Chapter 1 Pythagorean Theorem
1. Pythagorean theorem: the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse; Namely.
2. Proof of Pythagorean Theorem: Prove with the area relation of three squares (two methods).
3. Inverse theorem of Pythagorean theorem: If the lengths of three sides of a triangle are satisfied, then the triangle is a right triangle. Three positive integers are called Pythagoras numbers.
Chapter II Real Numbers
The concepts and properties of 1. square root and arithmetic square root;
(1) concept: If, then it is the square root of, and it is called the arithmetic square root.
(2) Properties: ① When ≥ 0, ≥ 0; When < 0, it is meaningless; ② = ; ③ 。
2. The concept and properties of cube root;
(1) concept: If, then it is a cube root, which is expressed as:
(2) Properties: ①; ② ; ③ =
3. The concept and classification of real numbers:
(1) Concept: Real number is a general term for rational number and irrational number;
(2) Classification: according to the definition, it can be divided into rational numbers and fractions that can be divided into integers; Divided into positive numbers, negative numbers and zero by nature. Irrational numbers are infinite cyclic decimals; Decimals can be divided into finite decimal, infinite cyclic decimal and infinite acyclic decimal. Among them, finite decimals and infinite circulating decimals are called fractions.
4. Concepts related to real numbers: In the range of real numbers, the meanings of reciprocal, reciprocal and absolute value are exactly the same as those in the range of rational numbers; In the range of real numbers, the algorithm of rational numbers is as effective as the algorithm law. Every real number can be represented by a point on the number axis; On the contrary, every point on the number axis represents a real number, that is, there is a one-to-one correspondence between the real number and the points on the number axis. So the number axis can be filled with real numbers.
5. Algorithm of arithmetic square root: (≥0, ≥ 0); ( ≥0, >0)。
Chapter III Translation and Rotation of Graphics
1. Translation: In a plane, a graphic moves a certain distance along a certain direction, and such graphic movement is called translation. Translation does not change the size and shape of the graph, but only changes the position of the graph; After translation, the line segments connected by the corresponding points are parallel and equal; The corresponding line segments are parallel and equal, and the corresponding angles are equal.
2. Rotation: In a plane, a graph rotates an angle in a certain direction around a fixed point, and such a graph movement is called rotation. This fixed point is called the center of rotation and the rotation angle is called the rotation angle. Rotation does not change the size and shape of the graph, but changes the position of the graph; After rotation, each point of the graphic point rotates by the same sum angle around the rotation center in the same direction; The angle formed by the connecting line of any pair of corresponding points and the rotation center is the rotation angle; The distance between the corresponding point and the center of rotation is equal.
3. Make a translation diagram and a rotation diagram.
The fourth chapter is the exploration of quadrilateral properties.
1. Classification of polygons:
2. Definition, properties and discrimination of parallelogram, rhombus, rectangle, square and isosceles trapezoid;
(1) parallelogram: Two groups of parallelograms with opposite sides are called parallelograms. The opposite sides of the parallelogram are parallel and equal; Diagonal angles are equal and adjacent angles are complementary; Divide diagonally. The quadrilateral with two diagonal bisections is a parallelogram; A set of quadrilaterals with parallel and equal opposite sides is a parallelogram; Two groups of quadrangles with equal opposite sides are parallelograms; Two groups of quadrangles with equal diagonal are parallelograms; Quadrilaterals whose diagonals bisect each other are parallelograms.
(2) Diamond: A group of parallelograms with equal adjacent sides is called a diamond. All four sides of the diamond are equal; Diagonal lines are bisected vertically, and each diagonal line bisects a set of diagonal lines. A quadrilateral with four equilateral sides is a diamond; Parallelograms with diagonal lines perpendicular to each other are diamonds; A set of parallelograms with equal adjacent sides is a diamond; Quadrilaterals whose diagonals are bisected and perpendicular to each other are diamonds. The area of the diamond is equal to half of the product of two diagonals (area calculation, i.e. S diamond =L 1*L2/2).
