take?

5. Parallelism of lines in space

Option 1

1. Write in a regular 4-gonal pyramid SABCD all pairs of parallel edges.

2. In the plane of two parallel lines a And b given point C, not belonging to these lines. Through the point C a direct line was drawn c. How can a straight line be positioned? c relative to straight lines a And b.

3. Through a point that does not belong to a given line, draw a line parallel to this one.

4. Find locus lines intersecting two given parallel lines.

Option 2

1. Write four pairs of parallel edges of the cube A...D 1.

2. Given three lines a,b And With. How can these straight lines be positioned so that a plane can be drawn containing all these straight lines?

3. Given two parallel lines a And b. Prove that any plane intersecting one of them will also intersect the other.

4. Find the locus of lines parallel to a given line and intersecting another line intersecting the first.

6. Crossing lines

Option 1

1. In a cube A...D 1 write down the edges crossing the edge AB.

2. Write pairs of crossing edges of a 4-gonal pyramid SABCD.

3. How are the lines located relative to each other? a And b in Figure 1? Justify your answer.

4. Given two skew lines a And b and a point that does not belong to them C. Construct a straight line c, passing through the point C and intersecting the lines a And b.

Option 2

1. Write down the edges that intersect with the edge S.A. regular 4-gonal pyramid SABCD.

2. Write down the edges that intersect the diagonal B 1D Cuba A...D 1.

c(Fig. 1). Straight a lies in plane a and intersects the line c. Is it possible to draw a line parallel to the line in plane b? a? Justify your answer.

4. Are there two parallel lines, each of which intersects two given skew lines? Justify your answer.

7. Parallelism of a straight line and a plane

Option 1

1. Write down the edges parallel to the plane of the face CC 1D 1D correct prism ABCDEFA 1B 1C 1D 1E 1F 1.

2. Direct a parallel to plane a; straight b intersects plane a at point B; straight c, intersecting the lines a And b respectively at points E And F, intersects plane a at point C a And b?

3. Planes a and b intersect in a straight line c. Dot A belongs to plane a, point B– plane b. Construct: a) a straight line a, lying in the plane a, passing through the point A and parallel to plane b; b) straight b, lying in the b plane passing through the point B and parallel to plane a. How will the straight lines be positioned relative to each other? a And b?

4. Points A And B belong to adjacent lateral faces of the pyramid. Draw two segments parallel to each other through these points on these faces.

Option 2

1. Write down the planes of the faces parallel to the edge CC 1 parallelepiped A...D 1.

2. Direct a parallel to plane a; straight b And c, intersecting the line a respectively at points B And C, intersect the plane a respectively at the points D And E. Make a drawing. How can straight lines be positioned relative to each other? a And b?

3. Planes a and b intersect in a straight line c. Straight a lies in plane a. Prove that if: a) a intersects plane b at point A, That A belongs to the line c; b) a is parallel to plane b, then it is parallel to the line c.

4. Points A And B belong to adjacent lateral faces of the prism. Draw two segments parallel to each other through these points on these faces.

8. Parallelism of two planes

Option 1

1. Write down the parallel planes of the parallelepiped A...D 1.

2. Are the statements true:

1) Through a point not belonging to a given plane, there passes a single plane parallel to the given one.

2) If two lines lying in one plane are respectively parallel to two lines lying in another plane, then these planes are parallel.

3) There are infinitely many lines parallel to a given plane and passing through a point not belonging to this plane.

4) If one of two given planes is parallel to two intersecting lines lying in the other plane, then these planes are parallel.

3. Prove that two planes parallel to the same third plane are parallel to each other.

4. Segments AB And CD lie in parallel planes a and b, respectively (Fig. 2). How can straight lines be positioned relative to each other? A.C. And BD? Can they be parallel?

Option 2

1. In a triangular pyramid SABC draw a plane parallel to its base ABC.

2. Are the statements true:

1) If a line lying in one plane is parallel to a line lying in another plane, then these planes are parallel.

2) If a plane intersects two given planes along parallel lines, then these planes are parallel.

3) There are infinitely many planes parallel to a given line and passing through a point not belonging to this line.

4) If two planes are parallel to the same line, then they are parallel.

3. Prove that if a plane intersects one of two parallel planes, then it also intersects the other.

4. Segments AB And CD lie in parallel planes a and b, respectively (Fig. 3). How can straight lines be positioned relative to each other? AD And B.C.? Can they intersect?

9. Vectors in space

Option 1

1. For given vector construct the vectors: a) -; b) 2; V) -.

2. How many vectors are defined by all possible pairs of points made up of the vertices of a regular quadrangular pyramid?

ABCD .

4. Given a parallelepiped A...D 1..gif" width="128" height="29 src=">.gif" width="15" height="19 src="> construct the vectors: a) 3; b) -2; V) .

2. How many vectors are defined by all possible pairs of points made up of the vertices of a triangular prism?

3. Draw regular tetrahedron ABCD and draw a vector: a) ; b) ; V) .

4. Given a parallelepiped A...D 1..gif" width="133" height="29 src=">.gif" width="15" height="17 src="> to get a vector in the same direction with and ||=1.

2. Given two oppositely directed vectors and , and || > ||..gif" width="15" height="19 src=">.

3. Given a tetrahedron ABCD. Write down three pairs of its vertices that define coplanar vectors.

4. Given a cube A...D 1. Write down the triplets not coplanar vectors with beginnings and ends at its vertices.

Option 2

1..gif" width="15" height="21">, oppositely directed with and ||=2.

2..gif" width="15" height="21 src=">.gif" width="15" height="21 src=">|. Find the direction and length of the vector +.

3. Given a tetrahedron ABCD. Write down three pairs of its vertices that define non-coplanar vectors.

4. Given a cube A...D 1. Write down triples of coplanar vectors with beginnings and ends at its vertices.

11. Parallel transfer

Option 1

1. Construct the figure that turns out parallel transfer straight a to vector if: a) E belongs a, F do not belong a; b) points E And F don't belong a.

2. Specify a parallel translation, which is the middle of the segment G.H. translates to some point M.

3. Construct a figure that is obtained from a square ABCD parallel transfer to a vector: a) https://pandia.ru/text/78/221/images/image025_45.gif" width="28" height="24 src=">.

4. Construct a figure that is obtained from a tetrahedron ABCD parallel transfer to a vector.

Option 2

1. Construct a figure that is obtained by parallel translation of a circle with the center at the point O to the vector https://pandia.ru/text/78/221/images/image024_45.gif" width="29" height="24 src=">.gif" width="29" height="24">.

12. Parallel design

Option 1

1. How many points will be obtained with parallel design two various points space? Make appropriate drawings and justification.

2. List the properties of a rectangle that are preserved during parallel design.

3. How should two straight lines be positioned so that they are projected onto a plane into a straight line and a point not belonging to this straight line?

4. Parallel lines a And b A,B And C are shown in Figure 4. Draw the fourth point D. Justify your answer.

Option 2

1. How many points will you get when designing three different points in space? Make appropriate drawings and justification.

2. List the properties of a rhombus that are preserved during parallel design.

3. How should a line and a point be located so that they are projected onto a plane into a line and a point belonging to this line?

4. Intersecting lines a And b intersect parallel planes a and b at four points. Three of them A,B And C are shown in Figure 5. Draw the fourth point D. Justify your answer.

13. Parallel projections of plane figures

Option 1

1. Draw a parallel projection of a right isosceles triangle lying in a plane parallel to the projection plane.

2. Draw a parallel projection of an equilateral triangle ABC and on it construct images of perpendiculars dropped from the point M– middle of the side AB to the sides A.C. And B.C..

ABCDEF, taking a rectangle as the original figure ABDE.

4. Draw a parallel projection of an equilateral triangle ABC and construct on it an image of a perpendicular drawn from the point K– midpoint of the segment B.O.(O– center of the triangle) to the side AB.

Option 2

1. Draw a parallel projection of an equilateral triangle lying in a plane parallel to the projection plane.

2. Draw a parallel projection of a square ABCD and on it construct an image of perpendiculars drawn from the point E– middle of the side B.C. to straight lines BD And AC.

3. Draw a parallel projection regular hexagon ABCDEF, taking as the original figure equilateral triangle ACE.

4. Draw a parallel projection of a rectangle ABCD, which one AD= 2AB. Construct an image of a perpendicular dropped from a vertex C to the diagonal BD.

14. Image of spatial figures

Option 1

1. Draw a regular quadrangular pyramid and its height.

2. Draw a cube whose two faces are parallel to the design plane.

3. Figure 6 shows a parallel projection of a cube A...D

4. Given a tetrahedron ABCD. The area of ​​its face ADC equal to S BDC to the plane ADC in the direction of a straight line AB.

Option 2

1. Draw the correct triangular pyramid and its height.

2. Draw a cube whose faces are not parallel to the design plane.

3. Figure 7 shows a parallel projection of a cube A...D 1. How is the cube located relative to the design plane?

4. Given a tetrahedron ABCD. The area of ​​its face ABD equal to Q. Find the projection area of ​​its face BDC to the plane A.D.B. in the direction of a straight line C.M., Where M– middle of the rib AB.

15. Sections of polyhedra

Option 1

1. In a hexagonal prism A...F 1 (Fig. 8) construct the point of intersection of the line PQ with plane ABC, where the points Q And P belong respectively to the lateral edges of the prism BB 1 and DD 1.

2. On the side ribs quadrangular prism A...D 1 three points are given K,L,M(Fig. 9). Construct a plane intersection line KLM with plane ABC.

3. Construct a section of the cube with a plane passing through the points X,Y,Z A.D.A.A. 1, BB 1 and such that AX:XD= 1:2, A 1Y:YA= 2:1, B 1Z:ZB = 1:2.

4. In the right pyramid SABCD construct a section passing through the side of the base AD and period M, belonging to the side edge S.B..

Option 2

1. On the side ribs BB 1 and E.E. 1 prism ABCDEA 1B 1C 1D 1E 1 points are given accordingly F And G(Fig. 10). Construct the point of intersection of the line FG with plane ABC.

2. Given a cube A...D 1. On his ribs A.A. 1, CC 1 and DD 1 three points are given respectively X,Y,Z(Fig. 11). Construct the line of intersection of the planes XYZ And ABC.

3. In a regular triangular prism A...C 1 construct a section passing through the points K,L And M, belonging respectively to the edges A.A. 1, A.C. And BB 1 and such that: AK =K.A. 1; AL:LC= 1:2 and BM =M.B. 1.

4. In the right pyramid SABCD construct a section passing through the diagonal A.C. base and parallel to the lateral edge SD.

16. Angle between straight lines in space. Perpendicularity of lines

Option 1

1. In a cube A...D AB And BB 1; b) BD And BB 1; V) AB 1 and CC 1; G) AB 1 and CD 1.

A...C 1 segment CD perpendicular to the edge AB CD And A.A. 1; b) CD And A 1B 1.

3. In the right way quadrangular pyramid SABCD with equal edges find the angle between the diagonal A.C. base and side edge S.C..

4. Find the angle between the crossing edges of a regular tetrahedron.

Option 2

1. In a cube A...D 1 find the angle between the lines: a) B.C. And BB 1; b) A 1C 1 and AD; V) BB 1 and BD; G) A 1D And B.C. 1.

