Chapter 1 Pythagorean Theorem
The sum of two right angles of a right triangle is equal to the square of the hypotenuse. I.e.:
The relationship between the sides is obtained from the right triangle.
A triangle is a right triangle if all three sides A, B and C are satisfied.
Three positive integers that meet the conditions are called Pythagoras numbers. Common Pythagorean arrays are: (3,4,5); (6,8, 10); (5, 12, 13); (8, 15, 17); (7,24,25); (20,2 1,29); (9,40,4 1); ..... (The multiples of these Pythagorean arrays are still Pythagorean numbers)
Chapter II Real Numbers
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, and is recorded as. The arithmetic square root of 0 is 0; By definition, A has an arithmetic square root only when a≥0.
Square root: Generally speaking, if the square root of a number x is equal to a, that is, x2=a, then this number x is called the square root of a. ※.
A positive number has two square roots (one positive and one negative). 0 has only one square root, which is itself. ※: Negative numbers have no square root.
The cube root of a positive number is a positive number. The cube root of 0 is 0. ※: The cube root of a negative number is a negative number.
Chapter III Translation and Rotation of Graphics
Translation: In a plane, a graphic moves a certain distance in a certain direction, and such graphic movement is called translation.
The basic properties of translation: after translation, the corresponding line segment and the corresponding angle are equal respectively; The line segments connected by the corresponding points are parallel and equal.
Rotation: in a plane, a figure rotates an angle in a certain direction around a fixed point, and such a figure movement is called rotation.
This fixed point is called the center of rotation and the rotation angle is called the rotation angle.
The nature of rotation: the size and shape of the rotated figure are the same as the original figure;
The distance between the corresponding points of the two graphs before and after rotation and the rotation center is equal;
The angles formed by the connecting lines from the corresponding points to the rotation center are equal to each other.
(Example: As shown in the figure, points D, E and F are the corresponding points of points A, B and C respectively. After rotation, every point on the graph rotates around the rotation center by the same angle, and the angle formed by the connecting line of any pair of corresponding points and the rotation center is the rotation angle, and the distance between the corresponding points and the rotation center is equal. )
The fourth chapter discusses the properties of Siping polygon.
Definition of four parallel sides: A quadrilateral with two opposite sides parallel to each other is called a parallelogram, and a line segment connected by two vertices of a parallelogram is called its diagonal. ※.
The nature of parallelogram: the opposite sides of parallelogram are equal, the diagonal is equal, and the diagonal is equally divided. ※.
Distinguishing method of parallelogram: Two groups of parallelograms with parallel opposite sides are parallelograms. ※.
Two sets of quadrilaterals with equal opposite sides are parallelograms.
A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.
A quadrilateral with two diagonal lines bisecting each other is a parallelogram.
Distance between parallel lines: If two lines are parallel to each other, the distance between any two points on one line and the other line is equal. This distance is called the distance between parallel lines. ※.
Definition of rhombus: A group of parallelograms with equal adjacent sides is called rhombus.
The nature of the diamond: it has the nature of a parallelogram, four sides are equal, two diagonals are bisected vertically, and each diagonal bisects a set of diagonals. ※.
The diamond is an axisymmetric figure, and the straight line where each diagonal line is located is the axis of symmetry.
The distinguishing method of rhombus: A group of parallelograms with equal adjacent sides is rhombus. ※.
Parallelograms with diagonal lines perpendicular to each other are diamonds.
A quadrilateral with four equilateral sides is a diamond.
Definition of rectangle: A parallelogram with a right angle is called a rectangle. A rectangle is a special parallelogram. ※.
The nature of rectangle: it has the nature of parallelogram, with equal diagonals and four corners at right angles. A rectangle is an axisymmetric figure with two axes of symmetry. ※. )
Determination of rectangle: A parallelogram with a right angle is called a rectangle (by definition). ※.
A parallelogram with equal diagonal lines is a rectangle.
A quadrilateral with four equal angles is a rectangle.
Inference: The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse. ※.
Definition of a square: A group of rectangles with equal adjacent sides is called a square.
Properties of Square: Square has all the properties of parallelogram, rectangle and diamond. A square is an axisymmetric figure with two axes of symmetry. ※.
Common judgment of square: ※:
A diamond with a right angle is a square;
A rectangle with equal adjacent sides is a square;
The rhombus with equal diagonal lines is a square;
A rectangle with diagonal lines perpendicular to each other is a square.
The relationship between squares, rectangles, diamonds and parallel edges (as shown in Figure 3):
Definition of trapezoid: A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are called trapezoid. ※.
Two trapezoid with equal waist are called isosceles trapezoid. ※.
A trapezoid with a vertical waist bottom is called a right-angled trapezoid. ※.
The nature of isosceles trapezoid: the two internal angles on the same bottom of isosceles trapezoid are equal and the diagonal lines are equal. ※.
Two trapeziums with equal internal angles on the same base are isosceles trapeziums.
Sum of polygon internal angles: the sum of n polygon internal angles is equal to (n-2)? ※? 6? 1 180
The sum of the outer angles of a polygon is equal to 360 degrees. ※
On the plane, a figure rotates around a certain point 180. ※. If the graphs before and after rotation coincide with each other, then this graph is called a centrosymmetric graph.
