3. In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, find the angle between the lines: AB and A 1 C. Solution: The desired angle is equal to the angle B 1 A 1 C. In the triangle B 1 A 1 C we draw height CD 1. In a right triangle A 1 CD 1 leg A 1 D 1 is equal to 0.5; hypotenuse A 1 C is equal. Hence,
Solution 1. Let O 1 be the center of the regular hexagon A 1 ...F 1. Then the straight line AO 1 is parallel to the straight line BC 1, and the desired angle between the straight lines AB 1 and BC 1 is equal to the angle B 1 AO 1. In the isosceles triangle B 1 AO 1 we have : O 1 B 1 = 1; AB 1 =AO 1 =. Applying the cosine theorem, we get.
Solution 2. Let's introduce a coordinate system, considering point A to be the origin of coordinates, point B to have coordinates (1, 0, 0), point A 1 to have coordinates (0, 0, 1). Then point C 1 has coordinates (1.5, 1). A vector has coordinates (1, 0, 1), a vector has coordinates (0.5, 1). Let's use the formula expressing the cosine of the angle between vectors through their scalar product and length. We have. Therefore, the cosine of the angle between straight lines AB 1 and BC 1 is 0.75.
Solution 2. Let's introduce a coordinate system, considering point A to be the origin of coordinates, point B to have coordinates (1, 0, 0), point A 1 to have coordinates (0, 0, 1). Then point D 1 has coordinates (1, 1). A vector has coordinates (1, 0, 1), a vector has coordinates (0, 1). Let's use the formula expressing the cosine of the angle between vectors through their scalar product and length. We have. Therefore, the cosine of the angle between straight lines AB 1 and BC 1 is equal.
Solution 1. Let us prove that the angle between straight lines AB 1 and BE 1 is equal to 90 degrees. To do this, we use the theorem of three perpendiculars. Namely, if the orthogonal projection of an inclined plane onto a plane is perpendicular to a straight line lying in this plane, then the inclined one itself is perpendicular to this straight line. The orthogonal projection of BE 1 onto the plane ABB 1 is the straight line A 1 B, perpendicular to AB 1. Consequently, the straight line BE 1 will also be perpendicular to the straight line AB 1, i.e. the desired angle is 90°.
Solution 2. Through point B we draw a line parallel to line AB 1, and denote G 1 its point of intersection with line A 1 B 1. The desired angle is equal to angle E 1 BG 1. Side BG 1 of triangle E 1 BG 1 is equal. In the right triangle BEE 1, the legs BE and EE 1 are equal to 2 and 1, respectively. Therefore, the hypotenuse of BE 1 is equal. In a right triangle G 1 A 1 E 1, the legs A 1 G 1 and A 1 E 1 are equal to 2 and respectively. Therefore, the hypotenuse G 1 E 1 is equal. Thus, in the triangle BE 1 G 1 we have: BG 1 =, BE 1 =, G 1 E 1 =. According to the theorem inverse to the Pythagorean theorem, we find that the angle E 1 BG 1 is equal to 90 degrees.
Solution 3. Let's introduce a coordinate system, considering point A to be the origin of coordinates, point B to have coordinates (1, 0, 0), point A 1 to have coordinates (0, 0, 1), point E to have coordinates (0, 0). Then the point E 1 has coordinates (0, 1), the Vector has coordinates (1, 0, 1), the vector has coordinates (-1, 1). Let's use the formula expressing the cosine of the angle between vectors through their scalar product and length. We have and, therefore, the angle between straight lines AB 1 and BE 1 is equal to 90 degrees.
13. In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, find the angle between the planes ABC and A 1 B 1 C. Solution: Let O, O 1 be the midpoints of the edges AB and A 1 B 1. The desired linear the angle will be angle OCO 1. In the right triangle OCO 1 we have OO 1 = 1; OC = Therefore
16. In the regular 6th prism A...F 1, the edges of which are equal to 1, find the angle between the planes CDF 1 and AFD 1. Answer: Solution: Let O be the center of the prism, G, G 1 the midpoints of the edges CD and C 1 D 1. The required angle is equal to the angle GOG 1. In the triangle GOG 1 we have: GG 1 = GO = G 1 O = 1. Therefore, = 60 o.
Cube 1 In cube A…D 1, find the angle between lines AC and BD 1. Answer. 90 o.
Cube 2 In cube A…D 1, find the angle between lines AB 1 and BD 1. Answer. 90 o.
