Chapter 11 congruent triangles

I. Definition

1. Consistency: Two drawings with the same shape and size can completely overlap.

Congruent triangles: Two triangles that can completely overlap.

Two. main points

1. Graphic congruence before and after translation, folding and rotation.

2. The nature of congruent triangles: congruent triangles has equal sides and congruent triangles has equal angles.

3. congruent triangles's judgment:

The three sides of SSS correspond to the coincidence of two equal triangles.

Two sides of SAS and their included angles correspond to two congruent triangles.

The two corners of ASA and their clamping edges correspond to the coincidence of two triangles.

Two angles of AAS and the opposite side of an angle are all equal to two triangles [edges and angles]

The hypotenuse and a right-angled side of h 1 correspond to the coincidence of two triangles.

4. The nature of the angular bisector: the distance from the point on the angular bisector to both sides of the angle is equal.

5. Determination of the bisector of the angle: the points with the same distance from the inside of the angle to both sides of the angle are on the bisector of the angle.

Three. attention

1. When two triangles are congruent, the letters representing the corresponding vertices are usually written in the corresponding positions.

Chapter 12 Axisymmetric

I. Definition

1. If a graph is folded along a straight line, the parts on both sides of the straight line can overlap each other, and this graph is called an axisymmetric graph. This straight line is its axis of symmetry. We also say that this graph is symmetrical about this straight line.

2. Fold a graph along a straight line. If it can overlap with another graph, the two graphs are said to be symmetrical about this straight line. This straight line is called the symmetry axis, and the overlapping point after folding is the corresponding point, which is called the corresponding point.

3. A straight line passing through the midpoint of a line segment and perpendicular to this line segment is called the median vertical line of this line segment.

If two figures are symmetrical about a straight line, then the symmetry axis is the middle perpendicular of the line segment connected by any pair of corresponding points.

The symmetry axis of an axisymmetric figure is the perpendicular bisector of a line segment connected by any pair of corresponding points.

A triangle with two equal sides is called an isosceles triangle.

A triangle with three equilateral sides is called an equilateral triangle.

Two. main points

1. Take two symmetrical figures as a whole, which is an axisymmetric figure.

2. Divide an axisymmetric figure into two figures along the axis of symmetry, and these two figures are symmetrical about this axis.

3. The nature of the midline: the distance between the point on the midline of a line segment and the two endpoints of the line segment is equal.

4. Judgment of the middle vertical line: The point where the two endpoints of a line segment are equidistant is on the middle vertical line of this line segment.

5. How to make an axis of symmetry: If two figures form an axis of symmetry, its axis of symmetry is the middle perpendicular of any pair of line segments connected by corresponding points. So we just need to find a pair of corresponding points and make a middle vertical line connecting them, and then we can get the symmetry axis of this figure.

Similarly, for axisymmetric graphics, as long as the median perpendicular of the line segment connected by any group of corresponding points is found, the symmetry axis of the graphics can be found.

6. Properties of axisymmetric graphics: When the direction and position of the axis of symmetry change, the direction and position of the resulting graphics will also change.

From a plane figure, we can get a figure that is symmetrical about a straight line, and this figure is exactly the same as the original figure in shape and size.

Every point on the new graph is a symmetrical point of a point on the original graph about a straight line.

The line segment connecting any pair of corresponding points is vertically bisected by the symmetry axis.

7. The nature of isosceles triangle: the two bottom angles of isosceles triangle are equal [equal corners].

The bisector of the top angle of an isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide [the three lines are one]

An isosceles triangle is an axisymmetric figure, and the straight line where the median line on the base (the bisector of the height and the vertex angle on the base) is located is its axis of symmetry.

The waist of an isosceles triangle is equal in height or midline.

The bisectors of the two base angles of an isosceles triangle are equal.

The sum of the distances from the top of the isosceles triangle base to the waist is equal to the distance from the bottom angle to the waist.

The bisector of the top angle of an isosceles triangle, the height on the bottom edge and the distance from the center line on the bottom edge to the waist are equal. ]

8. Determination method of isosceles triangle: If two angles of a triangle are equal, then the opposite sides of the two angles are also equal.

[If the bisector of the outer corner of a triangle is parallel to one side of the triangle, the triangle is an isosceles triangle. ]

9. Properties of equilateral triangle: All three internal angles of equilateral triangle are equal, and each angle is equal to 60.

10. Determination of an equilateral triangle: the three internal angles of an equilateral triangle are equal, and each angle is equal to 60.

A triangle with three equal angles is an equilateral triangle.

An isosceles triangle with an angle of 60 is an equilateral triangle.

1 1. One of the properties of a right triangle: in a right triangle, if an acute angle is equal to 30, then the right side it faces is equal to half of the hypotenuse.

12. In a triangle, if two sides are not equal, the angles they subtend are not equal, and the angles subtended by the long sides are larger.

Three. attention

1.(x, y) is symmetric about the origin (-x.-y)

On the Axisymmetry of X (x, -y)

On y-axis symmetry (-x, y)

2. Coordinate axis symmetry.

Chapter 13 Real Numbers

I. Definition

1. 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 A is called the root sign.

Generally speaking, if the square of a number is equal to a, then this number is called the square root or quadratic square root of a, and the operation of finding the square root of a number is called the square root.

Generally speaking, if the cube of a number is equal to a, then this number is called the cube root or cube root of a, and the operation of finding the cube root of a number is called the square root.

4. Any rational number can be written in the form of finite decimal or infinite cyclic decimal. Any finite decimal or infinite cyclic decimal is also a rational number.

