Senior one must have two mathematical knowledge.
1. Definition of a circle: The set of points whose distance to a point on a plane is equal to a fixed length is called a circle, the fixed point is the center of the circle, and the fixed length is the radius of the circle.
2. Equation of circle
(1) standard equation, center and radius r;
(2) General equation
At that time, the equation represented a circle. At this point, the center is and the radius is.
At that time, I said a point; At that time, the equation did not represent any graph.
(3) Method of solving cyclic equation:
Generally, the undetermined coefficient method is adopted: first determine, then solve. Determining a circle requires three independent conditions. If the standard equation of a circle is used,
Demand a, b, r; If you use general equations, you need to find d, e, F e, f;
In addition, we should pay more attention to the geometric properties of the circle: for example, the vertical line of a chord must pass through the origin, so as to determine the position of the center of the circle.
3. Summary of two compulsory knowledge points in high school mathematics: the positional relationship between straight line and circle:
The positional relationship between a straight line and a circle includes three situations: separation, tangency and intersection:
(1) Set a straight line and a circle, and the distance from the center of the circle to L is, then there is; ;
(2) Tangent to a point outside the circle: ①k does not exist, so verify the existence of ②k, establish an oblique equation, and solve k with the distance from the center of the circle to the straight line = radius, and get two solutions of the equation.
(3) The tangent equation of a point passing through a circle: circle (x-a)2+(y-b)2=r2, and a point on the circle is (x0, y0), then the tangent equation passing through that point is (x0-a) (x-a)+(y0-b) (y-b) =
4. The positional relationship between circles: it is determined by comparing the sum (difference) of the radii of two circles with the distance (d) between the center of the circle.
Set a circle,
The positional relationship between two circles is usually determined by comparing the sum (difference) of the radii of the two circles with the distance (d) between the center of the circle.
At that time, the two circles were separated, and there were four common tangents at this time;
At that time, the two circles were circumscribed, and the connection line crossed the tangent point, with two outer tangents and one inner common tangent;
At that time, the two circles intersect, and the connecting line bisects the common chord vertically, and there are two external tangents;
At that time, two circles were inscribed, and the connecting line passed through the tangent point, and there was only one common tangent;
At that time, two circles included; It was concentric circles.
Note: when two points on the circle are known, the center of the circle must be on the vertical line in the middle; It is known that two circles are tangent and two centers are tangent to the tangent point.
5, spatial point, straight line, plane position relationship
Axiom 1: If two points of a straight line are on a plane, then all points of the straight line are on this plane.
Application: judging whether a straight line is in a plane.
Express axiom1in symbolic language;
Axiom 2: If two non-coincident planes have a common point, then they have one and only one common straight line passing through the point.
Symbol: Plane α and β intersect, the intersection line is A, and it is denoted as α ∩ β = A. 。
Symbolic language:
The role of axiom 2:
It is a method to judge the intersection of two planes.
② Explain the relationship between the intersection line of two planes and the common point of two planes: the intersection line must pass through the common point.
③ It can be judged that a point is on a straight line, which is an important basis for proving several points.
Axiom 3: One and only one plane passes through three points that are not on the same straight line.
Inference: a straight line and a point outside the straight line determine a plane; Two intersecting straight lines define a plane; Two parallel lines define a plane.
Axiom 3 and its corollary: ① It is the basis for determining planes in space ② It is the basis for proving plane coincidence.
Axiom 4: Two lines parallel to the same line are parallel to each other.
Methods of learning mathematics well
First, pay attention to the lecture in class and review it in time after class.
Pay special attention to the study of basic knowledge and skills in class, and review in time after class, leaving no doubt.
First of all, before doing all kinds of exercises, you should recall the knowledge points that the teacher said and correctly grasp the reasoning process of various formulas. If you are not clear, you should try your best to recall them instead of turning to the book immediately. Finish your homework independently and be diligent in thinking. For some problems that are difficult to solve for a while because of their unclear thinking, let yourself calm down, analyze the problems carefully and find ways to solve them by yourself. At every learning stage, we should sort out and summarize, and combine the points, lines and surfaces of knowledge into a knowledge network and bring it into our own knowledge system.
Second, do more questions appropriately and develop good problem-solving habits.
1, if you want to learn mathematics well, you must do more problems and be familiar with the problem-solving ideas of various types of questions.
2. At first, we should start with the basic problems, take the exercises in the textbook as the standard, practice repeatedly, lay a good foundation, and then find some extracurricular exercises to help broaden our thinking, improve our ability to analyze and solve problems, and master the general rules of solving problems.
3. For some error-prone topics, you can prepare a set of wrong questions, write your own problem-solving ideas and correct problem-solving processes, and compare them to find out your own mistakes so as to correct them in time.
4. Develop good problem-solving habits at ordinary times. Let your energy be highly concentrated, make your brain excited, think quickly, enter the best state, and use it freely in the exam. Practice has proved that at the critical moment, your problem-solving habit is no different from your usual practice.
Senior one must have a complete set of mathematics knowledge.
① Definition of non-planar straight lines: two straight lines that are different from each other on any plane.
② Non-planar straight line properties: neither parallel nor intersecting.
③ Determination of out-of-plane straight line: A straight line passing through a point out of plane and a point in plane is an out-of-plane straight line.
(4) Angle formed by straight lines in different planes: when two straight lines are parallel, they form an acute angle or a right angle. The angle range formed by two straight lines in different planes is (0,90). If the angle formed by straight lines of two different planes is a right angle, we say that the straight lines of two different planes are perpendicular to each other.
