Chapter 1 Pythagorean Theorem

1, 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

2. Inverse theorem of Pythagorean theorem

If the lengths of three sides of triangle A, B and C are related, then this triangle is a right triangle.

3. Pythagoras number: Three positive integers that are satisfied, called Pythagoras number.

Chapter II Real Numbers

First, the concept and classification of real numbers

1, classification of real numbers

Positive rational number

Rational Numbers Zero Finite Decimals and Infinite Cyclic Decimals

Real negative rational number

Positive irrational number

Irrational number infinite acyclic decimal

Negative irrational number

2. Irrational number: Infinitely circulating decimals are called irrational numbers.

When understanding irrational numbers, we should grasp the moment of "infinite non-circulation", which can be summarized into four categories:

(1) An inexhaustible number, such as;

(2) Numbers with specific meanings, such as pi, or simplified numbers with pi, such as+8;

(3) Numbers with specific structures, such as 0.101001001… etc.

(4) Some trigonometric function values, such as sin60o, etc.

Second, the reciprocal, reciprocal and absolute value of real numbers.

1, reciprocal

A real number and its inverse are a pair of numbers (only two numbers with different signs are called inverse numbers, and the inverse of zero is zero). Seen from the number axis, the points corresponding to two opposite numbers are symmetrical about the origin. If a and b are opposites, then a+b=0, A =-B, and vice versa.

2. Absolute value

On the number axis, the distance between the point corresponding to a number and the origin is called the absolute value of the number. (|a|≥0). The absolute value of zero is itself and can also be regarded as its inverse. If |a|=a, then a ≥ 0; If |a|=-a, then a≤0.

Step 3 count down the seconds

If A and B are reciprocal, there is ab= 1, and vice versa. The numbers whose reciprocal equals itself are 1 and-1. Zero has no reciprocal.

4. Counting axes

The straight line that specifies the origin, positive direction and unit length is called the number axis (pay attention to the three elements specified above when drawing the number axis).

When solving problems, we should really master the idea of combining numbers with shapes, understand the one-to-one correspondence between real numbers and points on the number axis, and use them flexibly.

Step 5 estimate

Three, square root, arithmetic square root and cubic root

1, arithmetic square root: Generally speaking, if the square of a positive number X is equal to A, that is, x2=a, then this positive number X is called the arithmetic square root of A. In particular, the arithmetic square root of 0 is 0.

Representation: recorded as "",pronounced as the root number a.

Property: There is only one arithmetic square root of positive number and zero, and the arithmetic square root of zero is zero.

2. Square root: Generally speaking, if the square of a number X is equal to A, that is, x2=a, then this number X is called the square root (or quadratic root) of A. ..

Representation: the square root of a positive number is recorded as ""and pronounced as "positive and negative root sign A".

Property: a positive number has two square roots in opposite directions; The square root of zero is zero; Negative numbers have no square root.

Square root: The operation of finding the square root of a number is called square root.

Double nonnegativity of attention:

3. Cubic root

Generally speaking, if the cube of a number X is equal to A, that is, x3=a, then this number X is called the cube root (or cube root) of A. ..

Representation method: recorded as

Property: positive numbers have positive cubic roots; Negative numbers have negative cubic roots; The cube root of zero is zero.

Note: This shows that the negative sign in the cube root symbol can be moved outside the root sign.

Fourth, the comparison of real numbers.

1, real number comparison size: positive number is greater than zero, negative number is less than zero, and positive number is greater than all negative numbers; The number represented by two points on the number axis is always larger on the right than on the left; Two negative numbers, the larger one has the smaller absolute value.

2. Several common methods of comparing real numbers.

(1) axis comparison: the number on the right is always greater than the number on the left of the two numbers represented on the axis.

(2) difference comparison: let a and b be real numbers,

(3) quotient comparison method: let a and b be two positive real numbers,

(4) absolute value comparison method: let a and b be two negative real numbers, then.

(5) Flat method: Let A and B be two negative real numbers, then.