2. In a regular triangular prism A...C 1 A.M.– median of the base ABC. Find the angle between the lines: a) A.M. And C 1B 1; b) A.M. And A 1C 1.

3. In a regular tetrahedron ABCD dot M– middle of the rib C.B.. Find the angle between the lines A.M. And DC.

4. Find the angle between non-intersecting edges of a regular triangular pyramid.

17. Perpendicularity of a line and a plane

Option 1

1. Prove that the line perpendicular to the plane, intersects this plane.

2. Through the center O square ABCD a direct line was drawn OK, perpendicular to the plane of this square. Prove that the line A.K. perpendicular to a straight line BD.

3. Find the locus of points belonging to lines passing through a given point and perpendicular to a given line.

4. Point M belongs to the side face ABD triangular pyramid ABCD, in which AB =BD And AC =CD. Construct a section of this pyramid with a plane passing through the point M and perpendicular to the line AD.

Option 2

1. Direct a, perpendicular to plane a, intersects this plane at the point A. Prove that the line b, passing through the point A and perpendicular to the line a, lies in plane a.

2. Through the point M– middle of the side AB equilateral triangle ABC a direct line was drawn M.H., perpendicular to the plane of this triangle. Prove perpendicularity of lines AB And HC.

3. Given a straight line a and a point that does not belong to it A. Find the locus of lines passing through a point A and perpendicular to the line a.

4. In a rectangular parallelepiped A...D 1 construct a section passing through a point K, internal point diagonal section A.A. 1C 1C, and perpendicular to the line BB 1.

18. Perpendicular and oblique

Option 1

1. Given a plane a. From point A two inclined AB= 20 cm and A.C.= 15 cm. The projection of the first inclined plane onto this plane is 16 cm. Find the projection of the second inclined one.

2. From a point M, not belonging to the plane g, equal inclined slopes are drawn to it MA,M.B. And M.C.. Prove that the bases of the inclined ones belong to the same circle. Find its center.

3. From a point B two equal 2 cm inclined planes are drawn to plane b. The angle between them is 600, and between their projections is 900. Find the perpendicular dropped from the point B to plane b.

4. Given a triangle with sides 13 cm, 14 cm and 15 cm. Point M, not belonging to the plane of this triangle, is 5 cm away from the sides of the triangle. Find the perpendicular dropped from the point M to the plane of the given triangle.

Option 2

1. From a point A drawn to plane a inclined AB= 9 cm and perpendicular A.O.= 6 cm. Find the projection of this perpendicular onto the given inclined one.

2. Find the locus of points in space equidistant from all points on a given circle.

3. From a given point, two equal inclined slopes are drawn to a given plane, forming an angle of 600 between themselves. The angle between their projections is a straight line. Find the angle between each oblique and its projection.

4. Point M is removed from each vertex of a regular triangle by cm, and from each side by 2 cm. Find the perpendicular dropped from the point M to the plane of the triangle.

19. Angle between a straight line and a plane

Option 1

1. In a pyramid, the lateral ribs are equally inclined to the plane of the base. At what point is the top of the pyramid projected?

2. In a cube A...D A.A. 1 and plane AB 1D 1.

3. An inclined line is drawn to plane a M.H. (H belongs to plane a). Prove that if the projection is oblique M.H. forms equal angles with right angles A.H. And B.H., lying in the plane a, then the inclined M.H. forms equal angles with them.

4. Draw a straight line to the given plane through a given point on it, forming an angle of 900 with the plane.

Option 2

1. Prove that in a regular pyramid the lateral edges are equally inclined to the plane of the base.

2. In a cube A...D 1 find the cosine of the angle between the edge A 1D 1 and plane AB 1D 1.

3. An inclined line is drawn to plane b B.P. (P belongs to plane b), which forms equal angles with right angles P.E. And PF, lying in plane b. Prove that angles formed by straight lines P.E. And PF with oblique projection B.P. on the plane b are equal.

4. Through a point not belonging to a given plane, draw a straight line forming an angle of 900 with the plane.

20. Distance between points, lines and planes

Option 1

1. In a right triangle ABC(DIV_ADBLOCK16">

4. In a cube A...D 1 with rib a AB And B 1C 1.

Option 2

1. Legs of a right triangle ABC(C= 900) are equal to 15 cm and 20 cm. From the top C a perpendicular is drawn to the plane of the triangle CD equal to 5 cm. Find the distance from the point D to the hypotenuse AB.

2. In a unit cube A...D 1 find the distance between the vertex D 1 and: a) top B; b) edge AB; c) edge BB 1C 1C.

3. From a point K a perpendicular of length d and two inclined ones are drawn, the angles of which with the perpendicular are 300. The angle between the inclined ones is 600. Find the distance between the bases of the inclined ones.

4. In a cube A...D 1 with rib a find the distance between crossing edges DC And BB 1.

21. Dihedral angle

Option 1

a. Find orthogonal projection this inclined onto the plane if the angle between the inclined and the plane is 300.

2. Two points are taken on one face of a dihedral angle A And B. Perpendiculars are omitted from them A.A. 1, BB 1 to the other side and A.A. 2, BB 2 per edge of the dihedral angle. Find BB 2 if A.A. 1 = 6 cm, BB 1 = 3 cm, A.A. 2 = 24 cm.

3. Two equal rectangle have common side and their planes form an angle of 450. Find the ratio of the areas of two figures into which the orthogonal projection of the side of one rectangle divides the other.

4. Prove that the perpendiculars drawn from the points of a given line onto a plane lie in the same plane and the geometric location of the bases of these perpendiculars is the line of intersection of these planes.

Option 2

1. The inclined line drawn to the plane is equal to a. Find the orthogonal projection of this inclined plane onto the plane if the angle between the inclined and the plane is 600.

2. Two points are taken on one face of a dihedral angle, spaced 9 cm and 12 cm from its edge. The distance from the first point to the other face of the dihedral angle is 20 cm. Find the distance from this face to the second point.

3. Two isosceles triangle have common ground, and their planes form an angle of 600. The common base is 16 cm, side one triangle is 17 cm, and the sides of the other are perpendicular. Find the distance between the vertices of triangles lying opposite a common base.

4. Prove that the point of intersection of the orthogonal projections of two lines onto a plane is the orthogonal projection of the point of intersection of these lines onto the same plane.

22. Perpendicularity of planes

Option 1

1. Given a cube A...D 1. Prove the perpendicularity of the planes: a) ABD And DCC 1; b) AB 1C 1 and ABB 1.

2. Through a given line lying in a given plane, draw a plane perpendicular to this plane.

In a regular triangular pyramid SABC with top S the angle between the side edge and
base plane is equal to 60°, the side of the base is equal 1 , SH- height of the pyramid.
Find the cross-sectional area of ​​the pyramid by a plane passing through the point N
parallel to the ribs S.A. And B.C..

The base of the height of a regular pyramid is the center of the triangle ABC. First we will conduct
through the point N line segment RT, parallel to the edge Sun. Points P and T belong to the section.

In the plane of the face ACS through the point T let's draw a segment TK parallel to the edge AS.

In the plane of the face ABC through the point R let's draw a segment P.L. parallel to the edge AS.

Connecting the dots TO And L, we obtain the desired section. Let's prove that this is a rectangle.

Segments TK And P.L. not only parallel (each is parallel AS), but also equal.

So it's a quadrilateral KLPT- parallelogram on the basis of a parallelogram.
Besides, TK ⊥ TR, because AS⊥CB, and the sides TK And TR parallel AS And C.B..
Let's prove that AS⊥CB. You can use the three perpendicular theorem.
AS- inclined, AD projection of this oblique onto ABC, AD⊥CB, Means, AS⊥CB.

To find the area of ​​a rectangle, you need to find and multiply its sides.
Note that the side TR is two-thirds of the side of the base BC = 1.
Second side of the rectangle TK one third of the lateral rib AS.
We can find the side edge from the triangle SAH, in which ∠SAH = 60°
(angle between the side edge and the base) and ∠ASH = 30°, which means AS = 2·AN.

Find the length of the segment AN, knowing the side of the base, you can do it in different ways.
It’s better to do without formulas and consider right triangle ANF.

Let's go back to the triangle SAH and we'll find side rib pyramids:

It remains to multiply the found sides and obtain the cross-sectional area.

§ 2. INDEPENDENT WORK

1. Basic concepts and axioms of stereometry

Independent work N 1

Option 1

1. Draw a straight line a and dots A, B And C, not belonging to this line. Make the necessary notes.

2. Draw plane b, points E, F, belonging to her, period G, which does not belong to her. Make the necessary notes.

3. Draw a straight line a, lying in plane a. Make the necessary entry.

4. Draw two intersecting planes a and b. Make the necessary entry.

Option 2

1. Draw two intersecting at a point O straight a And b and dots A, B, C, and point A belongs to the line a, B belongs to the line b, dot C does not belong to the given lines.

2. Draw the plane g and points not belonging to it K, L and the point belonging to it M. Make the necessary notes.

3. Draw a straight line b, intersecting the plane b at the point O. Make the necessary entry.

4. Draw three intersecting lines a planes a, b and g. Make the necessary entry.

Independent work N 2

Option 1

1) The angles at the base of an isosceles triangle are equal.

2) A single straight line passes through two points in space.

3) Vertical angles are equal.

4) A parallelogram is a quadrilateral whose opposite sides are parallel in pairs.

2. Determine the relative position of planes a and b if they contain a triangle ABC. Justify your answer.

3. How many planes can pass through three points?

4. Find the greatest number of lines passing through different pairs of four points.

Option 2

1. From the following sentences, indicate axioms, definitions, theorems:

1) If two planes have a common point, then they intersect in a straight line.

2) The midline of a triangle is the segment connecting the midpoints of its two sides.

3) For straight lines and planes in space, the axioms of planimetry are satisfied.

4) The diagonals of a parallelogram are divided in half by the point of intersection.

2. Determine the relative position of two planes b and g if they contain points B And C. Justify your answer.

3. Find the greatest number of lines passing through different pairs of 5 points.

4. Find the greatest number of planes passing through different triplets of four points.

2. Corollaries from the axioms of stereometry

Option 1

1. In the plane of two intersecting lines a And b point given C, not belonging to these lines. Straight c, lying in a given plane, passes through a point C. How can a straight line be positioned? c relative to these straight lines?

2. Given three points that do not belong to the same line. Prove that all lines intersecting two of the three segments connecting these points lie in the same plane.

3. The plane is given by a straight line c and a point not belonging to it C a, different from the given line and not passing through the given point.

4. A plane is defined by two intersecting at a point O straight a And b. Draw a straight line c, which intersects these lines and does not lie in the given plane.

Option 2

1. Direct d, lying in the plane of the triangle ABC, crosses his side AB. What could be the relative position of the lines? d And B.C.?

2. Two parallel lines are drawn in plane a a And b. Prove that all lines intersecting these lines lie in the same plane.

3. A plane is defined by two intersecting at a point O straight m And n. Construct a straight line in this plane k, different from the given lines and not passing through the point O.

4. The plane is defined by three points D, E, F, not belonging to the same line. Draw a straight line a, which intersects the sides DE And DF triangle DEF and does not lie in this plane.