To find the angles formed by straight lines on different planes:
A. By defining the construction angle, one can be fixed, the other can be translated, or both can be translated to a special position at the same time, and the vertex can be selected at a special position. B, prove that the angle made is the angle to be found. C, using the triangle can find the angle.
(7) Equiangular theorem: If two sides of one angle and two sides of another angle are parallel respectively, then the two angles are equal or complementary.
(8) The positional relationship between spatial straight line and plane.
A straight line is in a plane-there are countless things in common.
Symbolic representation of three positional relationships: aαa∪α= Aa‖α.
(9) positional relationship between planes: parallel-no common point; α‖β
Intersection-there is a common straight line. α ∩ β = B。
2. Spatial parallelism.
Determination and properties of (1) parallel lines and planes
Theorem for judging that a straight line is parallel to a plane: if a straight line out of the plane is parallel to a straight line in the plane, the straight line is parallel to the plane.
Lines, lines, parallel lines, parallel planes.
Theorem of Parallelism between Straight Lines and Plane: If a straight line is parallel to a plane, the plane passing through the straight line intersects the plane.
Then this straight line is parallel to the intersection line. Line-plane parallel lines are parallel.
(2) The judgment and nature of parallelism between planes.
Theorem for judging the parallelism of two planes
(1) If two intersecting lines in one plane are parallel to the other plane, the two planes are parallel.
(parallel lines and planes → parallel planes),
(2) If two sets of intersecting straight lines are parallel in two planes, the two planes are parallel.
(parallel lines → parallel planes),
(3) Two planes perpendicular to the same straight line are parallel,
Property Theorem of Parallel Two Planes
(1) If two planes are parallel, a straight line in one plane is parallel to the other plane. (Plane parallel → Line parallel)
(2) If two parallel planes intersect with the third plane, their intersection lines are parallel (plane parallel → line parallel).
3. Vertical problem in space
(1) Definition of line, surface and line-surface verticality
(1) Perpendicularity of two straight lines with different planes: If the angle formed by two straight lines with different planes is a right angle, the two straight lines with different planes are said to be perpendicular to each other.
② Line-plane verticality: If a straight line is perpendicular to any straight line in a plane, it is said that the straight line is perpendicular to the plane.
③ Plane is perpendicular to the plane: if two planes intersect, the dihedral angle (the figure formed by two half planes starting from a straight line) is a straight dihedral angle (the plane angle is a right angle), which means that the two planes are perpendicular.
(2) Determination of vertical relation and property theorem.
(1) The judging theorem and property theorem of the perpendicularity between a straight line and a plane.
Decision theorem: If a straight line is perpendicular to two intersecting straight lines on a plane, then the straight line is perpendicular to the plane.
Property theorem: If two straight lines are perpendicular to a plane, then the two straight lines are parallel.
(2) The judgment theorem and property theorem of vertical plane.
Decision theorem: If one plane passes through the perpendicular of the other plane, then the two planes are perpendicular to each other.
Property Theorem: If two planes are perpendicular to each other, then the straight line perpendicular to their intersection in one plane is perpendicular to the other plane.
4. Spatial angle problem
(1) Angle between straight lines
① Angle formed by two parallel straight lines: specified as.
The angle formed by the intersection of two straight lines: the angle formed by the intersection of two straight lines is not greater than the right angle, which is called the angle formed by these two straight lines.
(3) Angle formed by two straight lines with different planes: When passing through any point o in space, make two straight lines with different planes A and B parallel to form two intersecting straight lines, and the angle formed by these two intersecting straight lines is called the angle formed by two straight lines with different planes.
(2) The angle formed by a straight line and a plane
(1) the angle between the parallel lines of the plane and the plane is specified as. (2) the angle between the perpendicular of the plane and the plane is specified as.
(3) The angle formed by the oblique line of the plane and the plane: the acute angle formed by the oblique line of the plane and its projection in the plane is called the angle formed by this straight line and this plane.
The idea of finding the included angle between oblique line and plane is similar to finding the included angle between straight lines on different planes: "one work, two proofs and three calculations".
When making an angle, project according to the definition key. From the definition of projection, the key lies in the point on the diagonal to the perpendicular to the surface.
When solving a problem, we should pay attention to mining two main information in the problem setting: (1) a point on the diagonal line to the perpendicular of the surface; (2) The diagonal or a point on the plane of the diagonal is perpendicular to the known surface, and the vertical line can be easily obtained from the vertical nature of the surface.
(3) The dihedral angle of dihedral angle and plane angle
① Definition of dihedral angle: The figure formed by two half planes starting from a straight line is called dihedral angle, this straight line is called the edge of dihedral angle, and these two half planes are called the faces of dihedral angle.
② Plane angle of dihedral angle: Take any point on the edge of dihedral angle as the vertex, and make two rays perpendicular to the edge in two planes respectively. The angle formed by these two rays is called the plane angle of dihedral angle.
Straight dihedral angle: A dihedral angle whose plane angle is a right angle is called a straight dihedral angle.
If the dihedral angle formed by two intersecting planes is a straight dihedral angle, then the two planes are vertical; On the contrary, if two planes are perpendicular, the dihedral angle formed is a straight dihedral angle.
(4) Calculation method of dihedral angle
Definition method: select the relevant point on the edge, and make a ray perpendicular to the edge in two planes through this point to get the plane angle.
Vertical plane method: when the vertical lines from one point to two surfaces in dihedral angle are known, the angle formed by the intersection of two vertical lines as the intersection of plane and two surfaces is the plane angle of dihedral angle.
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