Cube 3 In cube A…D 1, find the angle between lines DA 1 and BD 1. Answer. 90 o.
Cube 4 In the unit cube A...D 1, find the cosine of the angle between the lines AE and BE 1, where E and E 1 are the midpoints of the edges BC and B 1 C 1, respectively. Solution. Through point A we draw a line AF 1 parallel to BE 1. The desired angle is equal to the angle EAF 1. In the triangle AEF 1 AE = AF 1 = , EF 1 = . Using the cosine theorem we find the answer.
Cube 5 In cube A...D 1, find the angle between lines AE and BF 1, where E and F 1 are the midpoints of edges BC and C 1 D 1, respectively. Solution. From point F 1 we lower the perpendicular F 1 F to the straight line CD. Line AE is perpendicular to BF, therefore it is perpendicular to BF 1. Answer. 90 o.
Pyramid 1 In a regular tetrahedron ABCD, find the angle between lines AD and BC. Answer: 90 o.
Pyramid 1 In a regular tetrahedron ABCD, points E, F, G are the midpoints of edges AB, BD, CD. Find the angle EFG. Solution. Lines EF and FG are parallel to lines AD and BC, which are perpendicular. Therefore, the angle between them is 90 degrees. Answer: 90 o.
Pyramid 2 In a regular pyramid SABCD, all edges of which are equal to 1, point E is the middle of edge SC. Find the tangent of the angle between lines SA and BE. Solution. Through point E we draw a line parallel to SA. It will intersect the base at point O. The required angle is equal to angle OEB. In the right triangle OEB we have: OB = Answer: , OE = . Hence,
Pyramid 3 In a regular pyramid SABCD, all edges of which are equal to 1, points E, F are the midpoints of edges SB and SC. Find the cosine of the angle between lines AE and BF. Solution. Let G denote the midpoint of edge AD. Line GF is parallel to AE. The required angle is equal to angle BFG. In triangle BFG we have: BF = GF = , BG = . Using the cosine theorem we find the Answer:
Pyramid 4 In a regular pyramid SABCDEF, the base sides of which are equal to 1 and the side edges are equal to 2, find the angle between the lines SA and BF. Answer: 90 o.
Pyramid 5 In a regular pyramid SABCDEF, the base sides of which are equal to 1 and the side edges are equal to 2, point G is the middle of edge SC. Find the tangent of the angle between lines SA and BG. Solution. Let H denote the midpoint of the segment AC. Line GH is parallel to SA. The required angle is equal to angle BGH. In the triangle BGH we have: BH = 0, 5, GH = 1. Answer:
Prism 1 In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, find the cosine of the angle between straight lines AB 1 and BC 1. Solution: Let's build the prism to a 4-angled prism. Let us draw AD 1 parallel to BC 1. The desired angle will be equal to angle B 1 AD 1. In the triangle AB 1 D 1 Using the cosine theorem, we find
Prism 2 In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, points D, E are the midpoints of edges A 1 B 1 and B 1 C 1. Find the cosine of the angle between lines AD and BE. Solution. Let F denote the midpoint of the segment AC. Line EF is parallel to AD. The required angle is equal to angle BEF. In triangle BGH we have: Using the law of cosines we find the Answer.
Prism 3 In a regular 6th prism A…F 1, the edges of which are equal to 1, find the angle between straight lines AA 1 and BD 1. Solution: The required angle is equal to angle B 1 BD 1. In a right triangle B 1 BD 1 B 1 D 1 = ; B 1 B =1; BD 1=2. Therefore, the desired angle is 60°. Answer. 60 o.
Prism 4 In a regular 6th prism A...F 1, the edges of which are equal to 1, find the tangent of the angle between straight lines AA 1 and BE 1. Solution: The desired angle is equal to angle B 1 BE 1. In a right triangle B 1 BE 1 leg B 1 E 1 is equal to 2; side B 1 B is equal to 1. Therefore, Answer. 2.
Prism 5 In a regular 6th prism A…F 1, whose edges are equal to 1, find the angle between straight lines AC 1 and BE. Answer. 90 o.
Prism 6 In a regular 6th prism A…F 1, whose edges are equal to 1, find the angle between straight lines AD 1 and BF. Answer. 90 o.
Prism 7 In a regular 6th prism A…F 1, whose edges are equal to 1, find the angle between straight lines AB 1 and BE 1. Answer. 90 o.