3. Spatial figures

Option 1

1. Draw a pentagonal prism and divide it into tetrahedrons.

2. Determine the number of vertices, edges and faces: a) cube; b) 7-gonal prism; V) n-coal pyramid.

3. Determine the type of prism if it has: a) 10 vertices; b) 21 ribs; c) 5 faces.

4. How can the faces of a 4-gonal prism be colored so that neighboring (sharing a common edge) faces are painted in different colors? What is the smallest number of colors needed?

Option 2

1. Draw a pentagonal pyramid and divide it into tetrahedrons.

2. Determine the number of vertices, edges and faces: a) rectangular parallelepiped; b) 6-sided pyramid; V) n- carbon prism.

3. Determine the type of pyramid if it has: a) 5 vertices; b) 14 ribs; c) 9 faces.

4. How can the faces of an octahedron be colored so that neighboring (sharing a common edge) faces are painted in different colors. What is the smallest number of colors needed?

4. Modeling polyhedra

Option 1

1. Draw several nets of the cube.

2. Draw a figure consisting of four equal equilateral triangles, which is not a net of a regular tetrahedron.

3. Draw the development of a regular quadrangular pyramid and color it in such a way that when gluing adjacent faces have different colors. What is the smallest number of flowers you need to take?

4. Draw a development of a rectangular parallelepiped and color it in such a way that when gluing adjacent faces have different colors. What is the smallest number of flowers you need to take?

Option 2

1. Draw several nets of a regular tetrahedron.

2. Draw a figure consisting of six squares that is not a net of a cube.

3. Draw a cube development and color it in such a way that when gluing adjacent faces have different colors. What is the smallest number of flowers you need to take?

4. Draw the development of a regular 6-gonal pyramid and color it in such a way that when gluing adjacent faces have different colors. What is the smallest number of flowers you need to take?

5. Parallelism of lines in space

Option 1

1. Write in a regular 4-gonal pyramid SABCD all pairs of parallel edges.

2. In the plane of two parallel lines a And b given point C, not belonging to these lines. Through the point C a direct line was drawn c. How can a straight line be positioned? c relative to straight lines a And b.

3. Through a point that does not belong to a given line, draw a line parallel to this one.

4. Find the locus of the lines intersecting two given parallel lines.

Option 2

1. Write four pairs of parallel edges of the cube A… D 1 .

2. Given three lines a, b And With. How can these straight lines be positioned so that a plane can be drawn containing all these straight lines?

3. Given two parallel lines a And b. Prove that any plane intersecting one of them will also intersect the other.

4. Find the locus of lines parallel to a given line and intersecting another line intersecting the first.

6. Crossing lines

Option 1

1. In a cube A… D 1 write down the edges crossing the edge AB.

2. Write pairs of crossing edges of a 4-gonal pyramid SABCD.

3. How are the lines located relative to each other? a And b in Figure 1? Justify your answer.

4. Given two skew lines a And b and a point that does not belong to them C. Construct a straight line c, passing through the point C and intersecting the lines a And b.

Option 2

1. Write down the edges that intersect with the edge S.A. regular 4-gonal pyramid SABCD.

2. Write down the edges that intersect the diagonal B 1 D Cuba A…D 1 .

c(Fig. 1). Straight a lies in plane a and intersects the line c. Is it possible to draw a line parallel to the line in plane b? a? Justify your answer.

4. Are there two parallel lines, each of which intersects two given skew lines? Justify your answer.

7. Parallelism of a straight line and a plane

Option 1

1. Write down the edges parallel to the plane of the face CC 1 D 1 D correct prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1 .

2. Direct a parallel to plane a; straight b intersects plane a at point B; straight c, intersecting the lines a And b respectively at points E And F, intersects plane a at point C. Make a drawing. How can straight lines be positioned relative to each other? a And b?

3. Planes a and b intersect in a straight line c. Dot A belongs to plane a, point B– plane b. Construct: a) a straight line a, lying in the plane a, passing through the point A and parallel to plane b; b) straight b, lying in the b plane passing through the point B and parallel to plane a. How will the straight lines be positioned relative to each other? a And b?

4. Points A And B belong to adjacent lateral faces of the pyramid. Draw two segments parallel to each other through these points on these faces.

Option 2

1. Write down the planes of the faces parallel to the edge CC 1 parallelepiped A… D 1 .

2. Direct a parallel to plane a; straight b And c, intersecting the line a respectively at points B And C, intersect the plane a respectively at the points D And E. Make a drawing. How can straight lines be positioned relative to each other? a And b?

3. Planes a and b intersect in a straight line c. Straight a lies in plane a. Prove that if: a) a intersects plane b at point A, That A belongs to the line c; b) a is parallel to plane b, then it is parallel to the line c.

4. Points A And B belong to adjacent lateral faces of the prism. Draw two segments parallel to each other through these points on these faces.

8. Parallelism of two planes

Option 1

1. Write down the parallel planes of the parallelepiped A… D 1 .

2. Are the statements true:

1) Through a point not belonging to a given plane, there passes a single plane parallel to the given one.

2) If two lines lying in one plane are respectively parallel to two lines lying in another plane, then these planes are parallel.

3) There are infinitely many lines parallel to a given plane and passing through a point not belonging to this plane.

4) If one of two given planes is parallel to two intersecting lines lying in the other plane, then these planes are parallel.

3. Prove that two planes parallel to the same third plane are parallel to each other.

4. Segments AB And CD lie in parallel planes a and b, respectively (Fig. 2). How can straight lines be positioned relative to each other? A.C. And BD? Can they be parallel?

Option 2

1. In a triangular pyramid SABC draw a plane parallel to its base ABC.

2. Are the statements true:

1) If a line lying in one plane is parallel to a line lying in another plane, then these planes are parallel.

2) If a plane intersects two given planes along parallel lines, then these planes are parallel.

3) There are infinitely many planes parallel to a given line and passing through a point not belonging to this line.

4) If two planes are parallel to the same line, then they are parallel.

3. Prove that if a plane intersects one of two parallel planes, then it also intersects the other.

4. Segments AB And CD lie in parallel planes a and b, respectively (Fig. 3). How can straight lines be positioned relative to each other? AD And B.C.? Can they intersect?

9. Vectors in space

Option 1

1. For a given vector
construct the vectors: a) - ; b) 2; V) - .

2. How many vectors are defined by all possible pairs of points made up of the vertices of a regular quadrangular pyramid?

ABCD and draw a vector: a)
; b)
; V)
.

4. Given a parallelepiped A… D
; b)
; V)
.

Option 2

1. For a given vector construct the vectors: a) 3 ; b) -2; V) .

2. How many vectors are defined by all possible pairs of points made up of the vertices of a triangular prism?

3. Draw a regular tetrahedron ABCD and draw a vector: a)
; b)
; V)
.

4. Given a parallelepiped A… D 1 . Find the sum of vectors: a)
; b)
; V) .

10. Collinear and coplanar vectors

Option 1

to get the vector , identically directed with and | |=1.

2. Given two oppositely directed vectors and , and | | > | |. Find the direction and length of the vector +.

3. Given a tetrahedron ABCD. Write down three pairs of its vertices that define coplanar vectors.

4. Given a cube A… D 1 . Write triplets of non-coplanar vectors with beginnings and ends at its vertices.

Option 2

1. By what number should a non-zero vector be multiplied? to get the vector , oppositely directed with and | |=2.

2. Given two oppositely directed vectors and , and | | |. Find the direction and length of the vector +.

3. Given a tetrahedron ABCD. Write down three pairs of its vertices that define non-coplanar vectors.

4. Given a cube A… D 1 . Write triplets of coplanar vectors with beginnings and ends at its vertices.

11. Parallel transfer

Option 1

1. Construct a figure that is obtained by parallel translation of a line a to vector
, if: a) E belongs a, F do not belong a; b) points E And F don't belong a.

2. Specify a parallel translation, which is the middle of the segment G.H. translates to some point M.

3. Construct a figure that is obtained from a square ABCD parallel transfer to a vector: a)
; b) .

ABCD parallel transfer to a vector.

Option 2

1. Construct a figure that is obtained by parallel translation of a circle with the center at the point O to vector
, if: a) point K belongs to the circle; b) point K does not belong to the circle.

2. Specify the parallel translation, which is the intersection point O two straight lines a And b translates to some point N.

3. Construct a figure that is obtained from a regular triangle ABC parallel transfer to a vector: a) ; b) where is the point M– middle of the side B.C..

4. Construct a figure that is obtained from a tetrahedron ABCD parallel transfer to a vector
.

12. Parallel design

Option 1

1. How many points will be obtained by parallel projection of two different points in space? Make appropriate drawings and justification.

2. List the properties of a rectangle that are preserved during parallel design.

3. How should two straight lines be positioned so that they are projected onto a plane into a straight line and a point not belonging to this straight line?

4. Parallel lines a And b A, B And C are shown in Figure 4. Draw the fourth point D. Justify your answer.

Option 2

1. How many points will you get when designing three different points in space? Make appropriate drawings and justification.

2. List the properties of a rhombus that are preserved during parallel design.

3. How should a line and a point be located so that they are projected onto a plane into a line and a point belonging to this line?

4. Intersecting lines a And b intersect parallel planes a and b at four points. Three of them A, B And C are shown in Figure 5. Draw the fourth point D. Justify your answer.

13. Parallel projections of plane figures

Option 1

1. Draw a parallel projection of a right isosceles triangle lying in a plane parallel to the projection plane.

2. Draw a parallel projection of an equilateral triangle ABC and on it construct images of perpendiculars dropped from the point M– middle of the side AB to the sides A.C. And B.C..

ABCDEF, taking a rectangle as the original figure ABDE.

4. Draw a parallel projection of an equilateral triangle ABC and construct on it an image of a perpendicular drawn from the point K– midpoint of the segment B.O.(O– center of the triangle) to the side AB.

Option 2

1. Draw a parallel projection of an equilateral triangle lying in a plane parallel to the projection plane.

2. Draw a parallel projection of a square ABCD and on it construct an image of perpendiculars drawn from the point E– middle of the side B.C. to straight lines BD And A.C..

3. Draw a parallel projection of a regular hexagon ABCDEF, taking as the initial figure an equilateral triangle ACE.

4. Draw a parallel projection of a rectangle ABCD, which one AD = 2AB. Construct an image of a perpendicular dropped from a vertex C to the diagonal BD.

14. Image of spatial figures

Option 1

1. Draw a regular quadrangular pyramid and its height.

2. Draw a cube whose two faces are parallel to the design plane.

3. Figure 6 shows a parallel projection of a cube A… D

4. Given a tetrahedron ABCD. The area of ​​its face ADC equal to S BDC to the plane ADC in the direction of a straight line AB.

Option 2

1. Draw a regular triangular pyramid and its height.

2. Draw a cube whose faces are not parallel to the design plane.

3. Figure 7 shows a parallel projection of a cube A… D 1 . How is the cube located relative to the design plane?

4. Given a tetrahedron ABCD. The area of ​​its face ABD equal to Q. Find the projection area of ​​its face BDC to the plane A.D.B. in the direction of a straight line C.M., Where M– middle of the rib AB.

15. Sections of polyhedra

Option 1

1. In a hexagonal prism A… F 1 (Fig. 8) construct the point of intersection of the line PQ with plane ABC, where the points Q And P belong respectively to the lateral edges of the prism BB 1 and DD 1 .

