1. Basic knowledge 1. Number and Algebra A, Number and Formula: 1, Rational Number Rational Number: ① Integer → Positive Integer /0/ Negative Integer ② Fraction → Positive Fraction/Negative Fraction Number Axis: ① Draw a horizontal straight line, take a point on the straight line to represent 0 (origin), select a certain length as the unit length, and specify the right direction on the straight line as the positive direction. ② Any rational number can be represented by a point on the number axis. (3) If two numbers differ only in sign, then we call one of them the inverse of the other number, and we also call these two numbers the inverse of each other. On the number axis, two points representing the opposite number are located on both sides of the origin, and the distance from the origin is equal. The number represented by two points on the number axis is always larger on the right than on the left. Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers. Absolute value: ① On the number axis, the distance between the point corresponding to a number and the origin is called the absolute value of the number. (2) The absolute value of a positive number is itself, the absolute value of a negative number is its reciprocal, and the absolute value of 0 is 0. Comparing the sizes of two negative numbers, the absolute value is larger but smaller. Operation of rational numbers: addition: ① Add the same sign, take the same sign, and add the absolute values. ② When the absolute values are equal, the sum of different symbols is 0; When the absolute values are not equal, take the sign of the number with the larger absolute value and subtract the smaller absolute value from the larger absolute value. (3) A number and 0 add up unchanged. Subtraction: Subtracting a number equals adding the reciprocal of this number. Multiplication: ① Multiplication of two numbers, positive sign of the same sign, negative sign of different sign, absolute value. ② Multiply any number by 0 to get 0. ③ Two rational numbers whose product is 1 are reciprocal. Division: ① Dividing by a number equals multiplying the reciprocal of a number. ②0 is not divisible. Power: the operation of finding the product of n identical factors A is called power, the result of power is called power, A is called base, and N is called degree. Mixing order: multiply first, then multiply and divide, and finally add and subtract. If there are brackets, calculate first. 2. Real irrational numbers: Infinitely circulating decimals are called the square roots of irrational numbers: ① If the square of a positive number X is equal to A, then this positive number X is called the arithmetic square root of A ... If the square of a number X is equal to A, then this number X is called the square root of A (3) Positive numbers have two square roots /0 square roots are 0/ negative numbers have no square roots. (4) Find the square root of a number, which is called the square root, where a is called the square root. Cubic root: ① If the cube of a number X is equal to A, then this number X is called the cube root of A. ② The cube root of a positive number is positive, the cube root of 0 is 0, and the cube root of a negative number is negative. The operation of finding the cube root of a number is called square root, where a is called square root. Real numbers: ① Real numbers are divided into rational numbers and irrational numbers. ② In the real number range, the meanings of reciprocal, reciprocal and absolute value are exactly the same as those of reciprocal, reciprocal and absolute value in the rational number range. ③ Every real number can be represented by a point on the number axis. 3. Algebraic algebraic expression: A single number or letter is also algebraic. Merge similar items: ① Items with the same letters and the same letter index are called similar items. (2) Merging similar items into one item is called merging similar items. (3) When merging similar items, we add up the coefficients of similar items, and the indexes of letters and letters remain unchanged. 4. Algebraic expressions and fractional algebraic expressions: ① The algebraic expression of the product of numbers and letters is called a monomial, the sum of several monomials is called a polynomial, and monomials and polynomials are collectively called algebraic expressions. ② In a single item, the index sum of all letters is called the number of times of the item. ③ In a polynomial, the degree of the term with the highest degree is called the degree of this polynomial. Algebraic expression operation: when adding and subtracting, if you encounter brackets, remove them first, and then merge similar items. Power operation: am+an = a (m+n) (am) n = amn (a/b) n = an/bn division. Multiplication of algebraic expressions: ① Multiply the monomial with the monomial, respectively multiply their coefficients and the power of the same letter, and the remaining letters, together with their exponents, remain unchanged as the factors of the product. (2) Multiplying polynomial by monomial means multiplying each term of polynomial by monomial according to the distribution law, and then adding the products. (3) Polynomial multiplied by polynomial. Multiply each term of one polynomial by each term of another polynomial, and then add the products. There are two formulas: square difference formula/complete square formula algebraic division: ① monomial