1, related concepts of circle:

(1), determine that the elements of a circle are the center and radius.

(2)① A line segment connecting any two points on a circle is called a chord. ② The string passing through the center of the circle is called the diameter. The part between any two points on a circle is called an arc, or simply an arc. (4) An arc smaller than half a circle is called a bad arc. ⑤ An arc larger than half a circle is called an optimal arc. ⑥ In the same circle or equal circle, arcs that can coincide with each other are called equal arcs. ⑦ The vertex is on the circle, and the angle where both sides intersect with the circle is called the circumferential angle. You can draw a circle through the three vertices of a triangle, and only one can be drawn. The circle passing through the three vertices of a triangle is called the circumscribed circle of the triangle, the center of the circumscribed circle of the triangle is called the outer center of the triangle, the triangle is called the inscribed triangle of the circle, and the outer center is the intersection point of the vertical lines of each side of the triangle; The radius of the circumscribed circle of a right triangle is equal to half of the hypotenuse. The circle tangent to each side of the triangle is called the inscribed circle of the triangle, the center of the inscribed circle of the triangle is called the center of the triangle, and the triangle is called the circumscribed triangle. The center of the triangle is the intersection of the bisectors of the three internal angles of the triangle.

2. Related properties of circles

Theorem (1) In the same circle or equal circle, if the central angles are equal, then the arcs it faces are equal, the chords it faces are equal, and the chords it faces are equally spaced. It is inferred that in the same circle or equal circle, if one group of quantities of two central angles, two arcs, two chords or the distance between two chords are equal, then the other groups of quantities of their pairs are equal respectively.

(2) Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.

Inference 1: ① The diameter of bisecting the chord (not the diameter) is perpendicular to the chord and bisects the two arcs opposite the chord. (2) The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord. ③ bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and bisect another arc opposite to the chord.

Inference 2: The arcs sandwiched by two parallel chords of a circle are equal.

(3) Theorem of circumferential angle: the circumferential angle of an arc is equal to half the central angle of the arc. Inference 1 In the same circle or equal circle, the circumferential angles of the same arc or equal arc are equal, and so are the arcs with equal circumferential angles. Inference 2 The circumferential angles of semicircles or diameters are all equal, all equal to 90. The chord subtended by a circumferential angle of 90 is the diameter of the circle. Inference 3 If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.

(4) Determination and properties of the tangent: Determination theorem: The straight line passing through the outer end of the radius and perpendicular to this radius is the tangent of the circle. Property theorem: the tangent of a circle is perpendicular to the radius passing through the tangent point; A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point; A straight line perpendicular to the tangent through the tangent point must pass through the center of the circle.

(5) Theorem: Three points that are not on the same straight line determine a circle.

(6) The length of the line segment between a point on the tangent of a circle and the tangent point is called the tangent length from the point to the circle; Tangent Length Theorem: Two tangents of a circle can be drawn from a point outside the circle, and their tangents are equal in length. The connecting line between this point and the center of the circle bisects the included angle between the two tangents.

(7) The quadrangles inscribed in the circle are diagonally complementary, and one outer angle is equal to the inner diagonal; The sum of the opposite sides of the circumscribed quadrangle is equal;

(8) Chord angle theorem: the chord angle is equal to the circumferential angle of the arc pair it clamps.

(9) Proportional line segments related to a circle: the theorem of intersecting chords: the product of two intersecting chords in a circle is equal to the length of two lines divided by the intersection. If the chord intersects the diameter vertically, then half of the chord is the proportional average of two line segments formed by its separate diameters. Secant theorem: the tangent and secant of a circle are drawn from a point outside the circle, and the length of the tangent is the middle term in the length ratio of the two lines where this point intersects the secant. Draw two secants of a circle from a point outside the circle, and the product of the lengths of the two lines from that point to the intersection of each secant and the circle is equal.

(10) Two circles are tangent, and the connecting line intersects the tangent point; Two circles intersect, and the connecting line bisects the common chord vertically.

extreme

I. similar triangles (7 test sites)

Test center 1: similar triangles's concept, the meaning of similarity ratio, the enlargement and reduction of drawing graphics.