Five, the arithmetic square root calculation (quadratic root)

1, containing the quadratic root sign ""; The root sign a must be non-negative.

2. Nature:

( 1)

(2)

(3) ( )

(4) ( )

3. If the operation result contains the form of "",the following requirements must be met: (1) The factor of the root sign is an integer and the factor is an algebraic expression; (2) The number of square roots does not contain factors or factors that can be opened to the maximum.

Six, the operation of real numbers

(1) Six operations: addition, subtraction, multiplication, division, power and root.

(2) Operation sequence of real numbers

First calculate the power sum root, then multiply and divide, and finally add and subtract. If there are brackets, count them first.

(3) Operation law

Additive commutative law

associative law of addition

Commutative law of multiplication

Multiplicative associative law

Distribution law of multiplication to addition

Chapter III Translation and Rotation of Graphics

First, translation

1, definition

In a plane, the whole movement of a figure along a certain distance is called translation.

2. Nature

Before and after translation, the two figures are congruent figures, with corresponding points parallel and equal, corresponding line segments parallel and equal, and corresponding angles equal.

Second, rotation.

1, definition

On a plane, turning a figure around a fixed point in a certain direction by an angle is called rotation. This fixed point is called rotation center, and the rotation angle is called rotation angle.

2. Nature

The two figures before and after rotation are congruent figures, the distance between the corresponding point and the rotation center is equal, and the angle formed by the connecting line between the corresponding point and the rotation center is equal to the rotation angle.

The fourth chapter discusses the properties of quadrilateral.

First, the related concepts of quadrilateral

1, quadrilateral

On the same plane, a figure composed of four line segments that are not on the same straight line is called a quadrilateral.

2. The quadrilateral is unstable.

3. The interior angle sum theorem and exterior angle sum theorem of quadrilateral.

Theorem of the sum of quadrilateral internal angles: the sum of quadrilateral internal angles is equal to 360.

Theorem of the sum of quadrilateral external angles: the sum of quadrilateral external angles is equal to 360.

Inference: theorem of polygon interior angle sum: the sum of n polygon interior angles is equal to180;

Theorem of the sum of external angles of polygons: the sum of external angles of any polygon is equal to 360.

6. If the number of sides of a polygon is n, the diagonal of the polygon has * * *. Starting from a vertex of an n- polygon, (n-3) diagonal lines can be drawn, and the N-polygon can be divided into (n- 2) triangles.

Second, parallelogram

1, the definition of parallelogram

Two groups of parallelograms with parallel opposite sides are called parallelograms.

2. The properties of parallelogram

The opposite sides of the (1) parallelogram are parallel and equal.

(2) The adjacent angles of the parallelogram are complementary and the diagonals are equal.

(3) The diagonal of the parallelogram is equally divided.

(4) A parallelogram is a central symmetric figure, and the center of symmetry is the intersection of diagonals.

Common ground: (1) If a straight line passes through the intersection of two diagonals of a parallelogram, the midpoint of a line segment cut by a group of opposite sides is the intersection of the diagonals, and the straight line bisects the area of the parallelogram.

(2) Inference: The parallel segments sandwiched between two parallel lines are equal.

3. Determination of parallelogram

(1) Definition: Two groups of parallelograms with opposite sides are parallelograms.

(2) Theorem 1: Two groups of quadrangles with equal diagonals are parallelograms.

(3) Theorem 2: Two groups of quadrangles with equal opposite sides are parallelograms.

(4) Theorem 3: Quadrilaterals whose diagonals bisect each other are parallelograms.

(5) Theorem 4: A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.

4. The distance between two parallel lines

In two parallel lines, the distance between any point on a straight line and another straight line is called the distance between two parallel lines.

The distance between parallel lines is equal everywhere.

5. Area of parallelogram

S parallelogram = base length × height =ah

Third, rectangle.

1, the definition of rectangle

A parallelogram with a right angle is called a rectangle.