2. On the side edges of a quadrangular prism A… D 1 three points are given K, L, M(Fig. 9). Construct a plane intersection line KLM with plane ABC.

3. Construct a section of the cube with a plane passing through the points X, Y, Z, belonging respectively to the edges AD, A.A. 1 , BB 1 and such that AX:XD = 1:2, A 1 Y:YA= 2:1, B 1 Z:ZB = 1:2.

4. In the right pyramid SABCD construct a section passing through the side of the base AD and period M, belonging to the side edge S.B..

Option 2

1. On the side ribs BB 1 and E.E. 1 prism ABCDEA 1 B 1 C 1 D 1 E 1 points are given accordingly F And G(Fig. 10). Construct the point of intersection of the line FG with plane ABC.

2. Given a cube A… D 1 . On his ribs A.A. 1 , CC 1 and DD 1 three points are given respectively X, Y, Z(Fig. 11). Construct the line of intersection of the planes XYZ And ABC.

3. In a regular triangular prism A… C 1 construct a section passing through the points K, L And M, belonging respectively to the edges A.A. 1 , A.C. And BB 1 and such that: A.K. = K.A. 1 ; AL:L.C. = 1:2 and B.M. = M.B. 1 .

4. In the right pyramid SABCD construct a section passing through the diagonal A.C. base and parallel to the lateral edge SD.

16. Angle between straight lines in space. Perpendicularity of lines

Option 1

1. In a cube A… D AB And BB 1 ; b) BD And BB 1 ; V) AB 1 and CC 1 ; G) AB 1 and CD 1 .

A… C 1 segment CD perpendicular to the edge AB. Find the angle between the lines: a) CD And A.A. 1 ; b) CD And A 1 B 1 .

SABCD with equal edges find the angle between the diagonal A.C. base and side edge S.C..

4. Find the angle between the crossing edges of a regular tetrahedron.

Option 2

1. In a cube A… D 1 find the angle between the lines: a) B.C. And BB 1 ; b) A 1 C 1 And AD; V) BB 1 and BD; G) A 1 D And B.C. 1 .

2. In a regular triangular prism A… C 1 A.M.– median of the base ABC. Find the angle between the lines: a) A.M. And C 1 B 1 ; b) A.M. And A 1 C 1 .

3. In a regular tetrahedron ABCD dot M– middle of the rib C.B.. Find the angle between the lines A.M. And DC.

4. Find the angle between non-intersecting edges of a regular triangular pyramid.

17. Perpendicularity of a line and a plane

Option 1

1. Prove that a line perpendicular to a plane intersects this plane.

2. Through the center O square ABCD a direct line was drawn OK, perpendicular to the plane of this square. Prove that the line A.K. perpendicular to a straight line BD.

3. Find the locus of points belonging to lines passing through a given point and perpendicular to a given line.

4. Point M belongs to the side face ABD triangular pyramid ABCD, in which AB = BD And A.C. = CD. Construct a section of this pyramid with a plane passing through the point M and perpendicular to the line AD.

Option 2

1. Direct a, perpendicular to plane a, intersects this plane at the point A. Prove that the line b, passing through the point A and perpendicular to the line a, lies in plane a.

2. Through the point M– middle of the side AB equilateral triangle ABC a direct line was drawn M.H., perpendicular to the plane of this triangle. Prove perpendicularity of lines AB And HC.

3. Given a straight line a and a point that does not belong to it A. Find the locus of lines passing through a point A and perpendicular to the line a.

4. In a rectangular parallelepiped A… D 1 construct a section passing through a point K, internal point of the diagonal section A.A. 1 C 1 C, and perpendicular to the line BB 1 .

18. Perpendicular and oblique

Option 1

1. Given a plane a. From point A two inclined AB= 20 cm and A.C.= 15 cm. The projection of the first inclined plane onto this plane is 16 cm. Find the projection of the second inclined one.

2. From a point M, not belonging to the plane g, equal inclined slopes are drawn to it M.A., M.B. And M.C.. Prove that the bases of the inclined ones belong to the same circle. Find its center.

3. From a point B two equal 2 cm inclined planes are drawn to plane b. The angle between them is 60 0, and between their projections – 90 0. Find the perpendicular dropped from the point B to plane b.

4. Given a triangle with sides 13 cm, 14 cm and 15 cm. Point M, not belonging to the plane of this triangle, is 5 cm away from the sides of the triangle. Find the perpendicular dropped from the point M to the plane of the given triangle.

Option 2

1. From a point A drawn to plane a inclined AB= 9 cm and perpendicular A.O.= 6 cm. Find the projection of this perpendicular onto the given inclined one.

2. Find the locus of points in space equidistant from all points on a given circle.

3. From a given point, two equal inclined slopes are drawn to a given plane, forming an angle of 60 0 between them. The angle between their projections is straight. Find the angle between each oblique and its projection.

4. Point M distance from each vertex of a regular triangle by
cm, and from each side - 2 cm. Find the perpendicular dropped from the point M to the plane of the triangle.

19. Angle between a straight line and a plane

Option 1

1. In a pyramid, the lateral ribs are equally inclined to the plane of the base. At what point is the top of the pyramid projected?

2. In a cube A… D A.A. 1 and plane AB 1 D 1 .

3. An inclined line is drawn to plane a M.H. (H belongs to plane a). Prove that if the projection is oblique M.H. forms equal angles with right angles A.H. And B.H., lying in the plane a, then the inclined M.H. forms equal angles with them.

4. Draw a straight line to a given plane through a given point on it, forming an angle of 90 0 with the plane.

Option 2

1. Prove that in a regular pyramid the lateral edges are equally inclined to the plane of the base.

2. In a cube A… D 1 find the cosine of the angle between the edge A 1 D 1 and plane AB 1 D 1 .

3. An inclined line is drawn to plane b B.P. (P belongs to plane b), which forms equal angles with right angles P.E. And PF, lying in plane b. Prove that angles formed by straight lines P.E. And PF with oblique projection B.P. on the plane b are equal.

4. Through a point not belonging to a given plane, draw a straight line forming an angle of 90 0 with the plane.

20. Distance between points, lines and planes

Option 1

1. In a right triangle ABC(
C= 90 0) leg A.C. equal to 8 cm. From the top B a perpendicular is drawn to the plane of this triangle BD. Distance between points A And D equals 10 cm D to the leg A.C..

2. In a unit cube A… D A and: a) the top C 1 ; b) edge CC 1 ; c) edge BB 1 C 1 C.

3. Point M distance from all vertices of a right triangle a. The hypotenuse of a triangle is equal to c. Find the distance from the point M to the plane of the given triangle.

4. In a cube A… D 1 with rib a AB And B 1 C 1 .

Option 2

1. Legs of a right triangle ABC(C= 90 0) are equal to 15 cm and 20 cm. From the top C a perpendicular is drawn to the plane of the triangle CD equal to 5 cm. Find the distance from the point D to the hypotenuse AB.

2. In a unit cube A… D 1 find the distance between the vertex D 1 and: a) the top B; b) edge AB; c) edge BB 1 C 1 C.

3. From a point K a perpendicular of length d and two inclined ones are drawn, the angles of which with the perpendicular are 30 0. The angle between the inclined ones is 60 0. Find the distance between the bases of the inclined ones.

4. In a cube A… D 1 with rib a find the distance between crossing edges DC And BB 1 .

21. Dihedral angle

Option 1

a. Find the orthogonal projection of this inclined plane onto the plane if the angle between the inclined and the plane is 30 0.

2. Two points are taken on one face of a dihedral angle A And B. Perpendiculars are omitted from them A.A. 1 , BB 1 to the other side and A.A. 2 , BB 2 per edge of the dihedral angle. Find BB 2 if A.A. 1 = 6 cm, BB 1 = 3 cm, A.A. 2 = 24 cm.

3. Two equal rectangles have a common side and their planes form an angle of 45 0. Find the ratio of the areas of two figures into which the orthogonal projection of the side of one rectangle divides the other.

4. Prove that the perpendiculars drawn from the points of a given line onto a plane lie in the same plane and the geometric location of the bases of these perpendiculars is the line of intersection of these planes.

Option 2

1. The inclined line drawn to the plane is equal to a. Find the orthogonal projection of this inclined plane onto the plane if the angle between the inclined and the plane is 60 0.

2. Two points are taken on one face of a dihedral angle, spaced 9 cm and 12 cm from its edge. The distance from the first point to the other face of the dihedral angle is 20 cm. Find the distance from this face to the second point.

3. Two isosceles triangles have a common base, and their planes form an angle of 60 0. The common base is 16 cm, the side of one triangle is 17 cm, and the sides of the other are perpendicular. Find the distance between the vertices of triangles lying opposite a common base.

4. Prove that the point of intersection of the orthogonal projections of two lines onto a plane is the orthogonal projection of the point of intersection of these lines onto the same plane.

22. Perpendicularity of planes

Option 1

1. Given a cube A… D ABD And DCC 1 ; b) AB 1 C 1 and ABB 1 .

2. Through a given line lying in a given plane, draw a plane perpendicular to this plane.

3. Two perpendicular planes a and b intersect in a straight line AB. Straight CD lies in plane a, parallel AB and is located at a distance of 60 cm from it. Dot E belongs to plane b and is located at a distance of 91 cm from AB. Find the distance from the point E to a straight line CD.

4. Prove that the line a and plane a, perpendicular to the same plane b, are parallel if the straight line a does not lie in plane a.

Option 2

1. Given a cube A… D 1 . Prove the perpendicularity of the planes: a) A.A. 1 D 1 And D 1 B 1 C 1 ; b) A 1 B 1 D And BB 1 C 1 .

2. Through the inclined plane, draw a plane perpendicular to this plane.

3. Segment MN has ends on two perpendicular planes and makes equal angles with them. Prove that the points M And N equally distant from the line of intersection of these planes.

4. Prove that two planes a and b are parallel if they are perpendicular to the plane g and intersect it along parallel lines.

23*. Central design

Independent work N 1

Option 1

1. During central design, where does the straight line parallel to the design plane go?

2. Flat figure lies in a plane parallel to the design plane and is between the center and the design plane. How is the coefficient of similarity between a figure and its projection determined?

R. A plane parallel to the base is drawn through the middle of the height. Find the cross-sectional area.

4. In a triangular pyramid ABCD(Fig. 12) through points M And N, belonging respectively to the faces ABD And BCD, draw a section parallel to the edge A.C..

Option 2

1. In what case will the central projection of two lines be two parallel lines?

2. A plane figure lies in a plane parallel to the projection plane. The design plane is located between the design center and the plane of the given figure. How is the coefficient of similarity between a figure and its projection determined?

3. The radius of the base of the cone is equal to R. It is intersected by a plane parallel to the base and dividing the height of the cone in relation m:n, counting from the top. Find the cross-sectional area.

4. In a triangular pyramid ABCD(Fig. 13) through a point M, belonging to the height of the pyramid DO, draw a section parallel to the face BCD.

Independent work N 2

Option 1

1. Direct mS. Draw the central projection of a part of a given line located in the same half-space with the point S relative to plane p.

A… D A.A. 1 C 1 .

3. Draw the central projection of a regular hexagonal prism onto a plane parallel to its bases.

4. Given a regular quadrangular pyramid SABCD, whose dihedral angle at the base is equal to 60 0. Find the distance between the lines AB And S.C., If AB= 1.