division, which is divided by the coefficient and the same base power respectively, as the factor of quotient; For the letter only contained in the division formula, it is used as the factor of quotient together with its index. (2) Polynomial divided by single item, first divide each item of this polynomial by single item, and then add the obtained quotients. Factorization: transforming a polynomial into the product of several algebraic expressions. This change is called factorization of this polynomial. Methods: Common factor method, formula method, grouping decomposition method and cross multiplication were used. Fraction: ① Algebraic expression A is divided by algebraic expression B. If the divisor B contains a denominator, then this is a fraction. For any fraction, the denominator is not 0. ② The numerator and denominator of the fraction are multiplied or divided by the same algebraic expression that is not equal to 0, and the value of the fraction remains unchanged. Fractional operation: multiplication: take the product of molecular multiplication as the numerator of the product, and the product of denominator multiplication as the denominator of the product. Division: dividing by a fraction is equal to multiplying the reciprocal of this fraction. Addition and subtraction: ① Addition and subtraction with denominator fraction, denominator unchanged, numerator addition and subtraction. ② Fractions with different denominators shall be divided into fractions with the same denominator first, and then added and subtracted. Fractional equation: ① The equation with unknown number in denominator is called fractional equation. ② The solution whose denominator is 0 is called the root increase of the original equation. B. Equation and inequality 1, equation and system of equations One-dimensional linear equation: ① There is only one unknown in an equation, and the exponent of the unknown is 1. Such an equation is called a one-dimensional linear equation. ② Adding or subtracting or multiplying or dividing (non-0) an algebraic expression on both sides of the equation at the same time, the result is still an equation. Steps to solve a linear equation with one variable: remove the denominator, shift the term, merge the similar terms, and change the unknown coefficient into 1. Binary linear equation: An equation that contains two unknowns and all terms are 1 is called binary linear equation. Binary linear equations: The equations composed of two binary linear equations are called binary linear equations. A set of unknown values suitable for binary linear equation is called the solution of this binary linear equation. The common * * * solution of each equation in a binary linear system of equations is called the solution of this binary linear system of equations. Methods of solving binary linear equations: substitution elimination method/addition and subtraction elimination method. One-dimensional quadratic equation: an equation with only one unknown term and the highest coefficient of the unknown term is 2. 1) The relationship between quadratic functions of one-dimensional quadratic equation has been studied by everyone, and we have a deep understanding of it, such as the solution and the representation in the image. In fact, the quadratic equation of one variable can also be expressed by quadratic function. In fact, the quadratic equation of one variable is also a special case of quadratic function. Then, if expressed in a plane rectangular coordinate system, the quadratic equation of one variable is the intersection of the X axis in the image and the quadratic function. Which is the solution of the equation. 2) The solution of a quadratic equation. As we all know, the quadratic function has a vertex (-b/2a, 4ac-b2/4a), which is very important for everyone to remember, because as mentioned above, the quadratic equation with one variable is also a part of the quadratic function, so he also has his own solution, with which he can find all the solutions of the quadratic equation with one variable (65434). The same is true for solving quadratic equations with one variable. Using this, the equation is transformed into several products to solve (3) formula method. This method can also be used as a general method to solve quadratic equations with one variable. The root of the equation is x 1 = {-b+√ [B2-4ac]}/2a. X2 = {-b-√ [B2-4ac]}/2a3) Step of solving a quadratic equation with one variable: (1) Matching method Step: First, move the constant term to the right of the equation, then change the coefficient of the quadratic term into 1, and add the square of half the coefficient of 1 at the same time, and finally match. Formula method (here refers to the formula method in factorization) or cross multiplication, if possible, can be converted into the form of product. (3) Substitute the coefficient of quadratic equation by formula method, in which the coefficient of quadratic term is a, the coefficient of linear term is b, and the coefficient of constant term is c4. ) Vitta Theorem Understanding Vitta Theorem with Vitta Theorem In a quadratic equation, the sum of two roots is =-b/a, and all the coefficients in a quadratic equation can be obtained by using Vitta Theorem, which is commonly used in the topic. 