2, the nature of the rectangle

(1) The opposite sides of the rectangle are parallel and equal.

(2) All four corners of a rectangle are right angles.

(3) The diagonals of the rectangle are equal and equally divided.

(4) The rectangle is both a central symmetrical figure and an axisymmetric figure; The center of symmetry is the intersection of diagonal lines (the distance from the center of symmetry to the four vertices of the rectangle is equal); There are two symmetrical axes, which are straight lines connecting the midpoints of opposite sides.

3. Determination of rectangle

(1) Definition: A parallelogram with right angles is a rectangle.

(2) Theorem 1: A quadrilateral with three right angles is a rectangle.

(3) Theorem 2: A parallelogram with equal diagonals is a rectangle.

4, the area of the rectangle

S rectangle = length× width =ab

Fourth, diamonds.

1, the definition of diamond

A set of parallelograms with equal adjacent sides is called a diamond.

2, the nature of the diamond

The four sides of the (1) diamond are equal and the opposite sides are parallel.

(2) The adjacent angles of the rhombus are complementary and the diagonals are equal.

(3) The diagonal lines of the diamond are divided vertically, and each diagonal line divides a set of diagonal lines equally.

(4) The rhombus is both a central symmetrical figure and an axisymmetric figure; The center of symmetry is the intersection of diagonal lines (the distance from the center of symmetry to the four sides of the diamond is equal); There are two symmetrical axes, which are the straight lines where the diagonal line lies.

3. Determination of diamond shape

(1) Definition: A set of parallelograms with equal adjacent sides is a rhombus.

(2) Theorem 1: A quadrilateral with four equilateral sides is a diamond.

(3) Theorem 2: Parallelograms with diagonal lines perpendicular to each other are diamonds.

4, the area of the diamond

S diamond = base length × height = half of the product of two diagonal lines

V. Square (3~ 10)

Definition of 1 and square

A group of parallelograms with equal adjacent sides and a right angle is called a square.

2, the nature of the square

(1) All four sides of a square are equal and the opposite sides are parallel.

(2) All four corners of a square are right angles.

(3) The two diagonals of a square are equal and equally divided vertically, and each diagonal bisects a set of diagonals.

(4) The square is both a central symmetrical figure and an axisymmetric figure; The center of symmetry is the intersection of diagonal lines; There are four symmetry axes, which are the straight line where the diagonal line is located and the straight line connecting the midpoint of the opposite side.

3. Determination of the square

The main basis for judging whether a quadrilateral is a square is definition, and there are two ways:

Prove that it is a rectangle first, and then prove that it is a diamond.

Prove it is a diamond, and then prove it is a rectangle.

4, the area of the square

Let the side length of a square be a and the diagonal length be b.

S squared =

Six, trapezoidal

(A) 1, trapezoidal related concepts

A quadrilateral whose opposite sides are parallel but not parallel is called a trapezoid.

The parallel sides of a trapezoid are called the base of the trapezoid. Generally, the shorter bottom is called the upper bottom and the longer bottom is called the lower bottom.

The two sides of a trapezoid that are not parallel are called the waist of the trapezoid.

The distance between the two bases of a trapezoid is called the height of the trapezoid.

2. Determination of trapezoid

(1) Definition: A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are trapezoid.

(2) A set of quadrilaterals with parallel and unequal opposite sides is a trapezoid.

(2) Definition of right-angled trapezoid: A trapezoid whose waist is perpendicular to the bottom is called a right-angled trapezoid.

Generally speaking, trapezoid can be divided into the following categories:

General trapezoid

Trapezoidal right-angled trapezium

Special trapezoid

isosceles trapezoid

(3) isosceles trapezoid

1, the definition of isosceles trapezoid

An isosceles trapezoid is called an isosceles trapezoid.

2, the nature of the isosceles trapezoid

(1) The two waists of an isosceles trapezoid are equal and the two bottoms are parallel.

(2) The two angles on the same bottom of the isosceles trapezoid are equal, and the two angles on the same waist are complementary.