Option 2

1. Direct m intersects the design plane p and does not pass through the design center S. Draw the central projection of a part of a given line located in different half-spaces with a point S relative to plane p.

2. Draw the central projection of the cube A… D 1 on a plane parallel to the plane AB 1 C 1 .

3. Draw the central projection of a regular hexagonal prism onto a plane not parallel to its bases.

4. Given a regular triangular prism A… C 1, all edges of which are equal to 1. Find the distance between the lines A.A. 1 and B.C. 1 .

24. Polyhedral angles

Option 1

1. Write down under what conditions angles a, b and g can be plane angles of a trihedral angle.

2. B trihedral angle all plane angles are right angles. On its edges, segments of 2 cm, 4 cm, 6 cm are laid out from the top and a plane is drawn through their ends. Find the area of ​​the resulting section.

3. Along how many lines do the planes of all faces of a tetrahedral angle intersect in pairs?

Option 2

1. Two plane angles of a trihedral angle are equal to a and b, and a > b. Write down the limits within which the values ​​of the third plane angle g of a given trihedral angle are possible.

2. In a trihedral angle everything dihedral angles– straight. From the vertex of this angle in its internal region a segment is drawn whose projections onto the edges are equal a, b And c. Find this segment.

3. Along how many lines do the planes of all faces of a pentahedral angle intersect in pairs?

25*. Convex polyhedra

Option 1

n-coal prism: a) convex; b) non-convex.

2. Draw a convex polyhedron with 5 vertices.

3. In a convex polyhedron, the number of faces Г is known, and each face has the same number of sides n. Find the number of: a) plane angles (
); b) edges (P) of a given polyhedron. How are numbers and P related?

4. Convex polyhedron has B vertices, P edges and G faces. They cut him off m-faceted angle. Find the number of vertices, edges and faces of the resulting polyhedron.

Option 2

1. Determine the number of vertices (B), edges (P) and faces (D) n-coal pyramid: a) convex; b) non-convex.

2. Draw a convex polyhedron with 6 vertices.

3. In a convex polyhedron, the number of vertices B is known, and the same number of edges converge at each vertex m. Find the number of: a) plane angles (); b) edges of a given polyhedron (P). How are numbers and P related?

4. A convex polyhedron has B vertices, P edges and T faces. To his n- a pyramid was built on the coal face. Find the number of vertices, edges and faces of the new polyhedron.

26*. Euler's theorem

Option 1

1. Draw a non-convex polyhedron for which Euler’s theorem holds.

3. Prove that in any convex polyhedron with B vertices, P edges and G faces the following inequality holds: 3B – 6 R.

4. Find the side of the base of a regular triangular pyramid with height h and side edge b.

Option 2

1. Draw a non-convex polyhedron for which Euler’s theorem does not hold.

2. Prove that for any convex polyhedron the relation is true

3. Prove that in any convex polyhedron with B vertices, P edges and G faces the following inequality holds: 3G – 6 P.

4. Find the height of a regular triangular pyramid with the base side a and side edge height h.

27. Regular polyhedra

Option 1

1. Draw: a) the development of a tetrahedron; b) a polyhedron dual to a hexahedron.

2. Construct a section of the octahedron with a plane passing through one of its vertices and the midpoints of two parallel edges to which this vertex does not belong. Determine the type of section.

3. Into a tetrahedron ABCD a regular triangular prism with equal edges is inscribed in such a way that the vertices of one of its bases are on the side edges AD, BD, CD, and the other - in the plane ABC. The edge of a tetrahedron is a. Find the edge of the prism.

4. In a tetrahedron ABCD M– mid-height DO tetrahedron, parallel to the plane of the face ADC. Determine the type of section.

Option 2

1. Draw: a) the development of a cube; b) a polyhedron dual to a tetrahedron.

2. Construct a section of the octahedron with a plane passing through two of its parallel edges. Determine the type of section.

3. A cube is inscribed in an octahedron in such a way that its vertices are on the edges of the octahedron. The edge of the octahedron is a. Find the edge of the cube.

4. In a tetrahedron ABCD draw a section with a plane passing through the point M, belonging to the face ABC parallel to the face plane BCD. Determine the type of section.

28*. Semiregular polyhedra

Option 1

1. Find the number of vertices (B), edges (P) and faces (D) of a truncated hexahedron.

2. How can one obtain a 5-gonal antiprism?

3. Draw a polyhedron dual to a regular hexagonal prism.

4. Regular triangle ABC and another triangle ADC have a common side A.C. and are located in different planes, the angle between which is 30 0. Vertex D orthogonally projected onto the plane of the triangle ABC to its center. The height of a regular triangle is h. Find the side AD triangle ADC.

Option 2

1. Find the number of vertices (B), edges (P) and faces (D) of the truncated octahedron.

2. How can one obtain an octagonal antiprism?

3. Draw a polyhedron dual to the 6-gonal antiprism.

4. Square ABCD and triangle ABE have a common side AB and are located in different planes, the angle between which is 45 0. Vertex E triangle is orthogonally projected onto the plane of the square at its center O. Height E.H. triangle is equal h. Find the area of ​​the orthogonal projection of the triangle onto the plane of the square and the orthogonal projection of the segment O.E. to the plane of the triangle.

29*. Star polyhedra

Option 1

1. How to get a Kepler star from an octahedron?

2. Find the number of vertices (B), edges (P) and faces (D) of the small stellated dodecahedron.

3. How is a truncated cube obtained from a cube? What is its edge equal to if the edge of the cube is equal to a?

4. Prove that if a plane intersects a triangular pyramid and is parallel to its two intersecting edges, then the section will be a parallelogram.

Option 2

1. How to get a Kepler star from a hexahedron?

2. Find the number of vertices (B), edges (P) and faces (D) of the great dodecahedron.

3. How is a cuboctahedron obtained from a cube? What is its edge equal to if the edge of the cube is equal to a?

4. Prove that a regular tetrahedron can be intersected by a plane in such a way that the cross section results in a square.

thirty*. Crystals – natural polyhedrons

Option 1

1. Draw a rock crystal.

2. Draw a rhombic dodecahedron. What is the number of its vertices, edges and faces?

3. Find the sum of all planar angles of an Iceland spar crystal.

4. Find the sum of the areas of all faces of a diamond crystal (in the form of a cuboctahedron), if its edge is equal to a.

Option 2

1. Draw an Iceland spar crystal.

2. Draw a rhombic dodecahedron. Determine the number of its plane angles, dihedral angles; polyhedral angles and their types.

3. Find the sum of all planar angles of a garnet crystal.

4. Find the sum of the areas of all faces of a diamond crystal (in the form of a truncated octahedron), if its edge is equal to a.

31. Sphere and ball. The relative position of the sphere and the plane

Option 1

1. A sphere whose radius is 10 cm is intersected by a plane located at a distance of 9 cm from the center. Find the cross-sectional area.

2. Sections of a ball of radius R r 1 and r 2. Find the distance between these planes if they are located along different sides from the center.

3. The sides of the triangle touch the sphere. Find the distance from the center of the sphere to the plane of the triangle if the radius of the sphere is 5 cm and the sides of the triangle are 12 cm, 10 cm, 10 cm.

4. Each side of a rhombus touches a sphere of radius 10 cm. The plane of the rhombus is 8 cm away from the center of the sphere. Find the area of ​​the rhombus if its side is 12.5 cm.

Option 2

1. A plane is drawn perpendicular to it through the middle of the radius of the ball. How does the area of ​​the great circle of a given ball relate to the area of ​​the resulting section?

2. Sections of a ball of radius R two parallel planes have radii r 1 and r 2. Find the distance between these planes if they are located on the same side of the center.

3. The sides of the rhombus touch a sphere of radius 13 cm. Find the distance from the plane of the rhombus to the center of the sphere if the diagonals of the rhombus are 30 cm and 40 cm.

4. A plane is drawn through the end of the radius of the ball, making 30 0 with it. Find the cross-sectional area of ​​the sphere by this plane if the radius of the sphere is 6 cm.

32. Polyhedra inscribed in a sphere

Option 1

1. List the properties that a prism must satisfy in order to describe a sphere around it.

2. Figure 14 shows a triangular pyramid ABCD, which has an edge D.B. perpendicular to the plane ABC and angle ACB equals 90 0. Find the center of the sphere described around this pyramid.

3. In a regular quadrangular pyramid SABCD base side ABCD equal to 4 cm, dihedral angle at the base 45 0. Find the radius of the circumscribed sphere. Where will its center be?

4. The radius of a sphere circumscribed about a regular quadrangular prism is equal to R. Find the height of this prism, knowing that its diagonal forms an angle a with its side face.

Option 2

1. List the properties that a pyramid must satisfy in order to describe a sphere around it.

2. Figure 15 shows a pyramid ABCD, whose angles A.D.B., ADC And BDC straight. Find the center of the sphere described around this pyramid.

3. In a regular triangular pyramid SABC the center of the circumscribed sphere divides the height into parts equal to 6 cm and 3 cm. Find the side of the base ABC pyramids.

4. In a regular 4-angled prism, the diagonal of the base and the diagonal of the side face are 16 cm and 14 cm, respectively. Find the radius of the circumscribed sphere.

33. Polyhedra described around a sphere

Option 1

1. Is it possible to inscribe a sphere in a pyramid whose dihedral angles at the base are equal? Explain your answer.

2. A straight prism is described near the sphere, the base of which is a rhombus with diagonals of 6 cm and 8 cm. Find the area of ​​the base and the height of the prism.

3. The side of the base of a regular quadrangular pyramid is equal to a, the dihedral angle at the base is 60 0. Find the radius of the inscribed sphere.

4. The sides of the bases of a regular 4-gonal truncated pyramid are 1 cm and 7 cm. The side edge is inclined to the base at an angle of 45 0. Find the radius of the circumscribed sphere.

Option 2

1. What properties should a straight line have? triangular prism, so that a sphere can be inscribed in it?

2. At the base of the pyramid lies an isosceles triangle, each of whose equal angles is equal to a and whose base is equal to a. The side faces of the pyramid are inclined to the plane of the base at an angle b. Find the radius of the sphere inscribed in this pyramid.

3. Find the radius of the ball inscribed in correct pyramid, whose height is equal h, and the dihedral angle at the base is 45 0.

4. In a regular triangular truncated pyramid, the height is 17 cm, the radii of the circumscribed circles around the bases are 5 cm and 12 cm. Find the radius of the circumscribed sphere.

34. Cylinder. Cone

Option 1

1. A section is drawn in a cylinder whose base radius is 4 cm and height 6 cm, parallel to the axis. The distance between the cross-sectional diagonal and the cylinder axis is 2 cm. Find the cross-sectional area.

2. A section is drawn through the top of the cone at an angle of 60 0 to its base. Find the distance from the center of the base of the cone to the section plane if the height of the cone is 12 cm.

3. Point M belongs to the height of the cone. Dot N belongs to the plane of the base of the cone, but is located outside this base. Construct the point of intersection of the line MN with the surface of a cone.

4. Diagonals axial section truncated cone are perpendicular, height is 2 cm. Find the cross-sectional area of ​​a truncated cone drawn through the middle of the height parallel to the bases.