Prism 8 In the regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between straight lines BA 1 and FC 1. Solution: Through the middle O of the segment FC 1, draw a straight line PP 1, parallel to BA 1. The desired angle is equal to the angle POC 1. In triangle POC 1 we have: PO = ; OC 1= PC 1= Therefore, Answer. .
Prism 9 In the regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between straight lines AB 1 and BC 1. Solution: Let O 1 be the center of the regular 6th prism A 1...F 1. Then AO 1 is parallel BC 1, and the required angle is equal to angle B 1 AO 1. In an isosceles triangle B 1 AO 1 O 1 B 1=1; AB 1=AO 1= Applying the cosine theorem, we get
Prism 10 In the regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between the straight lines AB 1 and BD 1. Solution: The desired angle is equal to the angle B 1 AE 1. In the triangle B 1 AE 1 AB 1= ; B 1 E 1 = AE 1 = 2. Therefore,
Prism 11 In a regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between straight lines AB 1 and BF 1. Solution: Let O, O 1 be the centers of the bases of the prism. On the axis of the prism we plot O 1 O 2 = OO 1. Then F 1 O 2 will be parallel to AB 1, and the desired angle will be equal to the angle BF 1 O 2. In the triangle BF 1 O 2 BO 2 = BF 1 = 2; F 1 O 2 = By the cosine theorem, we have
Prism 12 In a regular 6th prism A…F 1, the edges of which are equal to 1, find the cosine of the angle between straight lines AB 1 and CD 1. Solution: The desired angle is equal to the angle CD 1 E. In the triangle CD 1 E CD 1= ED 1 = ; CE = By the cosine theorem, we have
Prism 13 In the regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between straight lines AB 1 and CE 1. Solution: Note that CE 1 is parallel to BF 1. Therefore, the required angle is equal to the angle between AB 1 and BF 1, which was found earlier. Namely,
Prism 14 In a regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between straight lines AB 1 and CF 1. Solution: Let O, O 1 be the centers of the bases of the prism. On the axis of the prism we plot O 1 O 2 = OO 1. Then F 1 O 2 will be parallel to AB 1, and the desired angle will be equal to the angle CF 1 O 2. In the triangle CF 1 O 2 CO 2= CF 1 = F 1 O 2 = Then
Prism 15 In the regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between the straight lines AB 1 and CA 1. Solution: In the continuation of BB 1, set aside B 1 B 2 = BB 1. Then A 1 B 2 will be parallel to AB 1, and the desired angle will be equal to angle CA 1 B 2. In a triangle CA 1 B 2 CA 1= 2; CB 2 = A 1 B 2 = Then
Prism 16 In the regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between straight lines AB 1 and DF 1. Solution: Note that DF 1 is parallel to CA 1. Therefore, the desired angle is equal to the angle between AB 1 and CA 1, which was found earlier. Namely,
Prism 17 In the regular 6th prism A...F 1, the edges of which are equal to 1, find the angle between straight lines AB 1 and DA 1. Solution: In the continuation of BB 1 we put aside B 1 B 2 = BB 1. Then A 1 B 2 will be parallel AB 1, and the required angle will be equal to the angle DA 1 B 2. In the triangle DA 1 B 2 DA 1= DB 2 = A 1 B 2 = Therefore, the required angle is 90 o.
Prism 18 In a regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between straight lines AB 1 and DC 1. Solution: Let O be the center of the base of the prism. The segments OC 1 and OB 1 will be equal and parallel to the segments AB 1 and DC 1, respectively. The desired angle will be equal to angle B 1 OC 1. In the triangle B 1 OC 1 OB 1 = OC 1 = ; B 1 C 1 = 1. Then, by the cosine theorem
Prism 19 In the regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between lines AC 1 and BD 1. Solution: Note that AE 1 is parallel to BD 1. Therefore, the desired angle is equal to angle C 1 AE 1 In triangle C 1 AE 1 AC 1 = AE 1 = 2; C 1 E 1 = By the cosine theorem, we have
Prism 20 In a regular 6th prism A...F 1, the edges of which are equal to 1, find the cosine of the angle between lines AC 1 and BE 1. Solution: Note that the segment GG 1 passing through the midpoints of the edges AF and C 1 D 1 is parallel and is equal to the segment AC 1. The required angle is equal to the angle G 1 OE 1. In the triangle G 1 OE 1 OG 1 = 1; OE 1 = ; G 1 E 1 = By the cosine theorem, we have.