Assessment requirements: (1) Understand the concept of similarity; (2) Grasping the characteristics of similar figures and the significance of similarity ratio, we can enlarge and reduce the known figures as needed.

Test site 2: the proportion theorem of parallel lines and the related theorem of parallel lines on one side of a triangle.

Examination requirements: Use the proportional theorem of parallel lines to understand and solve some geometric proofs and geometric calculations.

Note: An edge judged to be parallel cannot be used as the corresponding line segment in the condition in proportion.

Test site 3: similar triangles's concept

Evaluation requirements: Based on the concept of similar triangles, master the characteristics of similar triangles and understand the definition of similar triangles.

Test site 4: similar triangles's judgment, nature and application

Examination requirements: Master similar triangles's judgment theorem (including preliminary theorem, three judgment theorems, right triangle similarity judgment theorem) and its properties, and can apply it well.

Test site 5: the center of gravity of the triangle

Assessment requirements: know the definition of center of gravity and apply it initially.

Test site 6: related concepts of vectors

Test center 7: addition and subtraction of vectors, multiplication of real numbers and vectors, and linear operation of vectors.

Examination requirements: master the multiplication of real numbers and vectors and the linear operation of vectors.

Second, the acute triangle ratio (2 test sites)

Test site 8: the concept of acute angle triangle ratio (sine, cosine, tangent and cotangent of acute angle), triangle ratio of 30 degrees, 45 degrees and 60 degrees.

Test 9: Solving Right Triangle and Its Application

Assessment requirements: (1) Understand the meaning of solving right triangle; (2) We can use acute angle complementation, acute angle triangle ratio and Pythagorean theorem to solve right triangle and some simple practical problems, especially we should skillfully use the value of special acute angle triangle ratio to solve right triangle.

Cubic and quadratic functions (4 test sites)

Test center 10: function definition domain and function value, function expression, constant function and other functions and related concepts.

Assessment requirements: (1) Understand variables, independent variables and dependent variables through examples, and understand the concept of function, its definition domain and function value; (2) Know the constant function; (3) Know the representation of functions and the meaning of symbols.

Test center 1 1: Solving quadratic resolution function with undetermined coefficient method.

Assessment requirements: (1) Master the method of finding the resolution function; (2) Using the undetermined coefficient method skillfully to find the resolution function.

Pay attention to the steps of solving the resolution function: primary design, secondary generation, three columns and four returns.

Test center 12: Draw the image of quadratic function.

Examination requirements: (1) Knowing the meaning of function image, I will draw function image by drawing points in plane rectangular coordinate system; (2) Understand the image of quadratic function and realize the idea of combining numbers with shapes; (3) Can draw an approximate image of quadratic function.

Test center 13: the image of quadratic function and its basic properties

Assessment requirements: (1) Establish the relationship among linear function, binary linear equation and straight line with intuitive images, and understand and master the properties of linear function; (2) The vertex coordinates of quadratic function are obtained by collocation method, and the related properties of quadratic function are described.

Note: (1) When solving problems, you should combine numbers and shapes; (2) The translation of quadratic function should be transformed into vertex.

Fourth, the related concepts of circle (6 test sites)

Test center 14: the concepts of central angle, chord and chord center distance.

Examination requirements: clearly understand the concepts of central angle, chord and chord center distance, and make correct judgments with these concepts.

Test center 15: the relationship between central angle, arc, chord and chord center distance.

Examination requirements: Understand clearly the relationship among central angle, arc, chord and chord center distance. On the basis of understanding the theorem of the relationship among central angle, arc, chord and chord center distance and its inference, use this theorem to make preliminary geometric calculation and geometric proof.

Test center 16: vertical diameter theorem and its inference

Vertical diameter theorem and its inference is one of the most important knowledge points in circular plate.

Test center 17: the positional relationship between straight lines and circles, and the quantitative relationship between circles.

The positional relationship between a straight line and a circle can be reflected in two aspects: the relationship between a straight line and a circle and the number of intersections.

Test center 18: Related concepts and basic properties of regular polygons.

Examination requirements: be familiar with the related concepts of regular polygons (such as radius, telecentricity, central angle, external angle and sum), and skillfully use the basic properties of regular polygons for reasoning and calculation. In the calculation of regular polygons, right-angled triangles composed of radius, vertex and half length are often used, which transforms the calculation problem of regular polygons into the calculation problem of right-angled triangles.

