If A and B represent two algebraic expressions and B contains letters, then the formula A/B is called a fraction.
The numerator of a fraction is multiplied by the denominator or divided by an algebraic expression that is not equal to 0, and the value of the fraction remains the same.
Law of fractional multiplication: fractional multiplication, the product of molecules is the numerator of the product, and the product of denominator is the denominator.
Law of fractional division: a fraction is divided by a fraction, and the numerator and denominator of the divisor are in turn multiplied by the divisor.
Fractional power should be numerator and denominator respectively.
A-n = 1/A n (A ≠ 0) That is to say, A-n (A ≠ 0) is the reciprocal of A n.
Test method of fractional equation: bring the solution of the whole equation into the simplest common denominator. If the value of the simplest common denominator is not 0, the solution of the whole equation is the solution of the original fractional equation; Otherwise, this solution is not the solution of the original fractional equation.
Chapter 17 Inverse proportional function
A function in the form of y = k/x (where k is a constant and k≠0) is called an inverse proportional function.
The image of inverse proportional function belongs to hyperbola.
When k > 0, the two branches of hyperbola are located in the first quadrant and the third quadrant respectively, and the y value of each quadrant decreases with the increase of x value;
When k < 0, the two branches of hyperbola are located in the second quadrant and the fourth quadrant respectively, and the y value of each quadrant increases with the increase of x value.
Chapter 18 Pythagorean Theorem
Pythagorean Theorem: If the lengths of two right-angled sides of a right-angled triangle are A and B respectively and the length of the hypotenuse is C, then A 2+B 2 = C 2.
Inverse Theorem of Pythagorean Theorem: If the lengths of three sides of triangle A, B and C satisfy A 2+B 2 = C 2, then the triangle is a right triangle.
A proposition that is proved to be correct is called a theorem.
We call two propositions with opposite topics and conclusions reciprocal propositions. If one of them is called the original proposition, then the other is called its inverse proposition. (Example: Pythagorean Theorem and Pythagorean Theorem Inverse Theorem)
Chapter 19 Quadrilateral
A quadrilateral with two sets of parallel opposite sides is called a parallelogram.
The nature of parallelogram: the opposite sides of parallelogram are equal; Diagonal angles of parallelogram are equal. Diagonal bisection of parallelogram.
Determination of parallelogram;
1. Two groups of quadrangles with equal opposite sides are parallelograms;
2. The quadrilateral whose diagonal lines bisect each other is a parallelogram;
3. Two groups of quadrangles with equal diagonal are parallelograms;
4. A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.
The midline of the triangle is parallel to the third side of the triangle and equal to half of the third side.
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
The nature of the rectangle: all four corners of the rectangle are right angles; The diagonals of a rectangle are equally divided.
Rectangular judgment theorem;
1. A parallelogram with a right angle is called a rectangle.
2. Parallelograms with equal diagonals are rectangles.
A quadrilateral with three right angles is a rectangle.
The nature of the diamond: all four sides of the diamond are equal; The two diagonals of the diamond are perpendicular to each other, and each diagonal bisects a set of diagonals.
Judgement theorem of diamonds;
1. A set of parallelograms with equal adjacent sides is a diamond.
2. Parallelograms with diagonal lines perpendicular to each other are diamonds.
A quadrilateral with four equilateral sides is a diamond.
S diamond = 1/2×ab(a and B are two diagonal lines).
The essence of a square: all four sides are equal and all four corners are right angles.
A square is both a rectangle and a diamond.
Square judgment theorem;
1. A rectangle with equal adjacent sides is a square.
Diamonds with right angles are squares.
A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are called trapezoid.
The nature of isosceles trapezoid: the two angles on the same base of isosceles trapezoid are equal; The two diagonals of an isosceles trapezoid are equal.
Judgment theorem of isosceles trapezoid: two trapezoid with equal angles on the same base are isosceles trapezoid.
The center of gravity of the line segment is the midpoint of the line segment.
The center of gravity of a parallelogram is the intersection of its two diagonals.
The point of doubt when three center lines of a triangle meet is the center of gravity of the triangle.
A rectangle with an aspect ratio of (root number 5- 1)/2 (about 0.6 18) is called a golden rectangle.
Chapter 20 Data Analysis
Arrange a set of data in order from small to large (or from large to small). If the number of data is odd, the middle number is median); This set of data. If the number of data is even, the average of the middle two data is the median of this set of data.
The data with the highest frequency in a set of data is the pattern of this set of data.
The difference between the largest data and the smallest data in a set of data is called the range of this set of data.
The greater the variance, the greater the data fluctuation; The smaller the variance, the smaller the data fluctuation and the more stable it is.
Data collection and sorting steps: 1. Data collection II. Data arrangement 3. Data description 4. Data analysis. Investigation report writing. Communication.