Unified State Exam 2010. MATHEMATICS
Problem C2
Workbook
Edited by and
Publishing house MCNMO
2010
INTRODUCTION
This manual is intended to prepare you for completing task C2 of the Unified State Exam in mathematics. Its goals are:
– showing the approximate topics and level of difficulty of geometric problems included in the content of the Unified State Examination;
– checking the quality of students’ knowledge and skills in geometry, their readiness to take the Unified State Exam;
– development of students’ ideas about basic geometric figures and their properties, development of skills in working with drawings, and the ability to carry out additional constructions;
– improving the computing culture of students.
The manual contains problems on finding angles between straight lines in space, a straight line and a plane, two planes; finding the distances from a point to a line, from a point to a plane, between two lines. The presence of drawings helps to better understand the conditions of the problems, imagine the corresponding geometric situation, outline a solution plan, and carry out additional constructions and calculations.
To solve the proposed problems, knowledge of the definitions of trigonometric functions, formulas for finding the elements of a triangle, the Pythagorean theorem, the cosine theorem, the ability to carry out additional constructions, and knowledge of coordinate and vector methods of geometry are required.
Each task is scored based on two points. One point is awarded for correctly constructing or describing the required angle or distance. Also, one point is awarded for correctly performed calculations and the correct answer.
First, diagnostic work is proposed to find angles and distances for various polyhedra. For those who want to check the correctness of the solutions to the proposed problems or make sure that the answer received is correct, solutions to the problems are given, usually in two different ways, and the answers are given. Then, to consolidate the considered methods for solving problems, training work is proposed to find angles and distances for each of the types of figures considered in the diagnostic work.
If these tasks are successfully solved, you can move on to performing final diagnostic work containing tasks of different types.
At the end of the manual, answers to all problems are given.
Note that the best way to prepare for the Unified State Exam in geometry is to systematically study in a geometry textbook. This manual does not replace the textbook. It can be used as additional collection tasks when studying geometry in grades 10-11, as well as when organizing generalized repetition or independent geometry studies.
Diagnostic work
1.1. In a unit cube A…D 1 find the angle between the lines AB 1 and B.C. 1.
1.2.
In a unit cube A…D 1 find the angle between the lines D.A. 1 and BD 1.
1.3 . ABCA 1B 1C AD 1 and C.E. 1, where D 1 and E 1 – respectively, the middle of the ribs A 1C 1 and B 1C 1.
2.1. A…F A.F. and plane
2.2.
In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the angle between the line CC 1 and plane
2.3
. SABCD BE and plane S.A.D., Where E– middle of the rib S.C..
3.1.
In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the angle between the planes
AFF 1 and DEE 1.
3.2. In a unit cube A…D
ADD 1 and BDC 1.
3.3.
In a regular triangular prism ABCA 1B 1C 1D 1 ACB 1 and B.A. 1C 1.
4.1. In a regular hexagonal prism A…F A to a straight line D 1F 1.
4.2.
In a unit cube A…D A to a straight line BD 1.
4.3. SABCDEF F to a straight line B.G., Where G– middle of the rib S.C..
5.1. In a unit cube A…D 1 find the distance from the point A to plane BDA 1.
5.2.
In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the distance from the point A to plane SBC.
5.3.
In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance from the point A to plane B.F.E. 1.
6.1.
In a regular quadrangular pyramid SABCD S.A. And B.C..
6.2. In a unit cube A…D AB 1 and B.C. 1.
6.3.
In a regular hexagonal prism A…F A.A. 1 and CF 1.
Solutions to problems 1.1 – 1.3 of diagnostic work
1.1.
First solution. Straight AD 1 is parallel to the line B.C. 1 and therefore the angle between the lines AB 1 and B.C. 1 equals angle B 1AD 1. Triangle B 1AD 1 equilateral and therefore angle B 1AD 1 equals 60o.
Second solution A, coordinate axes – straight lines AB, AD, A.A. 1. Vector has coordinates (1, 0, 1). Vector has coordinates (0, 1, 1). Let's use the formula for finding the cosine of the angle between vectors And . We get and, therefore, the angle is 60°. Therefore, the desired angle between the lines AB 1 and B.C. 1 equals 60o.
Answer. 60o.