(3) Rectangle: A parallelogram with a right angle is called a rectangle. Diagonal lines of rectangles are equal; All four corners are right angles. Parallelograms with equal diagonals are rectangles; A parallelogram with right angles is a rectangle. The median line on the hypotenuse of a right triangle is equal to half the length of the hypotenuse; In a right triangle, the right side opposite to 30 is half of the hypotenuse.
(4) Square: A group of rectangles with equal adjacent sides is called a square. A square has all the properties of parallelogram, rhombus and rectangle.
(5) The two internal angles on the same bottom of the isosceles trapezoid are equal to the diagonal. Two trapeziums with equal internal angles on the same base are isosceles trapeziums; A trapezoid with equal diagonal lines is an isosceles trapezoid; Diagonally complementary trapezoid is isosceles trapezoid.
(6) The midline of triangle: the line segment connecting the key points on both sides of triangle. Property: parallel and equal to half of the third side.
3. The formula of polygon interior angle: (n-2) *180; The sum of the outer angles of a polygon is equal to.
4. Centrally symmetric figure: On a plane, a figure rotates around a certain point. If the figures before and after rotation coincide with each other, then this figure is called a centrally symmetric figure.
Chapter V Determination of Location
1. Cartesian coordinate system and related knowledge of coordinates.
2. Coordinate relation of points: if the abscissas of point A and point B are the same, then the ‖ axis; If the vertical coordinates of point A and point B are the same, then the ‖ axis.
3. Keep the vertical coordinate of the graph unchanged and double the horizontal coordinate, and the obtained graph is symmetrical with the original graph; The abscissa of the graph remains unchanged, and the ordinate is doubled, so that the obtained graph is symmetrical with the original graph; When the abscissa and ordinate of the graph are doubled, the obtained graph and the original graph are symmetrical about the origin.
Chapter VI Linear Functions
1. Definition of linear function: If the relationship between two variables can be expressed in the form of (constant,), it is called a linear function of. When we say it is. Proportional function is a special linear function.
2. Make a function image: list points, trace points, connect lines, and mark the corresponding function relationship.
3. Image attribute of proportional function: pass; > 0, passing through one or three quadrants; When < 0, it passes through the second and fourth quadrants.
4. Linear function image properties:
(1)> 0, it increases with the increase of, and the image shows an upward trend; When < 0, it decreases with the increase of, and the image shows a downward trend.
(2) The intersection of a straight line and an axis is, and the intersection of a straight line and an axis is.
(3) In the linear function: > 0, when > 0, the function image passes through the first, second and third quadrants; When > 0, < 0, the function image passes through the first, third and fourth quadrants; When < 0, > 0, the function image passes through the first, second and fourth quadrants; When < 0, < 0, the function image passes through the second, third and fourth quadrants.
(4) In two linear functions, when their values are equal, their images are parallel; When their values are not equal, their images intersect; When the product of their values is 0, their images are vertical.
4. The expression of the function is found once at any two points, and the expression of the function is found once according to the image.
5. Use the image of a linear function to solve practical problems.
Chapter VII Binary Linear Equations
1. Definition of binary linear equations and binary linear equations.
2. The basic idea of solving the equation is elimination, and the basic methods of elimination are: ① substitution elimination method; ② the method of addition, subtraction and elimination; ③ Image method.
3. The key to solving application problems with equations is to find the equivalence relation.
4. When solving application problems, it is divided into four steps: setting, sorting, solving and answering.
5. Every binary linear equation can be regarded as a linear function, and finding the solution of binary linear equations can be regarded as finding the intersection of two linear function images.
Chapter VIII Data Representation
1. The difference and connection between arithmetic average and weighted average: arithmetic average is a special case of weighted average (its special feature is that the weights of each item are equal). In practical problems, when the weights of each item are not equal, the weighted average should be used when calculating the average, and when the weights of each item are equal, the arithmetic average should be used when calculating the average.
2. Median and mode: Median refers to a piece of data (or the average of two data in the middle) arranged in order of size (from big to small or from small to big). Pattern refers to the data that appears most frequently in a set of data.