(3) The diagonal lines of the isosceles trapezoid are equal.

(4) The isosceles trapezoid is an axisymmetric figure with only one axis of symmetry, that is, the middle vertical line with two bottoms.

3. Determination of isosceles trapezoid

(1) Definition: An isosceles trapezoid is an isosceles trapezoid.

(2) Theorem: A trapezoid with two equal angles on the same base is an isosceles trapezoid.

(3) A trapezoid with equal diagonals is an isosceles trapezoid. (Multiple choice questions and fill-in-the-blank questions can be used directly)

(4) the area of the trapezoid

(1) as shown in the figure.

(2) The area of the figure in the trapezoid:

① ;

② ;

③

Seven, knowledge points about the midpoint quadrilateral problem:

(1) The quadrilateral obtained by connecting the midpoints of four sides of an arbitrary quadrilateral in turn is a parallelogram;

(2) The quadrangle obtained by connecting the midpoints of four sides of the rectangle in turn is a rhombus;

(3) The quadrilateral obtained by connecting the midpoints of four sides of the diamond in turn is a rectangle;

(4) The quadrangle obtained by connecting the midpoints of four sides of the isosceles trapezoid in turn is a rhombus;

(5) The quadrilateral obtained by connecting the midpoints of four sides of the quadrilateral with equal diagonal lines in turn is a rhombus;

(6) connecting the midpoints of four sides of a quadrilateral with mutually perpendicular diagonals in turn to obtain a quadrilateral that is a rectangle;

(7) A quadrilateral obtained by sequentially connecting the midpoints of four sides of a quadrilateral with mutually perpendicular and equal diagonals is a square;

Eight, the central symmetry figure

1, definition

On the plane, a figure rotates around a point 180. If the figures before and after rotation overlap, then this figure is called a central symmetric figure, and this point is called its symmetric center.

2. Nature

(1) Two graphs that are symmetric about the center are congruent.

(2) With regard to two graphs with symmetrical centers, the connecting lines of symmetrical points all pass through and are equally divided by the symmetrical centers.

(3) With respect to two figures with symmetrical centers, the corresponding line segments are parallel (or on the same straight line) and equal.

3. Judges

If a straight line connecting the corresponding points of two graphs passes through a point and is equally divided by the point, then the two graphs are symmetrical about the point.

Nine, quadrilateral, rectangle, diamond, square, trapezoid, isosceles trapezoid, right-angle ladder diagram:

Chapter V Determination of Location

First of all, in a plane, two data are usually needed to determine the position of an object.

Second, the plane rectangular coordinate system and related concepts

1, plane rectangular coordinate system

In a plane, two mutually perpendicular axes with a common origin form a plane rectangular coordinate system. Among them, the horizontal axis is called X axis or horizontal axis, and the right direction is the positive direction; The vertical axis is called Y axis or vertical axis, and the orientation is positive; The x-axis and y-axis are collectively referred to as coordinate axes. Their common origin o is called the origin of rectangular coordinate system; The plane on which the rectangular coordinate system is established is called the coordinate plane.

2. In order to describe the position of a point in the coordinate plane conveniently, the coordinate plane is divided into four parts, namely the first quadrant, the second quadrant, the third quadrant and the fourth quadrant.

Note: The points on the X axis and Y axis (points on the coordinate axis) do not belong to any quadrant.

3. The concept of point coordinates

For any point P on the plane, the intersection point P is perpendicular to the X-axis and Y-axis respectively, and the numbers A and B corresponding to the vertical feet on the X-axis and Y-axis are respectively called the abscissa and ordinate of the point P, and the ordered number pair (A, B) is called the coordinate of the point P. ..

The coordinates of points are represented by (a, b), and the order is abscissa before, ordinate after, and there is a ","in the middle. The positions of horizontal and vertical coordinates cannot be reversed. The coordinates of points on the plane are ordered real number pairs. When (a, b) and (b, a) are the coordinates of two different points.