1.2. First solution. Consider the orthogonal projection AD 1 straight BD 1 per plane ADD 1. Straight AD 1 and D.A. 1 are perpendicular. From the theorem about three perpendiculars it follows that straight lines D.A. 1 and BD 1 are also perpendicular, i.e. the desired angle between straight lines D.A. 1 and BD 1 equals 90o.
Second solution. Let us introduce a coordinate system, considering the point as the origin of coordinates A, coordinate axes – straight lines AB, AD, A.A. 1. Vector has coordinates (0, -1, 1). Vector has coordinates (-1, 1, 1). The scalar product of these vectors is equal to zero and, therefore, the desired angle between the lines D.A. 1 and BD 1 equals 90o.
Answer. 90o.
1.3 . First solution. Let's denote D And F 1 respectively the middle of the ribs A.C. And A 1B 1.
Direct DC 1 and DF 1 will be respectively parallel to straight lines AD 1 and C.E. 1. Therefore, the angle between the lines AD 1 and C.E. 1 will be equal to the angle C 1DF 1. Triangle C 1DF 1 isosceles, DC 1 = DF 1 = , C 1F 1 = . Using the cosine theorem, we get 
.
Second solution. Let us introduce a coordinate system, considering the point as the origin of coordinates A, as it shown on the picture. Dot C has coordinates, point D 1 has coordinates, point E 1 has coordinates . The vector has coordinates . The vector has coordinates 
. Cosine of the angle between lines AD 1 and C.E. 1 is equal to the cosine of the angle between the vectors and . Let's use the formula for finding the cosine of the angle between vectors. We'll get it.
Answer. 0.7.
Training work 1. Angle between straight lines
1.
Cubed A…D 1 find the cosine of the angle between the lines AB And C.A. 1.
2. In a regular tetrahedron ABCD dot E– middle of the rib CD. Find the cosine of the angle between the lines B.C. And A.E..
3. In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the cosine of the angle between the lines AB And C.A. 1.
4.
In a regular quadrangular pyramid SABCD E– middle of the rib SD S.B. And A.E..
5.
In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the cosine of the angle between the lines AB And F.E. 1.
6. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the cosine of the angle between the lines AB 1 and B.C. 1.
7. In a regular hexagonal pyramid SABCDEF S.B. And A.E..
8. In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the cosine of the angle between the lines S.B. And AD.
Solutions to problems 2.1 – 2.3 of diagnostic work
2.1. Solution. Let O– center of the lower base of the prism. Straight B.O. parallel A.F.. Since the plane ABC And BCC 1 are perpendicular, then the required angle will be the angle OBC. Since the triangle OBC equilateral, then this angle will be equal to 60°.
Answer. 60o.
2.2.
Solution. Since straight BB 1 and CC 1 are parallel, then the desired angle will be equal to the angle between the straight line BB 1 and plane BDE 1. Direct BD, through which the plane passes BDE 1, perpendicular to the plane ABB 1 and, therefore, a plane BDE 1 perpendicular to the plane ABB 1. Therefore, the desired angle will be equal to the angle A 1BB 1, i.e. equal to 45o.
Answer. 45o.
2.3. Solution. Through the top S draw a line parallel to the line AB, and plot a segment on it SF, equal to the segment AB. In a tetrahedron SBCF all edges are equal to 1 and the plane BCF parallel to the plane S.A.D.. Perpendicular E.H., dropped from the point E to the plane BCF, is equal to half the height of the tetrahedron, i.e. equal to . Angle between straight line BE and plane S.A.D. equal to angle EBH, whose sine is equal to .
Answer. .
Training work 2. Angle between a straight line and a plane
1.
Cubed A…D 1 find the tangent of the angle between the line A.C. 1 and plane
2.
Cubed A…D AB and plane
C.B. 1D 1.
3.
In a regular tetrahedron ABCD dot E– middle of the rib BD. Find the sine of the angle between the line A.E. and plane
4. In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the tangent of the angle between the line BB 1 and plane
AB 1C 1.
5. In a regular quadrangular pyramid SABCD, all edges of which are equal to 1, find the sine of the angle between the line BD and plane
6.
In a regular hexagonal pyramid SABCDEF B.C. and plane
7. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the angle between the line A.A. 1 and plane
8. In a regular hexagonal prism A…F B.C. 1 and plane
Solutions to problems 3.1 – 3.3 diagnostic work
3.1.
