The first chapter is the operation of algebraic expressions.
I. Algebraic expressions
1. Single item. ※
An algebraic expression consisting of the product of numbers and letters is called a monomial. A single number or letter is also a monomial.
(2) The coefficient of a single item is a numerical factor of a single item. As a coefficient of a monomial, the number must be preceded by an attribute symbol. If the monomial is just a product of letters, it is not without coefficients.
In a monomial, the sum of the exponents of all the letters is called the degree of the monomial.
2.※ Polynomial
The sum of several monomials is called polynomial. In polynomials, each monomial is called a polynomial term. Among them, items without letters are called constant items. In a polynomial, the degree of the term with the highest degree is called the degree of this polynomial.
(2) Both monomials and polynomials have degrees, monomials with letters have coefficients, and polynomials have no coefficients. Every term of a polynomial is a monomial, and the number of terms of a polynomial is the number of monomials with the polynomial as the addend. Each term in a polynomial has its own degree, but their degrees cannot all be regarded as the degree of this polynomial. Polynomials have only one degree, which is the highest degree of inclusion.
3. Algebraic expressions Monomials and polynomials are collectively called algebraic expressions. ※ 。
2. Addition and subtraction of algebraic expressions
164438+0. The addition and subtraction of algebraic expressions are essentially the combination of similar items after removing brackets, and the operation result is a polynomial or a single item.
There is a "-"sign before the brackets. When the brackets are deleted, the symbols of the items in the brackets should be changed. When a number is multiplied by a polynomial, the number should be multiplied by the items in brackets.
Three. Same base power multiplication
Same base powers's multiplication rule: (m, n are all positive numbers) is the most basic rule in power operation. Pay attention to the following points when applying regular operations. ※:
① The preconditions for using this rule are: when the bases of powers are the same and multiplied, the base a can be a specific numeric letter or a term or polynomial;
② When the index is 1, don't mistake it for no index;
③ Don't confuse multiplication with addition of algebraic expressions. Multiplication, as long as the base is the same, the indexes can be added; For addition, not only the radix is the same, but also the exponent needs to be added;
(4) When three or more bases are the same, the rule can be generalized as (where m, n and p are all positive numbers);
⑤ The formula can also be reversed: (M and n are positive integers)
4. Power and products.
1. power law: (m, n are both positive numbers) is derived from the power multiplication law, but the two cannot be confused. ※ 。
※2.。
3. When the base has a negative sign, it should be noted that when the base is a and (-a), it is not the same base, but it can be converted into the same base by power law. ※,
If (-a)3 is replaced by -a3.
4. The base sometimes has different forms, but it can be replaced with the same one. ※.
5. Pay attention to the difference between (ab)n and (a+b)n, and don't mistake (a+b) n = an+bn (both a and b are not zero). ※.
6. Power law of product: the power of product is equal to each factor of product multiplied by power respectively, that is, (n is a positive integer). ※.
7. Power and product power rules can be applied in reverse. ※.
Verb (abbreviation for verb) division with the same radix power
1. same base powers's division rule: same base powers divides, the base number is unchanged, and the exponent is subtracted, that is, (a≠0, m, n is a positive number, m >；; n)。
2. Pay attention to the following points when applying. ※:
(1) The prerequisite for using the rule is "divisible by same base powers" and 0 is not divisible, so a≠0 is included in the rule.
② Any number that is not equal to 0, whose power of 0 is equal to 1, that is, if (-2.50= 1), 00 is meaningless.
(3) The power of any number not equal to 0 is -p (p is a positive integer) which is equal to the reciprocal of the power of this number, that is, (a≠0, p is a positive integer), 0- 1, 0-3 is meaningless; When a>0, the value of a-p must be positive; When a<0, the value of a-p can be positive or negative, for example,
④ Pay attention to the operation sequence.
Multiplication of algebraic expressions of intransitive verbs
1. Multiplication rule of monomial: Multiply the monomial, respectively by its coefficient and the same letter. For a letter contained only in a monomial, together with its exponent, it will be a factor of product. ※.
When applying the monomial multiplication rule, we should pay attention to the following points:
① The coefficient of product is equal to the coefficient product of each factor. Determine the symbol first and then calculate the absolute value. At this time, it is easy to make mistakes, that is, the multiplication of coefficients and the addition of exponents are confused;
② Multiply the same letters, using the multiplication rule of the same base;
(3) The letters only contained in the monomial should be taken as the factors of the product together with their exponents;
④ The rule of monomial multiplication is also applicable to the multiplication of more than three monomials;
⑤ Single item multiplied by single item, the result is still single item.
2. Multiply the monomial with polynomial. ※
Polynomial multiplied by monomial is the distribution law of multiplication and addition, and when it is converted into monomial multiplied by monomial, it is polynomial multiplied by monomial, and then the products obtained are added.
When multiplying a monomial with a polynomial, please pay attention to the following points:
(1) multiply the monomial with the polynomial, and the product is a polynomial with the same number of terms as the polynomial;
(2) Pay attention to the sign of the product when operating, and each term of the polynomial contains the previous sign;
③ When mixing operations, pay attention to the operation sequence.