There is a one-to-one correspondence between points on the plane and ordered real number pairs.

4. Coordinate characteristics of different locations

(1), the coordinate characteristics of the midpoint of each quadrant.

Point P(x, y) is in the first quadrant.

Point P(x, y) is in the second quadrant.

Point P(x, y) is in the third quadrant.

Point P(x, y) is in the fourth quadrant.

(2) Characteristics of points on the coordinate axis

The point P(x, y) is on the X axis, and X is an arbitrary real number.

The point P(x, y) is on the y axis, and y is an arbitrary real number.

Point P(x, y) is on both X and Y axes, and both X and Y are zero, that is, the coordinate of point P is (0,0), that is, the origin.

(3) Coordinate characteristics of points on the bisector of two coordinate axes.

Point P(x, y) is equal to y (straight line y=x) on the bisector of the first and third quadrants.

Points P(x, y) are opposite to each other on the bisector of the second and fourth quadrants.

(4) Characteristics of the coordinates of points on a straight line parallel to the coordinate axis

The ordinate of each point on the straight line parallel to the X axis is the same.

The abscissa of each point on the straight line parallel to the Y axis is the same.

(5) Coordinate characteristics of points symmetrical about the X axis, Y axis or origin.

The abscissa of point P and point P' is equal to the X axis, and the ordinate is opposite, that is, the symmetrical point of point P(x, y) relative to the X axis is P'(x, -y).

The axisymmetrical ordinate of point P and point P' with respect to Y is equal, and the abscissa is opposite, that is, the symmetrical point of point P(x, y) with respect to Y axis is P'(-x, y).

Point P and point P' are symmetrical about the origin, and the abscissa and ordinate are opposite, that is, the symmetrical point of point P(x, y) about the origin is P'(-x, -y).

(6) Distance from point to coordinate axis and origin

Distance from point P(x, y) to coordinate axis and origin:

(1) The distance from the point P(x, y) to the X axis is equal to

(2) The distance from the point P(x, y) to the Y axis is equal to

(3) The distance from point P(x, y) to the origin is equal to

Third, the law of coordinate change and graphic change:

Changes in coordinates (x, y) Changes in graphs

X × a or y× a is stretched (compressed) horizontally or vertically to the original a times.

X × a and y× a are enlarged (reduced) by a factor.

X ×(-1) or Y× (- 1) is symmetrical about the y axis or the x axis.

X ×(-1) and Y× (- 1) are symmetrical about the center of the origin.

X +a or y+a translates one unit along the x or y axis.

X +a, y+a translate one unit along the x axis, and then translate one unit along the y axis.

Chapter VI Linear Functions

First, the function:

Generally speaking, there are two variables, X and Y, in a certain change process. If a value of X is given and a value of Y is determined accordingly, then we call Y a function of X, where X is the independent variable and Y is the dependent variable.

Second, the independent variable value range

The whole set of values of independent variables that make a function meaningful is called the range of independent variables. Generally speaking, we should consider algebraic expression (taking all real numbers), fraction (denominator is not 0), quadratic root (root is not negative) and practical significance.

Three representations of functions and their advantages and disadvantages

(1) relational expression (analysis) method

The functional relationship between two variables can sometimes be expressed by an equation containing these two variables and the symbols of digital operations. This representation is called relational (analytical) method.

(2) List method

A series of values of the independent variable x and the corresponding values of the function y are listed in a table to represent the functional relationship. This representation is called list method.

(3) Image method

The method of expressing functional relations with images is called image method.

Fourthly, the general steps of drawing its image by using functional relationship.

(1) List: List gives some corresponding values of independent variables and functions.

(2) Point tracking: Take each pair of corresponding values in the table as coordinates, and track the corresponding points on the coordinate plane.

(3) Connection: according to the order of independent variables from small to large, connect the tracked points with smooth curves.

Five, proportional function and linear function

The concepts of 1, proportional function and linear function

Generally speaking, if the relationship between two variables X and Y can be expressed in the form of (k, b is a constant, k 0), then Y is a linear function of X (X is an independent variable and Y is a dependent variable).