Option 2

1. The height of the cylinder is 15 cm, the radius of the base is 10 cm. Given a segment whose ends belong to the circles of both bases and whose length is 3
cm. Find the distance between this segment and the axis of the cylinder.

2. A section is drawn through the top of the cone at an angle of 30 0 to its height. Find the cross-sectional area if the height of the cone is 3
cm, and the radius of the base is 5 cm.

3. The axial section is specified in the cone. Points K And L belong to two generators of the cone that do not lie in this section. Construct the point of intersection of the line KL with the plane of a given axial section.

4. The radii of the bases of a truncated cone are in the ratio 1:3, the generatrix makes an angle of 45 0 with the plane of the base, the height is h. Find the area of ​​the bases.

35. Turn. Rotation figures

Option 1

1. Draw the shape that is obtained by rotating the square ABCD around a straight line a, passing through the vertex B BD.

2. Draw a figure that is obtained by rotating a circle around a tangent.

3. The curve is given by the equation y = sin x, 0 x p. Draw the shape that will result when this curve is rotated around an axis Oy.

4. The plane passes through the axis of the cylinder, and the area of ​​the axial section of the cylinder is related to the area of ​​its base as 4: p. Find the angle between the diagonals of the axial section.

Option 2

1. Draw the shape that is obtained by rotating the rhombus ABCD around a straight line a, passing through the vertex C and perpendicular diagonal A.C..

2. Draw a figure that is obtained by rotating a circle around a chord that is not a diameter.

3. The curve is given by the equation y =
, 0 x 4. Draw the shape that will be obtained by rotating this curve around the axis Ox.

4. The height of the cone is 20 cm, the angle between it and the generatrix is ​​60 0. Find the cross-sectional area drawn through two mutually perpendicular generatrices of the cone.

36. Inscribed and circumscribed cylinders

Option 1

1. A cylinder is inscribed in a sphere of radius 10 cm, the diagonal of the axial section of which is inclined to the plane of the base at an angle of 30 0 . Find the height of the cylinder and the radius of its base.

2. Find the radius of the base of the cylinder circumscribed about a sphere of radius R.

r, a regular triangular prism is inscribed. Find the cross-sectional area of ​​the prism passing through the axis of the cylinder and the side edge of the prism.

r, a regular quadrangular prism is described. Find the area of ​​its faces.

Option 2

1. A cylinder is inscribed in a sphere, the generatrix of which is 8 cm and the diagonal of the axial section is inclined to the plane of the base at an angle of 60 0. Find the radii of the sphere and the base of the cylinder.

2. Find the generatrix of a cylinder circumscribed about a sphere of radius R.

3. Into an equilateral cylinder (height equal to the diameter of the base), the radius of which is equal to r, a regular quadrangular prism is inscribed. Find the cross-sectional area of ​​the prism passing through the axis of the cylinder and the side edge of the prism.

4. Near an equilateral cylinder whose base radius is r, a regular triangular prism is described. Find the area of ​​its faces.

37*. Sections of a cylinder by a plane. Ellipse

Option 1

1. Draw a cylinder and an ellipse, which is the intersection of the side surface of the cylinder with a plane forming an angle of 45 0 with the base of the cylinder.

2. Side surface cylinder is intersected by a plane forming an angle of 30 0 with the cylinder axis. Find the major axis of the ellipse obtained in cross-section if the radius of the base of the cylinder is equal to R.

3. The plane intersects the side surface of the cylinder and forms an angle of 30 0 with the plane of the base. Find the distance between the foci of the ellipse obtained in cross-section if the radius of the base of the cylinder is 3 cm.

R, is intersected by a plane forming an angle of 45 0 with the base of the cylinder. Find the sum of the distances from the points of the ellipse obtained in the section to the foci.

Option 2

1. Draw a cylinder and an ellipse, which is the intersection of the side surface of the cylinder with a plane forming an angle of 60 0 with the base of the cylinder.

2. At what angle to the plane of the base of the cylinder must the plane be drawn in order to obtain an ellipse in the section of the side surface, the major axis of which is twice as large as the minor one?

3. The plane intersects the side surface of the cylinder and forms an angle of 45 0 with the plane of the base. Find the distance between the foci of the ellipse obtained in cross-section if the radius of the base of the cylinder is 2 cm.

4. A cylinder whose base radius is R, is intersected by a plane forming an angle of 30 0 with the base of the cylinder. Find the sum of the distances from the points of the ellipse obtained in section to the foci.

38. Inscribed and circumscribed cones

Option 1

1. A cone is inscribed in a sphere of radius 4 cm. Find the height of this cone and the radius of its base if the angle at the apex of the axial section is 60 0 .

2. The radius of the base of the cone is equal to r, the generatrix is ​​inclined to the plane of the base at an angle of 60 0. Find the radius of the sphere inscribed in the cone.

3. Is it possible to fit into a cone a 4-gonal pyramid whose base angles are consistently related as: a) 1:5:9:7; b) 4:2:5:7?

4. The base of the pyramid is an isosceles trapezoid with bases of 8 cm and 18 cm; The dihedral angles at the base of the pyramid are equal. A cone is inscribed in the pyramid. Find the radius of the base of the cone and its height if the smaller side edge of the pyramid makes an angle of 60 0 with the smaller side of the trapezoid.

Option 2

1. In a cone, the generatrix is ​​15 cm and makes an angle of 60 0 with the base. Find the radius of the circumscribed sphere.

2. A sphere is inscribed in a cone, the radius of which is R. Find the radius of the base of the cone if the angle at the apex of the axial section is 60 0 .

3. Is it possible to describe a 4-gonal pyramid near a cone, in which the sides of the base are consistently related as: a) 5: 6: 8: 7; b) 3:10:15:7?

4. The base of the pyramid is a right triangle; the side ribs are equal to each other, and the side faces passing through the legs make angles of 30 0 and 60 0 with the base. A cone is described around the pyramid in such a way that they have a common height. Find the radius of the base of the cone if the height of the pyramid is h.

39*. Conic sections

Option 1

1. The generatrix of the cone is inclined to the plane of its base at an angle of 60 0. The radius of the base of the cone is equal to R. A plane is drawn through the center of the base at an angle of 60 0 to the plane of the base. Find the radius of a sphere inscribed in a conical surface and tangent to this plane.

2. Draw a cone and a plane intersecting the conical surface along an ellipse.

3. The angle at the apex of the axial section of the cone is 90 0. At what angle to the plane of the base of the cone should the plane be drawn in order to obtain in the section of the conical surface: a) an ellipse; b) parabola; c) hyperbole?

4. The angle between the axis of the cone and its generatrix is ​​45 0. Through a point of the generatrix, spaced from the vertex of the cone at a distance a, a plane is drawn perpendicular to this generatrix. Find the distance between the focus and the directrix of the parabola resulting from the section of the conical surface by this plane.

Option 2

1. The angle at the apex of the axial section of the cone is 90 0. Through the point of the generatrix, spaced from the vertex of the cone at a distance a, a plane is drawn perpendicular to this generatrix. Find the radius of a sphere inscribed in a conical surface tangent to this plane.

2. Draw a cone and a plane intersecting the conical surface along a parabola.

3. The generatrix of the cone is inclined to the plane of its base at an angle of 60 0. At what angle to the plane of the base must the plane be drawn in order to obtain in the section of a conical surface: a) an ellipse; b) parabola; c) hyperbole?

4. The angle at the apex of the axial section of the cone is 30 0 . Through a point of the generatrix, spaced from the vertex at a distance b, a plane is drawn perpendicular to this generatrix. Find the major axis of the ellipse resulting from the section of the conical surface by this plane.

40. Symmetry of spatial figures

Option 1

1. For two points in space, find the point about which they are centrally symmetrical.

2. Construct a line that is mirror-symmetrical to the given line relative to the given plane a. Consider different cases.

3. Prove that with axial symmetry, a plane perpendicular to the axis transforms into itself.

4. Find the symmetry elements of a regular triangular prism.

Option 2

1. For two points in space, find the line relative to which they are symmetrical.

2. Construct a plane centrally symmetrical to the given plane relative to the point O. Consider different cases.

3. Prove that with axial symmetry, straight lines perpendicular to the axis transform into straight lines also perpendicular to the axis.

4. Find the symmetry elements of a regular 6-point pyramid.

41. Movements

Option 1

1. Prove that the composition of two movements (their sequential execution) is a movement.

A Cuba A… D 1 to the top C 1 .

A regular tetrahedron ABCD to the top C.

4. What kind of movement is the composition (sequential execution) of two axial symmetries with parallel axes?

Option 2

1. Prove that a transformation inverse to motion is also motion.

2. Find the movements that move the top B 1 cube A… D 1 to the top D.

3. Find the movements that move the top D regular tetrahedron ABCD to the top B.

4. What kind of movement is the composition (sequential execution) of two central symmetries?

42*. Surface orientation. Mobius strip

Option 1

1. How many sides does the surface have: a) pyramids; b) prisms; c) a twice twisted Möbius strip?

2. Draw a Mobius strip.

a, b(a b) by gluing the sides of the length a. What is the surface area of ​​a Mobius strip?

4. Is it possible to glue a one-sided surface from a hexagon?

Option 2

1. How many sides does the surface have: a) cone; b) cylinder; c) Moebius strip?

2. Draw a twice twisted Möbius strip.

3. The Mobius strip is obtained from a rectangle with sides a, b(a b) by gluing the sides of the length a. What is the length of the edge of a Mobius strip?

4. Is it possible to glue a one-sided surface from an octagon?

43. Volume of figures in space. Cylinder volume

Option 1

1. The axial section of a right circular cylinder is a square with a side of 3 cm. Find the volume of the cylinder.

2. From the cube A… D 1, whose edge is equal to 1, 4 triangular prisms are cut off by planes that pass through the midpoints of adjacent sides of the face ABCD, parallel to the edge A.A. 1 . Find the volume of the remaining part of the cube.

3. A right triangular prism is intersected by a plane that passes through the side edge and divides the area of ​​the side face opposite it in relation m:n. In what ratio is the volume of the prism divided?

4. The base of a right parallelepiped is a rhombus, the diagonals of which are in the ratio 5:2. Knowing that the diagonals of the parallelepiped are 17 dm and 10 dm, find the volume of the parallelepiped.

Option 2

1. The diagonal of the axial section of the cylinder is 2 cm and inclined to the plane of the base at an angle of 60 0. Find the volume of the cylinder.

2. Volume correct hexagonal prism equals V. Determine the volume of a prism whose vertices are the midpoints of the sides of the bases of this prism.

3. In what ratio is the volume of a right triangular prism divided by a plane passing through the midlines of the bases?

4. The base of a right parallelepiped is a rhombus, the diagonals of which are 1 dm and 7 dm. Knowing that the diagonals of a parallelepiped are in the ratio 13:17, find the volume of the parallelepiped.

44. Cavalieri principle

Option 1

1. Is it true that two cones that have equal bases and heights are equal in size?

1. Find the volume of an inclined prism whose base area is equal to S, and the side edge b inclined to the plane of the base at an angle of 60 0.

3. In an inclined parallelepiped, two lateral faces have areas S 1 and S 2, their common edge is equal a, and they form a dihedral angle of 150 0 between themselves. Find the volume of the parallelepiped.