Test center 19: Draw regular triangles, quadrangles and hexagons.

Examination requirements: You can use basic drawing tools to correctly make regular triangles, quadrilaterals and hexagons.

Tisso

The fifth chapter equation (group)

Focus on the solution of linear equations of one variable, quadratic equations of one variable and linear equations of two dimensions; Related application problems of the equation (especially travel and engineering problems)

☆ Summary ☆

I. Basic concepts

1. equation, its solution (root), its solution, its solution (group)

2. Classification:

Second, the basis of solving the equation-the nature of equality

1.a=b←→a+c=b+c

2.a=b←→ac=bc(c≠0)

Third, the solution

1. Solution of linear equation with one variable: remove denominator → remove brackets → move terms → merge similar terms →

The coefficient becomes 1→ solution.

2. Solution of linear equations: ① Basic idea: "elimination method" ② Method: ① Replacement method.

② addition and subtraction

Fourth, a quadratic equation

1. Definition and general form:

2. Solution: (1) direct leveling method (pay attention to characteristics)

(2) Matching method (pay attention to the steps-knock down the root formula)

(3) Formula method:

(4) factorization method (feature: left =0)

3. The discriminant of the root:

4. The relationship between root and coefficient top:

Inverse theorem: If, then the quadratic equation of the root is:.

5. Common equation:

5. Equations that can be transformed into quadratic equations

1. Fractional equation

(1) definition

(2) Basic ideas:

⑶ Basic solution: ① Denominator removal ② Substitution method (such as).

(4) Root test and method

2. Unreasonable equation

(1) definition

(2) Basic ideas:

(3) Basic solution: ① Multiplication method (pay attention to skills! ! (2) substitution method (example), (4) root test and method.

3. Simple binary quadratic equation

A binary quadratic equation consisting of a binary linear equation and a binary quadratic equation can be solved by method of substitution.

Six, column equation (group) to solve application problems

summary

Solving practical problems by using equations (groups) is an important aspect of middle school mathematics department. The specific steps are as follows:

(1) review the questions. Understand the meaning of the question. Find out what is a known quantity, what is an unknown quantity, and what is the equivalent relationship between problems and problems.

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⑵ Set an element (unknown). ① Direct unknowns ② Indirect unknowns (often both). Generally speaking, the more unknowns, the easier it is to list the equations, but the more difficult it is to solve them.

⑶ Use algebraic expressions containing unknowns to express related quantities.

(4) Find the equation (some are given by the topic, some are related to this topic) and make the equation. Generally speaking, the number of unknowns is the same as the number of equations.

5] Solving equations and testing.

[6] answer.

To sum up, the essence of solving application problems by column equations (groups) is to first transform practical problems into mathematical problems (setting elements and column equations), and then the solutions of practical problems (column equations and writing answers) are caused by the solutions of mathematical problems. In this process, the column equation plays a role of connecting the past with the future. Therefore, the column equation is the key to solve the application problem.

Two commonly used equality relations

1. Travel problem (uniform motion)

Basic relationship: s=vt

(1) Meeting problem (at the same time):

(2) Follow-up questions (start at the same time):

If A starts in t hours, B starts, and then catches up with A at B, then

(3) sailing in the water:

2. batching problem: solute = solution × concentration

Solution = solute+solvent

3. Growth rate:

4. Engineering problems: Basic relationship: workload = working efficiency × working time (workload is often considered as "1").

5. Geometric problems: Pythagorean theorem, area and volume formulas of geometric bodies, similar shapes and related proportional properties.

Third, pay attention to the relationship between language and analytical formula.

For example, more, less, increase, increase to (to), at the same time, expand to (to), expand, ...

Another example is a three-digit number, where A has 100 digits, B has 10 digits and C has one digit. Then this three-digit number is: 100a+ 10b+c, not abc.

Fourth, pay attention to writing equal relations from the language narrative.

For example, if X is greater than Y by 3, then x-y=3 or x=y+3 or X-3 = Y, and if the difference between X and Y is 3, then x-y=3. Pay attention to unit conversion

Such as the conversion of "hours" and "minutes"; Consistency of s, v and t units, etc.