Let a plane and a point not lying on it be given:
A perpendicular dropped from a given point onto a given plane is a segment connecting a given point with a point on the plane and lying on a straight line perpendicular to the plane;
- the end of this segment lying in the plane is called the base of the perpendicular;
- the distance from a point to a plane is the length of a perpendicular drawn from this point to the plane;
An inclined line drawn from a given point to a given plane is any segment connecting a given point with a point on the plane that is not perpendicular to the plane;
- the end of the segment lying in the plane is called the inclined base;
A segment connecting the bases of a perpendicular and an oblique drawn from the same point is called an oblique projection.
In the figure, from point A, perpendicular AB and inclined AC are drawn to the plane. Point B is the base of the perpendicular, point C is the base of the inclined one, BC is the projection of the inclined AC onto the plane.
Three Perpendicular Theorem:
If a straight line drawn on a plane through inclined base, perpendicular to it projections, then it is perpendicular inclined. And vice versa: If a straight line in a plane is perpendicular to an inclined one, then it is perpendicular and oblique projection.
Two intersecting planes are called perpendicular if the third plane, perpendicular to the line of intersection of these planes, intersects them along perpendicular lines.
Example #1
A straight line is drawn through the center of a circle inscribed in a triangle, perpendicular to the plane of the triangle. Prove that each point on this line is equidistant from the sides of the triangle.
Let A, B, C be the points of contact of the sides of the triangle with the circle, O be the center of the circle and S be the point on the perpendicular. Since the radius OA is perpendicular to the side of the triangle, then, according to the theorem of three perpendiculars, the segment SA is perpendicular to this side, and its length is the distance from the point S to the side of the triangle. According to the Pythagorean theorem SA= , where r is the radius of the inscribed circle. Similarly we find: 
, i.e. all distances from point S to the sides of the triangle are equal.
Control questions:
- What is a perpendicular dropped from a given point onto a plane?
- What is oblique projection?
Practical part:
1. Given a straight line a and a plane. Draw through a line a a plane perpendicular to the plane.
2. Prove that if a line is parallel to a plane, then all its points are at the same distance from the plane.
3. Two inclined ones are drawn from a point to a plane, one of which is 20 cm larger than the other. The inclined projections are 10 cm and 30 cm. Find the inclined ones.
4. The side of a square is 4 cm. The point equidistant from all vertices of the square is at a distance of 6 cm from the point of intersection of its diagonals. Find the distance from this point to the vertices of the square.
5. Two inclined slopes are drawn from a point to a plane, equal to 10 cm and 17 cm. The difference in the projections of these inclined ones is 9 cm. Find the projections of the inclined ones.
6. Two inclined slopes are drawn from a point to a plane, equal to 23 cm and 33 cm. Find the distance from this point to the plane if the projections of the inclined ones are in the ratio 2:3.
8. Line a is perpendicular to plane ABC. MD = 13. AC = 15, BC = 20. AC BC, MD AB. Find MC.
9. The legs of right triangle ABC (C = 90°) are equal to 4 cm and 3 cm. Point M is located at a distance of √6 cm from the plane of triangle ABC and at the same distance from all its vertices. Find the distance from point M to the vertices of the triangle.
Literature:
1. Mathematics: textbook for institutions beginning. and Wednesday prof. education / M.I. Bashmakov. –M.: Publishing Center “Academy”, 2010.
Independent work No. 5.
Solving problems involving counting the number of placements and permutations.
Purpose of the lesson: to master methods for solving problems involving calculating the number of samples
Theoretical part:
Combinatorics is a part of mathematics that is devoted to solving problems of choosing and arranging elements of a certain finite set in accordance with given rules, i.e. combinatorics solves the problem of selecting elements from a finite set and arranging these elements in some order.
Arrangements of n - elements by m - elements () are combinations made up of given n - elements by m - elements that differ from each other either in the elements themselves or in the order of the elements.
N(n-1)(n-2)…(n-m+1)
Example No. 1. How many three-digit numbers can be made from the numbers 1...9?
Permutations of n - elements are the number of placements of these n - elements by n - elements.