※ The line segment connected by each pair of corresponding points on the central symmetric figure is divided into two by the symmetric center.
Chapter V Determination of Location
Concept of plane rectangular coordinate system: in a plane, two mutually perpendicular number axes with a common origin form a plane rectangular coordinate system, and the horizontal number axis is called X axis or horizontal axis. The vertical axis is called Y axis or vertical axis, and the intersection o of the two axes is called the origin. ※.
Point coordinates: a point P on the plane, the X axis and Y axis passing through P are vertical lines respectively, and the corresponding numbers A and B on the X axis and Y axis are called the abscissa and ordinate of the point P respectively, then the ordered real number pairs (A and B) are called the coordinates of the point P. ※.
How to find this point according to its coordinates in rectangular coordinate system (as shown in Figure 4)? The method is to find point A with coordinate A on the X axis, and then find point B with coordinate B on the Y axis. The intersection of two perpendicular lines is the found point P. ※.
How to establish an appropriate rectangular coordinate system according to known conditions? ※?
The requirement of establishing coordinate system according to known conditions is to make the calculation as convenient as possible. Generally, there is no definite method, but there are several commonly used methods: ① Take a known point as the origin and make its coordinates (0,0); ② Take the straight line of a line segment in the graph as the X axis (or Y axis); ③ Take the midpoint of the known line segment as the origin; ④ Take the intersection of two straight lines as the origin; ⑤ Use the symmetry axis of the graph and take the symmetry axis as the y axis.
Variation law of "vertical and horizontal expansion" of graphics. ※:
A, when the ordinate of the coordinates of each point on the graph remains unchanged and the abscissa becomes n times of the original coordinate, the obtained graph is the cross section of the original graph: ① when n >; At 1, the elongation is n times of the original; ② When 0
B, when the horizontal coordinate of each point on the graph is unchanged and the vertical coordinate is changed to n times of the original one, the obtained graph is more vertical than the original graph: ① when n >; At 1, the elongation is n times of the original; ② When 0
Variation law of "vertical and horizontal position" of graphics. ※:
A, keeping the ordinate of the coordinates of each point on the diagram unchanged, and adding a to the abscissa respectively, the shape and size of the obtained graph remain unchanged, but the position is to the right (a >；); 0) or to the left (a
B, keeping the abscissa of the coordinates of each point on the diagram unchanged, and respectively adding b on the ordinate to keep the shape and size of the obtained graph unchanged, with b > position upward (b >); 0) or down (b
Variation law of "inversion and symmetry" of graphics. ※:
A, the abscissa of each point on the graph is unchanged, and the ordinate is multiplied by-1 respectively, so that the graph obtained is symmetrical to the original graph about X axis.
B, the ordinate of each point on the graph is unchanged, and the abscissa is multiplied by-1 respectively, so that the graph obtained in this way is symmetrical to the original graph about the y axis.
The law of "expansion" of graphics. ※:
Change the vertical and horizontal coordinates of each point on the diagram to the original n times (n >；; 0), the shape of the obtained graph is unchanged compared with the original graph; (1) when n >; When 1, the size of the corresponding line segment is expanded to n times of the original; ② When 0
Chapter VI Linear Functions
If the relationship between two variables X and Y can be expressed in the form of y=kx+b(k≠0), then Y is a linear function of X (X is the independent variable and Y is the dependent variable). In particular, when b=0, y is said to be a proportional function of x.
※ The image with the proportional function y=kx is a straight line passing through the origin (0,0).
※ in the linear function y=kx+b: when k >; 0, y increases with the increase of x; When k < 0, y decreases with the increase of x.
Chapter VII Binary Linear Equations
An equation with two unknowns whose degree is 1 is called a binary linear equation. ※. An equation group consisting of two linear equations is called a binary linear equation group.
Solving binary linear equations: ① Substituting elimination method; (2) addition and subtraction elimination method (whether it is substitution elimination method or addition and subtraction elimination method, its purpose is to change "binary linear equation" into "unitary linear equation", which is called "elimination").
When solving practical problems with equations, there are mainly two steps: ① setting unknowns (when setting unknowns, in most cases, only problems with the purpose of X or Y are solved; But sometimes it must be considered according to known conditions, equivalence relations and other aspects); (2) Find equivalence relation (a general topic will contain a sentence expressing equivalence relation, and only need to find this sentence to list equations according to it).
The process of solving the problem can be further summarized as follows: ※:
Chapter VIII Data Representation
Weighted average: when the weight of a set of data is added to, it is called the weighted average of these n numbers. (For example, a student's grades in mathematics, Chinese and science are 72, 50 and 88 respectively, and the "weights" of the three grades are 4, 3 and 1 respectively, so the weighted average is:)
Generally, n data are 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. ※.
※ The data with the highest frequency in a set of data is called the pattern of this set of data.
The mode focuses on the frequency of each data, and the median should first arrange the data in order of size. Note that when the number of data is odd, the middle data is the median. ※ When the number of data is even, the average value of the middle two data is the median. Pay special attention to the fact that the average and median of a set of data are unique, but the pattern is not necessarily unique.