4. In an inclined triangular prism, the area of ​​one of the side faces is equal to Q, and the distance from it to the opposite edge is d. Find the volume of the prism.

Option 2

1. Is it true that two pyramids with equal bases and equal heights are equal in size?

2. Find the volume of an inclined cylinder whose base radius is R, and the side edge b inclined to the plane of the base at an angle of 45 0.

3. In an inclined parallelepiped, the base and side face are rectangles and their areas are 20 cm 2 and 24 cm 2, respectively. The angle between their planes is 30 0. Another face of the parallelepiped has an area of ​​15 cm 2. Find the volume of the parallelepiped.

4. In an inclined triangular prism, two lateral faces are perpendicular and have a common edge equal to a. The areas of these faces are equal S 1 and S 2. Find the volume of the prism.

45. Volume of the pyramid

Option 1

1. A pyramid whose volume is equal to V, and at the base lies a rectangle, intersected by four planes, each of which passes through the top of the pyramid and the midpoints of adjacent sides of the base. Find the volume of the remaining part of the pyramid.

2. The base of the pyramid is an equilateral triangle with a side equal to 1. Two of its side faces are perpendicular to the plane of the base, and the third forms an angle of 60 0 with the base. Find the volume of the pyramid.

3. At the base of the primordium lies a right triangle, one of the legs of which is 3 cm, and the adjacent one sharp corner equals 30 0 . All lateral edges of the pyramid are inclined to the plane of the base at an angle of 60 0. Find the volume of the pyramid.

4. Centers of the faces of a cube whose edge is equal to 2 a, serve as the vertices of the octahedron. Find its volume.

Option 2

1. Find the volume of a regular quadrangular pyramid if its diagonal section is a regular triangle with side equal to 1.

2. The base of the pyramid is a rectangle, one side face is perpendicular to the base plane, and the other three side faces are inclined to the base plane at an angle of 60 0. The height of the pyramid is 3 cm. Find the volume of the pyramid.

3. The lateral faces of the pyramid, at the base of which lies a rhombus, are inclined to the plane of the base at an angle of 30 0. The diagonals of a rhombus are 10 cm and 24 cm. Find the volume of the pyramid.

4. Into a cube with an edge equal to a, a regular tetrahedron is inscribed in such a way that its vertices coincide with the four vertices of the cube. Find the volume of the tetrahedron.

46. ​​Volume of a cone

Option 1

1. The diameter of the base of the cone is 12 cm, and the angle at the apex of the axial section is 90 0. Find the volume of the cone.

2. Two cones have a common height and parallel bases. Find the volume of their common part if the volume of each cone is equal to V.

3. Into a cone whose volume is equal to V, a cylinder is inscribed. Find the volume of the cylinder if the ratio of the diameters of the bases of the cone and the cylinder is 10:9.

4. Each edge of a regular 4-gonal pyramid is equal to a. A plane parallel to the plane of the base of the pyramid cuts off the truncated pyramid from it. Find the volume of a truncated pyramid if the side of the section is equal to b.

Option 2

1. The axial section of the cone is an isosceles right triangle with an area of ​​9 cm 2. Find the volume of the cone.

2. Another cone is inscribed into a cone in such a way that the center of the base of the inscribed cone divides the height of this cone in the ratio 3:2, counting from the vertex of the cone, and the vertex of the inscribed cone is located in the center of the base of this cone. Find the ratio of the volumes of the given and inscribed cones.

3. Prove that if two equal cones have a common height and parallel base planes, then the volume of their common part is equal to the volume of each of them.

4. The radii of the bases of the truncated cone are 3 cm and 5 cm. Find the ratio of the volumes of the parts of the truncated cone into which it is divided by the middle section.

47. Volume of the ball and its parts

Option 1

1. Find the ratio of the volume of the sphere to the volume of the cube inscribed in it.

2. Find the ratio of the volume of the sphere to the volume of the octahedron described around it.

3. A plane is drawn in the ball, perpendicular to the diameter and dividing it into parts equal to 3 cm and 9 cm. Find the volumes of the parts of the ball.

4. Radius of the spherical sector R, angle in the axial section is 120 0. Find the volume of the spherical sector.

Option 2

1. Find the ratio of the volume of the sphere to the volume of the octahedron inscribed in it.

2. Find the ratio of the volume of the sphere to the volume of the cube circumscribed around it.

3. In a ball of radius 13 cm, two equal parallel sections of radius 5 cm are drawn on opposite sides of the center. Find the volume of the resulting spherical layer.

4. Find the volume of a spherical sector if the radius of its base circle is 60 cm and the radius of the sphere is 75 cm.

48. Surface area

Option 1

1. A plane passing through the side of the base of a regular triangular prism and the middle of the opposite edge forms an angle of 45 0 with the base, and the side of the base is equal to a. Find the lateral and total surface area of ​​the prism.

2. The base of the pyramid is a square whose side is equal to a. Two faces of the pyramid are perpendicular to the base, and the remaining two side faces are inclined to it at an angle of 60 0. Find the lateral surface area of ​​the pyramid.

3. In a regular quadrangular prism, the side of the base is equal to b; the section drawn through the opposite sides of the bases makes an angle j with the plane of the base. Find the lateral surface area of ​​the cylinder circumscribed around the given prism.

4. The angle at the apex of the axial section of the cone is 60 0; square great circle, inscribed in this cone of the ball, is equal to Q

Option 2

1. In a regular 4-angled prism, the side of the base is equal to a. A plane drawn through opposite sides of the bases makes an angle of 60 0 with one of them. Find the lateral and total surface area of ​​the prism.

2. The two side faces of a triangular pyramid are perpendicular to its base; the height of the pyramid is h; plane angles at the vertex are 60 0, 60 0 and 90 0. Find the lateral surface area of ​​the pyramid.

3. In a regular triangular prism, the lateral edge is equal to b; the segment connecting the middle of the side edge with the center of the base makes an angle j with the base. Find the lateral surface area of ​​the cylinder inscribed in this prism.

4. In a cone, the generatrix makes an angle of 60 0 with the base; The area of ​​the great circle of a circumscribed ball is Q. Find the total surface area of ​​the cone.

49. Surface area of ​​the ball and its parts

Option 1

1. Prove that the total surface area of ​​an equilateral cone (axial section is an equilateral triangle) is equal to the surface area of ​​a ball with a diameter of the height of the cone.

2. Find the surface area of ​​a ball inscribed in an equilateral cylinder (the axial section is a square), the diagonal of the axial section of which is equal to a.

3. The radii of the bases of the spherical belt are 10 cm and 12 cm, and its height is 11 cm. Find the surface area of ​​the spherical belt.

4. The radius of the ball segment is equal to R, the arc of the axial section is 90 0. Find the total surface area of ​​the segment.

Option 2

1. Prove that if an equilateral cone (axial section is an equilateral triangle) and a hemisphere have a common base, then the area of ​​the lateral surface of the cone is equal to the surface area of ​​the hemisphere.

2. Find the ratio of the surface areas of two spheres, one of which is inscribed, and the second is circumscribed about an equilateral cylinder (the axial section is a square).

3. The radius of the ball is 25 cm. Find the areas of the parts into which the surface of the ball is divided by a section whose area is 49p cm 2.

4. The height of the ball segment is h, the arc of the axial section is equal to 120 0. Find the total surface area of ​​the segment.

50. Rectangular coordinate system in space

Option 1

1. Construct points using coordinates: A(1,2,3); B(-2,0,3); C(0,0,-4); D(3,-1,0).

2. Among these points K(-6,0,0), L(10,-5,0), M(0,6,0), N(7,-8,0), P(0,0,-20), Q(0,11,-2) find those that belong to: a) axis Oy; b) axes Oz; c) plane Oxy; d) planes Oyz.

3. Find the coordinates of the bases of the perpendiculars omitted from the given points E(6,-2,8) and F(-3,2,-5) on: a) axis Ox; b) plane Oxz.

G.H., If G(2,-3,5), H(4,1,-3).

U(8,0,6), V(20.-14.0) relative to: a) plane Oyz; b) axes Ox.

Option 2

1. 1. Construct points using coordinates: E(-1,2,0); F(1,0,-4); G(2,3,-1); H(0,-2,0).

2. Among the points A(0,-1,0), B(0,1,-3), C(4,0,0), D(0,0,-5), E(-1,0,7), F(0,10,10) find those that belong to: a) axis Ox; b) axes Oy; c) plane Oyz; d) planes Oxz.

3. Find the coordinates of the bases of the perpendiculars dropped from the points M(9,-1,-6) and N(-12,5,8) on: a) axis Oz; b) plane Oxy.

4. Find the coordinates of the midpoint of the segment G.H., If G(3,-2,4), H(5,2,-6).

5. Find the coordinates of points symmetrical to the points P(0,0,5), V(0,-1,-2) relative to: a) plane Oxy; b) axes Oy.

51. Distance between points in space

Option 1

A(2,3,4), B(1,2,3), C(3,4,5) by the vertices of the triangle.

Oz M(-1,-2,0) and N(3,0,4).

C(-2,0,3) and: a) radius; b) passing through a point K(1,-4,3).

x 2 + 8y + y 2 + z 2 – 6x =0.

5. Sphere x 2 + y 2 + z 2 +4x – 2y=0 intersected by a plane Oyz

Option 2

1. Determine whether the points are E(-4,-5,-6), F(-1,-2,-3), G(-2,-3,-4) by the vertices of the triangle.

2. Find the coordinates of a point belonging to the axis Oy and equally distant from the points K(1,3,0) and L(4,-1,3).

3. Write down the equation of a sphere with center at point C(0,-5,6) and: a) radius 10; b) passing through a point H(2,-3,5).

4. Find the coordinates of the center and radius of the sphere given by the equation x 2 + y 2 + z 2 – 8z - 20 =0.

5. Sphere x 2 + y 2 + z 2 +2x – 6z=0 intersected by a plane Oxy. Find the coordinates of the center and the radius of the circle lying in the section.

52. Vector coordinates

Option 1

1. Find the coordinates of the vector: a) 2 + 3 - 4 ; b) -5 + 10 ; c) - + .

2. Find the length of the vector: a) (1,-2,10); b) if A(0,-5,1), B(2,0,-8); c) + if (6,2,-6), (2,-2,0).

3. Find the coordinates of the point C, if: a)
(-5,6,8), D(0,-1,2); b) D(-13, ,6),
(-5,0,0).

4. Find the numbers x, y, z, so that the equality holds =
, if (5,-2,0), (0,2,-6), (-5,0,-8), (-5,2,-4).

Option 2

1. Find the coordinates of the vector: a) 3 - 4 + 2 ; b) -2 - ; V) - .

2. Find the length of the vector: a) (0,-3,2); b) if M(0,-5,1), N(2,0,-8); c) - if (0,-2,6), (-5,0,3).

3. Find the coordinates of the point E, if: a) (0,-3,11), F(5,-1,0); b) F(5,0,-9),
(-2,4,-6).

4. Find the numbers u, v, w, so that the equality =
, if (-30.6,-12), (5,-6.0), (10,-3,2), (0,1,2).

53. Dot product of vectors

Option 1

1. Identify the sign dot product vectors and , if the angle between them satisfies the inequalities: a) 0 0

2. The angle between the vectors and is equal to 90 0. Why equal to the angle between vectors: a) - and ; b) - and ?

+
+
= 0.