First solution. Since the plane FCC 1 parallel to the plane DEE AFF 1 and FCC 1. Since the plane AFF 1 and FCC 1 perpendicular to the plane ABC A.F.C., which is equal to 60o.
Second solution. Since the plane AFF 1 parallel to the plane BEE 1, then the desired angle is equal to the angle between the planes BEE 1 and DEE 1. Since the plane BEE 1 and DEE 1 perpendicular to the plane ABC, then the corresponding linear angle will be the angle BED, which is equal to 60o.
Answer. 60o.
3.2.
Solution. Since the plane ADD 1 parallel to the plane BCC 1, then the desired angle is equal to the angle between the planes BCC 1 and BDC 1. Let E– the middle of the segment B.C. 1. Then straight C.E. And DE will be perpendicular to the line B.C. 1 and therefore the angle CED will be the linear angle between the planes BCC 1 and BDC 1. Triangle CED rectangular, leg CD equals 1, leg C.E. equal to . Hence, 
.
3.3. Let DE– line of intersection of these planes, F– the middle of the segment DE, G– the middle of the segment A 1C 1. Angle GFB 1 is the linear angle between these planes. In a triangle GFB 1 we have: FG =
FB 1 = , G.B. 1 = . Using the cosine theorem we find 
.
Answer. .
Training work 3. Angle between two planes
1.
Cubed A…D 1 find the tangent of the angle between the planes
ABC And C.B. 1D 1.
2.
Cubed A…D B
A 1C 1 and AB 1D 1.
3.
In a regular triangular prism ABCA 1B 1C
ABC And C.A. 1B 1.
4. In a regular quadrangular pyramid SABCD, all edges of which are equal to 1, find the cosine of the angle between the planes S
AD And SBC.
5. In a regular quadrangular pyramid SABCD, all edges of which are equal to 1, find the cosine of the dihedral angle formed by the faces
SBC And SCD.
6.
In a regular hexagonal pyramid SABCDEF
SBC And S.E.F..
7. In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1, and the side edges are equal to 2, find the cosine of the angle between the planes
SAF And SBC.
8. In a regular hexagonal prism A… F 1, all edges of which are equal to 1, find the tangent of the angle between the planes
ABC And D.B. 1F 1.
Solutions to problems 4.1 – 4.3 of diagnostic work
4.1. Solution. Since it's straight D 1F 1 perpendicular to the plane AFF 1, then the segment A.F. 1 will be the required perpendicular dropped from the point A directly D 1F 1. Its length is .
4.2. First solution A.H. right triangle ABD 1, in which AB = 1, AD 1 = , BD 1 = . For area S . Where do we find it from? A.H. = .
Second solution. The required perpendicular is the height A.H. right triangle ABD 1, in which AB = 1, AD 1 = , BD 1 = . Triangles BAD 1 and B.H.A. AD 1:BD 1 = A.H.:AB. Where do we find it from? A.H. = .
Third solution. The required perpendicular is the height A.H. right triangle ABD 1, in which AB = 1, AD 1 = , BD 1 = . Where 
and therefore 
Answer. .
4.3. The required distance from the point F to a straight line B.G. equal to height FH triangle FBG, in which FB = FG = , B.G.= . Using the Pythagorean theorem we find FH = .
Training work 4. Distance from a point to a line
1.
In a unit cube A…D 1 find the distance from the point B to a straight line D.A. 1.
2.
In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the distance from the point B to a straight line A.C. 1.
3. In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the distance from the point S to a straight line B.F..
4.
In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the distance from the point B to a straight line S.A..
5.
In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance from the point B to a straight line A 1F 1.
6. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance from the point B to a straight line A 1D 1.
7.
In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance from the point B to a straight line F.E. 1.
8. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance from the point B to a straight line AD 1.
Solutions to problems 5.1 – 5.3 of diagnostic work
5.1.
First solution. Let O– the middle of the segment BD. Straight BD perpendicular to the plane AOA 1. Therefore, planes BDA 1 and AOA A to the plane BDA 1, is the height A.H. right triangle AOA 1, in which A.A. 1 = 1, A.O. = , O.A. 1 = . For area S of this triangle the equalities hold . Where do we find it from? A.H. = .