5) The root of the quadratic equation of one variable is understood by the discriminant of the root, which can be written as "δ" and read as "Tiao ta", and δ= B2-4ac, which can be divided into three cases: i When δ> 0, the quadratic equation of one variable has two unequal real roots; II When △=0, the quadratic equation of one variable has two identical real roots; III When △, =, and 0, pass through quadrant124; When k > 0 and b < 0, pass through quadrant134; When k > 0 and b > 0, pass through quadrant 123. ④ When k > 0, y value increases with the increase of x value, and when x < 0, y value decreases with the increase of x value. Second, space and graphics A. Understanding of graphics 1, points, lines, points, lines and surfaces: ① Graphics are composed of points, lines and surfaces. (2) Lines intersecting face to face and points where lines intersect. (3) Points move into lines, lines move into planes, and planes move into adults. Unfolding and folding: ① In a prism, the intersection of any two adjacent faces is called an edge, and the side edge is the intersection of two adjacent edges. All sides of the prism are equal in length, the upper and lower bottom surfaces of the prism are the same in shape, and the side surfaces are cuboids. (2) N prism is a prism with N faces on its bottom. Cutting a geometric figure: cutting a figure with a plane, and the cutting surface is called a section. Views: main view, left view and top view. Polygon: It is a closed figure composed of some line segments that are not on the same straight line. Arc and sector: ① A figure consisting of an arc and two radii passing through the end of the arc is called a sector. ② The circle can be divided into several sectors. 2. Angle line: ① The line segment has two endpoints. (2) The line segment extends infinitely in one direction to form a ray. A ray has only one endpoint. ③ A straight line is formed by the infinite extension of both ends of a line segment. A straight line has no end. Only one straight line passes through two points. Comparison length: ① Of all the connecting lines between two points, the line segment is the shortest. ② The length of the line segment between two points is called the distance between these two points. Measurement and expression of angle: ① An angle consists of two rays with a common endpoint, and the common endpoint of the two rays is the vertex of the angle. ② One degree of 1/60 is one minute, and one minute of1/60 is one second. Comparison of angles: ① An angle can also be regarded as a light rotating around its endpoint. (2) The ray rotates around its endpoint. When the ending edge and the starting edge are on a straight line, the angle formed is called a right angle. The starting edge continues to rotate, and when it coincides with the starting edge again, the angle formed is called fillet. (3) The ray from the vertex of an angle divides the angle into two equal angles, and this ray is called the bisector of the angle. Parallelism: ① Two straight lines that do not intersect in the same plane are called parallel lines. ② One and only one straight line is parallel to this straight line after passing through a point outside the straight line. If both lines are parallel to the third line, then the two lines are parallel to each other. Perpendicular: ① If two lines intersect at right angles, they are perpendicular to each other. (2) The intersection of two mutually perpendicular straight lines is called vertical foot. ③ On the plane, there is one and only one straight line perpendicular to the known straight line at one point. Perpendicular bisector: A straight line perpendicular to and bisecting a line segment is called perpendicular bisector. The perpendicular bisector in perpendicular bisector must be a line segment, not a ray or a straight line, which is related to the infinite extension of rays and straight lines. Look at the back, the middle vertical line is a straight line, so when drawing the middle vertical line, the line segment should pass through two points and then two points (about drawing, we will talk about it later). Midline theorem: property theorem: the distance from a point on the midline to both ends of the line segment is equal; Decision theorem: The point with the same distance from the endpoint of line segment 2 is the bisector of an angle on the vertical line of this line segment: the ray bisected by an angle is called the bisector of that angle. There are several points to note in the definition, that is, the bisector of an angle is a ray, not a line segment or a straight line. Many times there will be a straight line in the topic, which is the symmetry axis of the bisector, which also involves the problem of trajectory. The bisector of an angle is a point property theorem with equal distance to both sides of the angle: a point on the bisector of an angle is equal to both sides of the angle; The point with equal distance to both sides of the angle is the square on the bisector of the angle; A group of rectangles with equal adjacent sides are square properties: a square has all the properties of parallelogram, rhombus and rectangle: 1, rhombus with equal diagonal lines and rectangle with equal adjacent sides; Fundamental theorem 65442. Only one straight line passes through two points. 2. The shortest line segment between two points. 3. The same angle or the complementary angle of the same angle is equal. 4. The same angle or the complementary angle of the same angle is equal. 