4. In a regular tetrahedron ABCD with edge equal to 1, find the scalar product: a)
; b)
; V)
, Where H And QA.C. And BD.

Option 2

1. Determine in what interval the angle between the vectors and is located if: a) > 0.

2. Angle between vectors and
equals 90 0. What is the angle between the vectors: a) and - ; b) - and - ?

3. Prove the equality: a) ; b)
=
.

4. In a regular tetrahedron ABCD with an edge equal to a, find the scalar product: a)
; b) ; c) where E And F– the middle of the ribs, respectively B.C. And AD.

54. Equation of a plane in space

Option 1

H(-3,0,7) and perpendicular to the vector with coordinates (1,-1,3).

2. Find the coordinates of the intersection point of plane 2 x – y + 3z– 1 = 0 with axis: a) abscissa; b) ordinate.

B(3,-2,2) and: a) parallel to the plane Oyz; b) perpendicular to the axis Ox.

M(5,-1,3) and perpendicular to the vector if N(0,-2,1).

Option 2

1. Write the equation of the plane passing through the point P(5,-1,0) and perpendicular to the vector with coordinates (0,-6,10).

2. Find the coordinates of the intersection point of the plane x + 4y - 6z– 7 = 0 with the axis: a) ordinate; b) applicate.

3. Write the equation of the plane if it passes through the point C(2,-4,-3) and: a) parallel to the plane Oxz; b) perpendicular to the axis Oy.

4. Write the equation of the plane that passes through the point E and perpendicular to the vector (4,-5,0), if F(3,-1,6).

55*. Equation of a line in space

Option 1

1. Find the value d, for which the straight line

crosses the axis Oz.

in order for the straight line: a) to be parallel to the axis Ox; b) lay in a plane Oxz; c) crossed the axis Oy.

with coordinate planes.

Option 2

1. Find the values b And d, for which the straight line

intersects the plane Oxy.

2. Find the conditions that the coefficients in the equations of the line must satisfy

in order for the straight line: a) to coincide with the axis Oz; b) was parallel to the plane Oyz; c) passed through the origin.

3. Find the coordinates of the points of intersection of the line

with coordinate planes.

4. Write down the parametric equations of the line

56. Analytical assignment of spatial figures

Option 1

x 2 + y 2 +z 2 = 1; b) x 2 = 1; V) xyz = 0.

A)
b)

3. Points are given A(2,5,12), B(1,0,0), C(-1,-5,4) and planes
And , given respectively by equations 2 x – y + z+1 = 0 and x – 5y –13z+1 = 0. For each of these planes, find among the given points those that lie on the same side of the plane as the origin.

4. Given plane 3 x – y +4z O(0,0,0) and D(2,1,0); b) E(1,2,1) and F(5,15,-1)?

Option 2

1. Find out which geometric figure sets the equation: a) x 2 + y 2 +(z+1) 2 = 1; b) x 2 – y 2 = 0; V) x 2 = 0.

2. Find out what geometric figure the system defines:

A)
b)

3. Points are given E(-14,22,0), F(1,-5,12), G(0,0,5) and planes And , given respectively by the equations x – 2z+12 = 0 and x + 5y + z+25 = 0. For each of these planes, find among the given points those that lie on the same side of the plane as the origin.

4. Given plane 3 x – y +4z+1 = 0. Do the points lie on the same side of it: a) A(-1,2,-5) and B(-15,1,0); b) K(1,
,5) and L(1,15,-15)?

57*. Polyhedra in optimization problems

Option 1

1. The vertices of the tetrahedron have the following coordinates: O(0,0,0), A(1,1,0), B(0,2,0),C(1,5,7). Write down the inequalities that characterize the interior region of this tetrahedron.

2. Find the area defined by the following system of inequalities:

a) b)

Picture her.

3. Write down a system of inequalities that determines the internal region of a right triangular prism OABO 1 A 1 B 1 if O(0,0,0), A(0,2,0), B(0,0,2), O 1 (5,0,0). Draw it and find its volume.

u = x + y – 2z + 1 on the triangular prism from the previous problem.

Option 2

1. Given the vertices of a tetrahedron A(-1,1,0), B(-2,2,0), C(-2,0,0), D(-1,5,7). Which of the points M(2,3,-1), N(- , , ), P(0,0,1), H(- , , ) belong to the internal region of this tetrahedron?

2. Find the area defined by the following system of inequalities: a) b)

3. Write down the system of inequalities that determine the internal region of the tetrahedron OABC, If O(0,0,0), A(5,0,0), B(0,3,0), C(0,0,6). Draw it and find its volume.

4. Find the largest and smallest values ​​of the linear function u= x – y + z – 1 on the tetrahedron from the previous problem.

58*. Polar coordinates on a plane

Option 1

A(2, ), B(1, ), C( , ), D(3, ), E(4, ), F( , ).

2. Write down the Cartesian coordinates of the points G(2, ), H( , ), P(5, ), Q(3,- ).

3. Find the polar coordinates of the vertices and the intersection points of the diagonals of a unit square, taking one of its vertices as the origin of coordinates, and the side that passes through the selected vertex as the polar axis.

M(1, ), N(3, ), P( ,- ), Q(, ) relative to: a) the polar axis; b) the origin of coordinates.

Option 2

1. Draw in polar system point coordinates A(3, ), B(5, ), C( , ), D(6, ), E(2, ), F( , ).

2. Write down the polar coordinates of the points K(0,6), L(-2,0), M(-1,1), N( ,1).

3. Find the polar coordinates of the vertices of a regular hexagon whose side is equal to 1, taking one of its vertices as the origin, and the side that passes through the selected vertex as the polar axis.

4. Find the polar coordinates of points symmetrical to the points G(2, ), H(3, ), R(3,- ), S( , ) relative to: a) the origin of coordinates; b) polar axis.

59*. Spherical coordinates in space

Option 1

1. Find the Cartesian coordinates of the following points in space, specified by spherical coordinates: (1.45 0 .60 0), (2.30 0 .90 0), (1.90 0 , 20 0).

2. Find the spherical coordinates of the following points in space, given by Cartesian coordinates: A(1,1, ), B(1,0,1), C(0,0,1).

3. Find the geometric locus of points in space whose spherical coordinates satisfy the conditions: a) y = 45 0 ; b) j= 60 0 .

r 2; b) r 1, y 0?

Option 2

1. Find the Cartesian coordinates of the following points in space, specified by spherical coordinates: (1,-45 0,60 0), (2,30 0,-90 0), (3,-90 0, 50 0).

2. Find the spherical coordinates of the following points in space, given by Cartesian coordinates: A(2,2 ), B(-1,0,1), C(0,0,-1).

3. Find the geometric locus of points in space whose spherical coordinates satisfy the conditions: a) y= 30 0 ; b) j = 90 0 .

4. Which figure in space is given by the inequalities: a) r 1; b) r 1, - j 0?

60*. Using the computer program "Mathematics" to depict spatial figures

Option 1

1. Obtain an image of a tetrahedron.

2. Perform the operation of truncation of the tetrahedron and obtain an octahedron.

3. How to get a Kepler star from an octahedron?

z = xy.

Option 2

1. Get an image of a cube.

2. Perform the operation of truncating the cube and obtain a cuboctahedron.

3. How to get a rhombic dodecahedron from a cube?

4. Obtain an image of the surface z = cos x cos y.

ANSWERS

Independent work N 2

IN 1. 4. 6. B2. 3. 10. 4. 4.

IN 1. 2. a) B=8, P=12, D=6; b) V=14, P=21, D=9; c) B= n+1, Р=2 n, Г= n+1. 3. a) 5-gonal; b) 7-gonal; c) 3-gonal. 4. Three colors. AT 2. 2. a) B=8, P=12, D=6; b) B=7, P=12, G=7; c) B=2 n, Р=3 n, Г= n+2. 3. a) tetragonal; b) 7-gonal; c) octagonal. 4. Two colors.

IN 1. 3. 3. 4. 3. B2. 3. 3. 4. 3.

IN 1. 3. They interbreed. AT 2. 3. No. 4. No.

IN 1. 3. Parallel.

IN 1. 2. Statements 1), 3), 4) are true. 4. If AB || CD, That A.C.|| BD; If AB crosses with CD, That A.C. crosses with BD. AT 2. 2. Statement 3) is true. 4. If AB || CD, That AD And B.C. intersect; If AB And CD cross, then AD And B.C. interbreed.

IN 1. 2. 26. 3. a) ; b)
; V)
, Where M– middle B.C.. 4. a)
; b) ; V)
. AT 2. 2. 24. 3. a)
; b) ; V)
, Where M– middle B.A.. 4. a)
; b) ; V) .

IN 1. 1. . 2. Vector + has the same direction as vector ; | + | = | | - | |. AT 2. 1.
. 2. Vector + has the same direction as vector ; | + |=| | - | |.

IN 1. 1. One, if the straight line passing through them is parallel to the design direction; two in otherwise. 2. Parallelism and equality of opposite sides; bisecting the diagonals at the point of intersection. 3. The straight lines intersect and one of them is parallel to the design direction. AT 2. 1. One, if all points belong to one straight line parallel to the design direction; two, if the line passing through any two of these points is parallel to the design direction, and the third point does not belong to this line; three in other cases. 2. Parallelism and equality of opposite sides; bisecting the diagonals at the point of intersection. 3. The straight line is not parallel to the design direction and the point belongs to the line or the plane passing through this point and the line is parallel to the design direction.

IN 1. 3. The faces of the cube are not parallel to the design plane and the design direction is parallel to the diagonal B.D. 4. . AT 2. 1. H=24, P=36, D=14. 4. . 3. Rb; b) y B. 4. a) Yes; b) no. AT 2. 1. a) Sphere with center at point (0,0,-1) and radius 1; b) two intersecting planes; c) plane Oyz. 2. a) Rectangular parallelepiped; b) two intersecting lines lying in a plane Oxy. 3. For : point F; for : points E, F, G. 4. a) Yes; b) no.

IN 1. 1.

2. a) Internal region of a tetrahedron with vertices (0,0,0), (1,0,0). (0,1,0), (0,0,1); b) the internal region of a rectangular parallelepiped with vertices (5,5,0), (5,3,0), (7,3,0), (7,5,0), (5,5,10), (5 ,3,10), (7,3,10), (7,5,10).

3.
V = 20. 4. 8 – greatest; 3 is the smallest.

AT 2. 1. Points N And H. 2. a) The area between two parallel planes; b) the internal region of a right triangular prism with vertices (0,0,0), (0,3,0). (0,0,3), (-2,0,0), (-2,3,0), (-2, 0.3).

(); (); (0, stereometry. Give examples of real objects... polyhedra. Development. List basic concepts And axioms stereometry. Give examples of real objects...

Guidelines

46 - 2 Introduction. Item stereometry. Basic concepts And axioms stereometry. First corollaries from axioms 2 2 ... and icosahedron) 1 § 3*. Axioms, laws, rules 2 9. Axioms stereometry Basic concepts stereometry(point, straight line, plane, ...

Work program of the training course "Geometry"

Working programm

... stereometry. Axioms stereometry. Some corollaries from the axioms. Main The goal is to form students’ ideas about main concepts And axioms stereometry, their...