Second solution. Let O– the middle of the segment BD. Straight BD perpendicular to the plane AOA 1. Therefore, planes BDA 1 and AOA 1 are perpendicular. The required perpendicular dropped from the point A to the plane BDA 1, is the height A.H. right triangle AOA 1, in which A.A. 1 = 1, A.O. = , O.A. 1 = . Triangles AOA 1 and HOA similar at three angles. Hence, A.A. 1:O.A. 1 = A.H.:A.O.. Where do we find it from? A.H. = .
Third solution. Let O– the middle of the segment BD. Straight BD perpendicular to the plane AOA 1. Therefore, planes BDA 1 and AOA 1 are perpendicular. The required perpendicular dropped from the point A to the plane BDA 1, is the height A.H. right triangle AOA 1, in which A.A. 1 = 1, A.O. = , O.A. 1 = . Where 
and therefore 
Answer. .
5.2.
First solution. Let O A.O. parallel to the line B.C. SBC O to plane SBC. Let G– the middle of the segment B.C.. Then straight O.G. perpendicular B.C. O to the plane SBC, is the height OH right triangle SOG. In this triangle O.G. = , S.G. = , SO= . For area S of this triangle the equalities hold . Where do we find it from? OH = .
Second solution. Let O– the center of the base of the pyramid. Straight A.O. parallel to the line B.C. and therefore parallel to the plane SBC. Therefore, the required distance is equal to the distance from the point O to plane SBC. Let G– the middle of the segment B.C.. Then straight O.G. perpendicular B.C. and the desired perpendicular dropped from the point O to the plane SBC, is the height OH right triangle SOG. In this triangle O.G. = , S.G. = , SO= . Triangles SOG And OHG similar at three angles. Hence, SO:S.G. = OH:O.G.. Where do we find it from? OH = .
Answer. .
5.3. First solution. Let O And O 1 – centers of the prism bases. Straight A.O. 1 parallel to the plane B.F.E. 1 and therefore the distance from the point A to plane B.F.E. 1 is equal to the distance from the line A.O. 1 to plane B.F.E. 1. Plane AOO 1 perpendicular to the plane B.F.E. 1 and therefore the distance from the straight line A.O. 1 to plane B.F.E. 1 is equal to the distance from the line A.O. 1 to the intersection line GG 1 planes AOO 1 and B.F.E. 1. Triangle AOO 1 rectangular, A.O. =
O.O. 1 = 1, GG 1 – its midline. Therefore, the distance between the lines A.O. 1 and GG 1 equals half the height OH triangle AOO 1, i.e. equal to .
Second solution. Let G– point of intersection of lines AD And B.F.. Angle between straight line AD and plane B.F.E. 1 is equal to the angle between the lines B.C. And B.C. 1 and equals 45o. Perpendicular A.H., dropped from the point A to the plane B.F.E. 1, equal to . Because A.G. = 0.5, then A.H. = .
Answer. .
Training work 5. Distance from point to plane
1.
In a unit cube A…D 1 find the distance from the point A to plane C.B. 1D 1.
2.
In a unit cube A…D 1 find the distance from the point A to plane BDC 1.
3.
In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the distance from the point A to plane B.C.A. 1.
4.
In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the distance from the point A to plane C.A. 1B 1.
5. In a regular quadrangular pyramid SABCD, all edges of which are equal to 1, find the distance from the point A to plane SCD.
6. In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the distance from the point A to plane SDE.
7. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance from the point A to plane D.E.A. 1.
8. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance from the point A to plane DEF 1.
Solutions to problems 6.1 – 6.3 of diagnostic work
6.1. Solution. Straight B.C. parallel to the plane S.A.D., which contains the straight line S.A.. Therefore, the distance between the lines S.A. And B.C. equal to the distance from the straight line B.C. to plane S.A.D..
Let E And F respectively the middle of the ribs AD And B.C.. Then the required perpendicular will be the height FH triangle S.E.F.. In a triangle S.E.F. we have: E.F. = 1, S.E. = SF= , height SO equal to . For area S triangle S.E.F. the equalities hold, from which we obtain .
6.2.
Solution. Planes AB 1D 1 and BDC 1, in which these lines lie, are parallel. Therefore, the distance between these straight lines is equal to the distance between the corresponding planes.