N(n-1)(n-2)…1=n!
Example No. 2. In how many ways can 5 books be arranged on a shelf?
Combinations of n - elements by m - elements are combinations made up of given n - elements by m - elements that differ from each other by at least one element.
Example No. 3. There are 30 students in a group. To pass the test, they must be divided into three groups. In how many ways can this be done?
Control questions:
1. Outline the goals of combinatorics.
2. What is the number of combinations of n elements of m called?
3. What is the number of placements of n elements into m called?
4. What is called a permutation of n elements?
Practical part:
1. In how many ways can a group of 25 people send 4 students to a scientific and practical conference?
2. Ten students shook hands. How many handshakes were there?
3. In how many ways can a three-color striped flag be made from seven pieces of material of different colors?
4. How many dictionaries must be published in order to be able to translate from any of the five languages into any of them?
5. Calculate:
6. Calculate:
7. Calculate: 5! + 6!
8. Find the number of arrangements of 10 elements of 4.
9. Calculate:
10. Thirty students exchanged photographs. How many photographs were there in total?
11. In how many ways can three people be selected from eight candidates for three positions?
12. Solve the equation:
13. Calculate the value of the expression:
14. Calculate the value of the expression.
5. Parallel lines
Two straight lines are called parallel, if, being in the same plane, they do not intersect.
Parallelism of lines is indicated by the sign || (for example AB||CD).
Theorem. Two perpendiculars to the same line are parallel.
Proof: If the perpendiculars intersected at some point, then two perpendiculars would be drawn from this point onto a straight line, which is impossible.
Names of angles obtained when two straight lines intersect with a third
Signs of parallelism.
If, when two straight lines intersect with a third straight line:
any corresponding angles are equal,
or some crosswise angles are equal,
or the sum of any two internal or two external one-sided angles is equal to 180 degrees,
then the two lines are parallel.
Axiom of parallel lines.
Through the same point it is impossible to draw two different lines parallel to the same line.Corollary 1. If a line intersects one of the parallel lines, then it also intersects the other.
Corollary 2. Two lines parallel to a third are parallel.
Angles with respectively parallel or perpendicular sides.
Theorem. If the sides of one angle are respectively parallel to the sides of another angle, then such angles are either equal or add up to two right angles.
Theorem. If the sides of one angle are respectively perpendicular to the sides of another angle, then such angles are either equal or add up to two right angles.
The sum of the angles of a triangle and a polygon.
Theorem. The sum of the angles of a triangle is equal to two right angles.
Consequences
:1. Every exterior angle of a triangle is equal to the sum of two interior angles.
2. If two angles of one triangle are equal to two angles of another triangle, then the third angles are also equal.
3. The sum of two acute angles of a right triangle is equal to a right angle.
Theorem. Sum of angles
n-gon is 180*(n-2) degrees.Theorem. The sum of the exterior angles of a polygon is equal to four right angles.
2. Given two lines intersecting at point C. Does any third line lie with them in the same plane, having a common point with each of these lines?
3.
4. The distance between two parallel planes is 8 cm. A straight segment, the length of which is 17 cm, is located between them so that its ends belong to the planes. Find the projection of this segment onto each of the planes.
5. Complete the sentence to make the correct statement:
D) I don’t know
6. Lines a and b are perpendicular. Points A and B belong to straight line a, points C and D belong to straight line b. Do straight lines AC and BD lie in the same plane?