5. There is only one straight line perpendicular to the known straight line. 6. Of all the line segments connecting the points on the straight line and those outside the straight line, the vertical line segment is the shortest. 7. The parallel axiom passes through a point outside the straight line, and there is only one straight line parallel to this straight line. 8. If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other, with the same included angle of 10 and the same internal angle. 1 1 is complementary to the internal angle of the same side, which is 12 and13,65438 respectively. The inner angles of the same side are complementary to each other 15, the sum of the two sides of the theorem triangle is greater than the third side 16, the difference between the two sides of the inference triangle is less than the third side 17, the sum of the inner angles of the triangle and the three inner angles of the theorem triangle is equal to 180 18, and the two acute angles of the inference triangle are mutual. Inference 2: One external angle of a triangle is equal to the sum of two non-adjacent internal angles; Inference 3: One outer angle of a triangle is larger than any inner angle that is not adjacent to it; 2 1; The corresponding edge of congruent triangles is equal to the corresponding angle; 22; Angular axiom (SAS) has two triangles with equal angles; 23; Corner axiom (ASA). The congruence of two triangles with two angles corresponding to their clamping sides is 24, the congruence of two triangles with two angles corresponding to the opposite sides of an angle is 25 in inference, the congruence of two triangles with three sides is 26 in the side-by-side axiom, and the congruence of right triangles with two hypotenuses and a right-angled side is 27 in the hypotenuse and right-angled side axiom (HL). Theorem 1 bisector of an angle. The bisector of an angle is the set of all points with equal distance to both sides of the angle 30. The property theorem of isosceles triangle: the two base angles of isosceles triangle are equal (that is, equal corners) 3 1. It is inferred that the bisector of the top angle of the isosceles triangle bisects the bottom surface and is perpendicular to the bottom surface 32, and the bisector of the top angle of the isosceles triangle, the midline on the bottom surface and the height on the bottom surface coincide. Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60 34. Decision Theorem of Isosceles Triangle If a triangle has two equal angles, then the opposite sides of the two angles are equal (equal sides) 35. Inference 1 A triangle with three equal angles is an equilateral triangle 36. Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle 37. If the acute angle is equal to 30, the right-angled side it faces is equal to half of the hypotenuse 38, the median line on the hypotenuse of a right triangle is equal to half of the hypotenuse 39, the distance between the point on the perpendicular line of a theorem line segment and the two endpoints of the line segment is equal to 40, and the inverse theorem and the point with the same distance between the two endpoints of a line segment are 4 1 on the perpendicular line segment. The middle vertical line of the line segment can be regarded as the set of all points with equal distance from both ends of the line segment. Theorem 1 Two figures symmetrical about a straight line are conformal. Theorem 2 If two figures are symmetrical about a straight line, the symmetry axis is perpendicular bisector 44 connecting the corresponding points. Theorem 3 Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, the intersection point is on the symmetry axis. 45. Inverse Theorem If the connecting line of the corresponding points of two graphs is vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line. 46. Pythagorean Theorem The sum of squares of two right-angled sides A and B of a right-angled triangle is equal to the square of hypotenuse C, that is, a2+b2=c247 and the inverse theorem of Pythagorean Theorem. If the three sides of a triangle are related to A, B and C, and a2+b2=c2, then this triangle is a right triangle 48. The sum of the inner angles of quadrilateral is equal to 360 49, the sum of the outer angles of quadrilateral is equal to 360 50, and the sum of the inner angles of polygon and theorem N is equal to (n-2) ×1805/kloc-. It is inferred that the parallel line segment sandwiched between two parallel lines is equal to 55. The parallelogram property theorem 3 The diagonal of the parallelogram is bisected by 56. Parallelogram decision theorem 1 Two sets of quadrilaterals with equal diagonals are parallelograms 57. Parallelogram Decision Theorem 2 Two groups of quadrangles with equal opposite sides are parallelograms 58. Parallelogram Decision Theorem 3 The quadrangles whose diagonals are bisected are parallelogram 59 and parallelogram Decision Theorem 4. A set of parallelograms with parallel and equal opposite sides is parallelogram 60, rectangle property theorem 1, rectangle property theorem 2, rectangle diagonal equality 62, rectangle judgment theorem 1, and a quadrilateral with three right angles is rectangle 63. Rectangular Decision Theorem 2 A parallelogram with equal diagonals is a rectangle 64, and all four sides of the rhombus property theorem 1 rhombus are equal to 65. The rhombus property theorem 2 has diagonal lines perpendicular to each other, and each diagonal line bisects a set of diagonal lines.