Diagonal C.A. 1 cube is perpendicular to these planes. Let's denote E And F diagonal intersection points C.A. 1 respectively with planes AB 1D 1 and BDC 1. Length of the segment E.F. will be equal to the distance between the lines AB 1 and B.C. 1. Let O And O 1 respectively, the centers of the faces ABCD And A 1B 1C 1D 1 cube. In a triangle ACE line segment OF parallel A.E. and goes through the middle A.C.. Hence, OF ACE and, therefore, E.F. = F.C.. Similarly, it is proved that O 1E– midline of the triangle A 1C 1F and, therefore, A 1E = E.F.. Thus, E.F. is one third of the diagonal C.A. 1, i.e. E.F. = .
Answer. .
6.3. Solution. Distance between lines A.A. 1 and CF 1 is equal to the distance between parallel planes ABB 1 and CFF 1 in which these lines lie. It is equal.
Training work 6. Distance between two straight lines
1.
In a unit cube A…D 1 find the distance between the lines B.A. 1 and D.B. 1.
2.
In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the distance between the lines CC 1 and AB.
3.
In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the distance between the lines AB And C.B. 1.
4.
In a regular quadrangular pyramid SABCD, all edges of which are equal to 1, find the distance between the lines S.B. And A.C..
5.
In a regular quadrangular pyramid SABCD, all edges of which are equal to 1, find the distance between the lines S.A. And CD.
6.
In a regular hexagonal pyramid SABCDEF S.B. And A.F..
7.
In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the distance between the lines S.B. And A.E..
8.
In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance between the lines BB 1 and E.F. 1.
Diagnostic work 1
1. Cubed A…D 1 find the angle between the lines B.A. 1 and B 1D 1.
2. In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the cosine of the angle between the lines AB 1 and B.C. 1.
3.
In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the cosine of the angle between the lines AB 1 and DC 1.
4. Cubed A…D 1 find the sine of the angle between the line A 1 D 1 and plane
5. In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the sine of the angle between the line AB and plane
6.
In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the sine of the angle between the line A.F. 1 and plane
7. In a regular quadrangular pyramid SABCD, all edges of which are equal to 1, find the cosine of the angle between the planes
ABC And SCD.
8.
In a regular hexagonal prism A…F
AFF 1 and BCC 1.
9. Cubed A…D 1 find the cosine of the angle between the planes
AB 1D 1 and C.B. 1D 1.
10. In a unit cube A…D 1 find the distance from the point B to a straight line D.A. 1.
11. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance from the point A to a straight line E.B. 1.
12.
In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the distance from the point A to a straight line SD.
13. In a unit cube A…D 1 find the distance from the point B to plane D.A. 1C 1.
14. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance from the point A to plane B.F.A. 1.
15.
In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the distance from the point A to plane S.C.E..
16.
In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the distance between the lines A.A. 1 and B.C..
17. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance between the lines BB 1 and CD 1.
18. In a unit cube A…D 1 find the distance between the lines AB 1 and BD 1.
Diagnostic work 2
1. Cubed A…D 1 find the angle between the lines AB 1 and BD 1.
2. In a regular quadrangular pyramid SABCD, all edges of which are equal to 1, point E– middle of the rib S.B.. Find the tangent of the angle between the lines S.A. And BE.
3. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the cosine of the angle between the lines AB 1 and BD 1.
4. Cubed A…D 1 find the sine of the angle between the line DD 1 and plane
5. In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the sine of the angle between the line A.F. and plane
6. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the sine of the angle between the line B.C. 1 and plane
7.
In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1, and the side edges are equal to 2, find the cosine of the angle between the planes
ABC And S.E.F..
8.
In a regular hexagonal prism A…F 1 find the angle between the planes
AFF 1 and BDD 1.
9. Cubed A…D 1 find the tangent of the angle between the planes
ABC And D.A. 1C 1.
10.
In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the distance from the point A to a straight line C.B. 1.
11.
In a regular hexagonal prism A … F 1, all edges of which are equal to 1, find the distance from the point A to a straight line BE 1.
12. In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the distance from the point A to a straight line S.C..
13.
In a unit cube A…D 1 find the distance from the point B to plane AB 1D 1.
14.
In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance from the point A to plane CEF 1.
15.
In a regular hexagonal pyramid SABCDEF, the sides of the base are equal to 1 and the side edges are equal to 2, find the distance from the point A to plane SBF.
16.
In a regular triangular prism ABCA 1B 1C 1, all edges of which are equal to 1, find the distance between the lines A.A. 1 and B.C. 1.
17. In a regular hexagonal prism A…F 1, all edges of which are equal to 1, find the distance between the lines BB 1 and F.E. 1.