In particular, when b=0 in a linear function (that is, k is a constant, k 0), y is said to be a proportional function of x.

2. Images of linear functions: All images of linear functions are straight lines.

3. Main features of linear function and proportional function images:

The image of a linear function is a straight line passing through point (0, b); The image of the proportional function is a straight line passing through the origin (0,0).

Symbolic function image characteristics of symbol b of k

k & gt0b & gt0y

0 x

The image passes through the first, second and third quadrants, and y increases with the increase of x.

b & lt0y

0 x

The image passes through the first, third and fourth quadrants, and y increases with the increase of x.

K & lt0b & gt0y

0 x

The image passes through the first, second and fourth quadrants, and y decreases with the increase of x.

b & lt0

y

0 x

The image passes through two, three and four quadrants, and y decreases with the increase of x.

Note: When b=0, the linear function becomes a proportional function, which is a special case of linear function.

4, the nature of the proportional function

Generally speaking, the proportional function has the following properties:

(1) When k >: 0, the image passes through the first and third quadrants, and y increases with the increase of x;

(2) when k < 0, the image passes through the second and fourth quadrants, and y decreases with the increase of x.

5. Properties of linear functions

Generally, linear functions have the following characteristics:

(1) when k >: 0, y increases with the increase of x.

(2) when k < 0, y decreases with the increase of x.

6. Determination of the proportional function and the first resolution function.

Determining a proportional function is to determine the constant k(k0) in the definition of proportional function. To determine a linear function, we need to determine the constants K and b(k0) in the definition of linear function. The general method to solve this kind of problem is the undetermined coefficient method.

7, the relationship between linear function and linear equation:

Any one-dimensional linear equation can be transformed into the form of kx+b=0(k, b is constant, k≠0), and the linear resolution function is exactly y=kx+b(k, b is constant, k ≠ 0). When the function value is 0, that is, kx+b=0, it is exactly the same as the one-dimensional linear equation.

Conclusion: Since any one-dimensional linear equation can be transformed into kx+b=0 (both K and B are constants, k≠0), solving one-dimensional linear equation can be transformed into finding the value of the corresponding independent variable when the linear function value is 0.

From the image point of view, this is equivalent to determining the abscissa value of the straight line y=kx+b when it is known.

Chapter VII Binary Linear Equations

1, binary linear equation

An integral equation with two unknowns whose terms are 1 is called a binary linear equation.

2, the solution of binary linear equation

A set of unknown values suitable for binary linear equation is called the solution of this binary linear equation.

3. Binary linear equation

An equation group consisting of two linear equations with two unknowns is called a binary linear equation group.

Solutions of four binary linear equations

The common * * * solution of each equation in a binary linear system of equations is called the solution of this binary linear system of equations.

5, the solution of binary linear equations

(1) substitution (elimination) method (2) addition and subtraction (elimination)

6, the relationship between linear function and binary linear equation (group):

The relationship between (1) linear function and binary linear equation;

The coordinate y=kx+b of any point on a straight line is the solution of its corresponding binary linear equation kx- y+b=0.

(2) The relationship between linear function and binary linear equations:

The solution of binary linear equations can be regarded as two linear functions.

The intersection of the images of and.

When the function images intersect, it means that the corresponding binary linear equations have solutions; When the function images (straight lines) are parallel, that is, there is no intersection point, it means that the corresponding binary linear equations have no solution.

Chapter VIII Data Representation

1. Describes the number of trends (average level) in the data set: average, mode and median.

2. Average

(1) average: Generally speaking, for n numbers, we call it the arithmetic average of these n numbers, which is called the average for short.

(2) Weighted average:

Step 3: Ways

The data with the highest frequency in a set of data is called the pattern of this set of data.

4. Median value

Generally, a group of data is arranged in order of size, and the data in the middle position (or the average of the two data in the middle) is called the median of this group of data.