7. In the cube ABCDA1B1C1D1 the diagonals of the faces AC and B1D1 are drawn. what is their relative position?
8. The edge of the cube ABCDA1B1C1D1 is equal to m. Find the distance between straight lines AB and CC1.
A) 2m B) 1/2m C) m D) I don’t know
9. Determine if the statement is true:
A) yes B) no C) not always D) don’t know
10. In the cube ABCDA1B1C1D1, find the angle between the planes BCD and ВСС1В1.
A) 90° B) 45° C) 0° D) 60°
11. Is there a prism with only one side face perpendicular to the base?
A) yes B) no C) I don’t know
12. Can the diagonal of a rectangular parallelepiped be less than its side edge?
A) yes B) no C) I don’t know
13. What is the lateral surface area of a cube with edge 10?
A) 40 B) 400 C) 100 D) 200
14. What is the total surface area of a cube if its diagonal is d?
A) 2d2 B) 6d2 B) 3d2 D) 4d2
15. How many planes of symmetry does a regular quadrangular pyramid have?
A) 2 B) 3 C) 4 D) 6
16. What is the axial section of any regular pyramid?
A) equilateral triangle
B) rectangle
B) trapezoid
D) isosceles triangle
please help me solve the test1. How many common lines can two different non-coinciding planes have?
A) 1 B) 2 C) an infinite number of D) none E) I don’t know
2. Given two lines intersecting at point C. Does any third line lie with them in the same plane, having a common point with each of these lines?
A) always yes B) always no C) lies, but not always D) I don’t know
3. Determine whether the statement is true:
Two planes are parallel if they are parallel to the same line.
A) yes B) no C) don’t know D) not always
4. The distance between two parallel planes is 8 cm. A straight segment, the length of which is 17 cm, is located between them so that its ends belong to the planes. Find the projection of this segment onto each of the planes.
A) 15 cm B) 9 cm C) 25 cm D) I don’t know
5. Complete the phrase to make the correct statement:
If a straight line lying in one of two perpendicular planes is perpendicular to their line of intersection, then it...
A) parallel to another plane
B) intersects with another plane
B) perpendicular to another plane
D) I don’t know
6. Lines a and b are perpendicular. Points A and B belong to straight line a, points C and D belong to straight line b. Do straight lines AC and BD lie in the same plane?
A) yes B) no C) not always D) don’t know
7. In the cube ABCDA1B1C1D1 the diagonals of the faces AC and B1D1 are drawn. what is their relative position?
A) intersect B) intersect C) parallel D) don’t know
8. The edge of the cube ABCDA1B1C1D1 is equal to m. Find the distance between straight lines AB and CC1.
A) 2m B) B) m D) I don’t know
9. Determine whether the statement is true:
If two straight lines form equal angles with the same plane, then they are parallel.
A) yes B) no C) not always D) don’t know
10. In the cube ABCDA1B1C1D1, find the angle between the planes BCD and ВСС1В1.
A) 90 B) 45 C) 0 D) 60
11. Is there a prism with only one side face perpendicular to the base?
A) yes B) no C) I don’t know
12. Can the diagonal of a rectangular parallelepiped be less than its side edge?
A) yes B) no C) I don’t know
13. What is the area of the lateral surface of a cube with edge 10?
A) 40 B) 400 C) 100 D) 200
14. What is the total surface area of a cube if its diagonal is d?
A) 2d2 B) 6d2 B) 3d2 D) 4d2
15. How many planes of symmetry does a regular quadrangular pyramid have?
A) 2 B) 3 C) 4 D) 6
16. What is the axial section of any regular pyramid?
A) equilateral triangle
B) rectangle
B) trapezoid
D) isosceles triangle
points that do not lie on the same line?
2. Can two different planes have only two common points?
Direct a andb intersect at a point M. A straight line c not passing through the point M intersects the lines A And b. Do all these three lines lie in the same plane? What is the relative position of the lines: 1) A 1 D And MN; 2) A 1 D And V 1C; 3) MN And A 1B1(Fig. 1). Direct A And b crossed with a straight line With. Can straight A And b be parallel? Two lines are parallel to the same plane. Can we say that these lines are parallel to each other? If not, what is their relative position? In Figure 2 there are straight lines type parallel. Points A And IN respectively belong to the direct type; b lies in a plane α, a\\b. What is the relative position of lines b and c? Given a quadrilateral ABCD and plane α. Its diagonals AC And BD parallel to the plane α. What is the mutual position AB and planes α? Planes α and β are parallel. Intersecting at a point M straight A And b intersect the plane α respectively at points IN And A, and the plane β - at points E And F Find an attitude
10. Flatness α passes through the diagonal of the base of the parallelepiped and the middle of one of the sides of the upper base. Determine the type of section.