5. Infinitely circulating decimals are also called irrational numbers.

6. Rational numbers and irrational numbers are collectively called real numbers.

7. There is a one-to-one correspondence between points on the number axis and real numbers, and there is also a one-to-one correspondence between ordered real number pairs in the plane rectangular coordinate system.

Two. main points

1. sum of squares and squares are reciprocal operations.

2. A positive number has two square roots, the two square roots are in opposite directions, and the positive square root is the arithmetic square root of this number.

3. When the decimal point of the square root is moved two places to the right, the decimal point of its arithmetic square root is moved one place to the right.

When the square decimal point moves three places to the right, its cube root decimal point moves one place to the right.

5. The inverse of the number A is -a[a is an arbitrary real number], the absolute value of a positive real number is itself, and the absolute value of a negative real number is its inverse; The absolute value of 0 is 0.

Three. attention

The number of 1. roots must be non-negative.

The arithmetic square root of 2. 0, 1 is itself; The square root of 0 is 0, and negative numbers have no square root; The cube root of a positive number is positive, the cube root of a negative number is negative, and the cube root of 0 is 0.

3. Integer multiples or fractions of irrational numbers with roots are still irrational numbers; A number with a root sign is rational if it is opened; Any rational number can be written as a fraction.

Chapter 14 Linear Functions

I. Definition

1. In the process of changing according to a certain law, the amount of numerical change is variable and the constant is constant.

2. Generally speaking, in a change process, if there are two variables X and Y, and for each definite value of X, Y has a unique definite value corresponding to it, then X is an independent variable and Y is a function of X. If y=b when x=a, then B is called a function value when the independent variable value is A. 。

3. Generally, the function in the form of y=kx[k is a constant and k≠0] is called the proportional function, where k is called the proportional coefficient. [Form of product of a number and an independent variable]

4. A function in the form of y = kx+b [k, b is a constant, k≠0] is called a linear function.

Two. main points

1. Value range of independent variable:

(1) Algebraic expression type y=3x+ 1── all real numbers.

(2) Fractional type-make the denominator not 0.

(3) Radical type-make the number of roots non-negative.

(4) comprehensive type

2. The general steps of making functional images:

(1) list

(2) Tracking point

(3) Connection

3. The image of general proportional function y=kx[k is constant, k≠0] is a straight line passing through the origin, which we call straight line y=kx. When k >; 0, the straight line y=kx passes through the first three quadrants, and y increases with the increase of x; When k < 0, the straight line y=kx passes through the second quadrant, and y decreases with the increase of x 。

4. Application of undetermined coefficient method.

5. See the solution of linear equation with function image. [2x+5= 17]

Solution: The original equation is 2x- 12=0.

Draw an image with y=2x- 12.

…

According to the image, the intersection of the straight line y=x- 12 and the x axis is (6,0).

So x=6.

6. Look at the unary linear inequality [5x+6 >; 3x+ 10]

Solution 1: the original inequality is transformed into 2x-4 >: 0.

Draw an image of the function y=2x-4.

…

As can be seen from the figure, when x>2 is at o'clock, the image of straight line y=2x-4 is above the X axis.

So the solution set of inequality 2x-4 >:0 is x>2.

So the solution set of the original inequality is x>2.

Solution 2: Draw the image of the function y 1=5x+6 and y2=x+ 10.

…

It can be seen from the figure that when x>2 is at o'clock, the image of the straight line y 1 is above y2, that is, Y 1 >: y2.

So the inequality 5x+6 >；; The solution set of 3x+ 10 is x >；; 2

7. See the image of binary linear equation and function.

Solution: transform the original equations into {[the form of y is expressed by the formula containing x].

Draw an image of the sum of functions

…

According to the image, the intersection of the lines and is (1, 1).

So the solution of the system of equations {…} is {x= 1, y= 1.

So the solutions of the original equations are {x= 1, y= 1.

Three. attention

1. Constants and variables are not immutable.

2. The image of inverse proportional function is hyperbola.

3. The proportional function is a special linear function.

4. Choose a scheme.

Chapter XV Multiplication, Division and Factorization of Algebraic Expressions

I. Definition

1 algebraic expression multiplication

(1). Me? An=am+n[m, n are all positive integers]

Power with the same base, the base is constant, and the index is added.

(2).(am)n=amn[m, n are all positive integers]

Power, constant radix, exponential multiplication.

(3).(ab)n=anbn[n is a positive integer]

The power of a product is equal to multiplying the factors of the product, and then multiplying the power.

(4).ac5？ bc2=(a？ b)？ (c5？ c2)=abc5+2=abc7

Multiply the monomial with the monomial, respectively by their coefficients and the same letters. For letters contained only in the monomial, they are used as a factor of the product together with its index.

(5).m(a+b+c)=ma+mb+mc

Multiplying a polynomial by a monomial is to multiply each term of a polynomial by a monomial, and then add the products.

(6).(a+b)(m+n)=am+an+bm+bn

Multiply each term of one polynomial by each term of another polynomial, and then multiply the products.

2. Multiplication formula

( 1).(a+b)(a-b)=a2-b2

Square difference formula: the product of the sum of two numbers and the difference between these two numbers is equal to the square difference between these two numbers.

(2).(a b)2=a2 2ab+b2

Complete square formula: the square of the sum [or difference] of two numbers is equal to the sum of their squares, plus [or minus] twice their product.

3. Algebraic expression division

(1) am ÷ an = am-n [a ≠ 0, m and n are positive integers, m >；; n]

Same base powers divides, the base remains the same, and the exponent is subtracted.

(2)a0= 1[a≠0]