3. Multiply polynomials with polynomials. ※
Multiply polynomials by multiplying each term in one polynomial by each term in another polynomial, and then add the products.
Polynomial multiplication should pay attention to the following points:
(1) Polynomials should be multiplied by polynomials to prevent missing items. The checking method is as follows: before merging similar terms, the number of terms of the product should be equal to the product of the original two polynomial terms;
② Attention should be paid to the similar items in the results of merging polynomial multiplication;
(3) Multiply two same-letter linear binomials with linear coefficient of 1 and quadratic coefficient of 1. The linear coefficient is equal to the sum of the constant terms in the two factors, and the constant term is the product of the constant terms in the two factors. For the multiplication of two linear binomials (mx+a) and (nx+b) whose linear coefficients are not 1, we can get
Seven. Variance formula
1.square difference formula: the product of the sum of two numbers and the difference between these two numbers is equal to their square difference.
That be it. ※.
(1644) Its structural features are:
① The left side of the formula is the multiplication of two binomials, in which the first term is the same and the second term is the opposite number;
② On the right side of the formula is the square difference of two terms, that is, the square difference of the same term and the square difference of the opposite term.
Eight. perfect square trinomial
1.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.
That is;
Oral decision: the first side, the last side, the middle 2 times product;
2. Structural features:
① The left side of the formula is the complete square of binomial;
② There are three terms on the right side of formula * *, which is the sum of squares of two terms in binomial formula, plus or minus twice the product of these two terms.
3. When using the complete square formula, we should pay attention to the sign of the item on the right side of the formula to avoid such mistakes.
Nine. Division in algebraic expressions
1. Single division
Single division, which is divided by the coefficient and the same base, is the factor of quotient. For letters only included in the division formula, it is the factor of quotient together with its index;
2. Polynomial divided by monomial
When a polynomial is divided by a single term, each term of the polynomial is divided by the single term, and then the obtained quotients are added. It is characterized by dividing a polynomial by a monomial and converting it into a monomial divided by a monomial, and the number of terms obtained is the same as that of the original polynomial. In addition, pay special attention to symbols.
Chapter II Parallel Lines and Intersecting Lines
A corner on the pool table
1. Concepts and properties of complementary angle and complementary angle. ※
If the sum of two angles is 90 degrees (or right angle), then the two angles are complementary;
If the sum of two angles is 180 (or flat angle), then the two angles are complementary;
Note: these two concepts are aimed at two angles, emphasizing the quantitative relationship between the two angles, regardless of the mutual position of the two angles.
Their main properties are: the same angle or the complementary angle of the same angle is equal;
The same angle or the complementary angle of the same angle is equal.
2. Explore the conditions of parallel lines
There are three conditions for two straight lines to be parallel to each other, that is, the judgment theorem of two straight lines being parallel to each other. ※:
(1) Same angle, two straight lines are parallel;
② The internal dislocation angles are equal and the two straight lines are parallel;
③ The internal angles on the same side are complementary, and the two straight lines are parallel.
Three. Characteristics of parallel lines
The characteristic of parallel lines is the property theorem of parallel lines, which has three * * *. ※:
(1) Two straight lines are parallel and the same angle is equal;
② Two straight lines are parallel and the internal dislocation angles are equal;
③ The two straight lines are parallel and complementary.
4. Use rulers as line segments and angles.
1. About ruler painting. ※
Drawing with a ruler means drawing with only compasses and a ruler without scales.
2. About the role of the ruler. ※
The function of ruler is to connect the line segment between two points; A line segment extend in two directions.
The function of compasses is to make a circle with any point as the center and any length as the radius; Draw an arc with any point as the center and any length as the radius.
Chapter III Data in Life
1. scientific notation: any positive number can be written in the form of a× 10n, where 1 ≤ A < 10, and n is an integer. This notation is called scientific notation. ※.
2. When taking the divisor of a number by rounding, it is said that the divisor is accurate to which place; For a divisor, all the numbers from the first non-zero number on the left to the most accurate number are called the significant digits of this number.
3. Statistical work includes:
(1) Set goals; ② Collecting data; (3) sorting out data; ④ Expression and description of data; ⑤ Analysis results.
Chapter IV Probability
+0. The probability of occurrence and non-occurrence of random events is not always 50%.
2. There are a lot of uncertain events in real life, and probability is a subject to study uncertain events. ※.
3. Understand the probability of inevitable events and impossible events. ※.
The probability of inevitable events is 1, that is, p (inevitable events) =1; The probability of an impossible event is 0, that is, p (impossible event) = 0; If a is an uncertain event, then 0
4. Understand the calculation method of geometric probability. ※
Event occurrence probability =
The fifth chapter triangle
I. Understanding triangles
The Concept of 1. Triangle and Its Angle Classification
A figure composed of three line segments that are not on the same line end to end is called a triangle.
There are two points to note here:
① The three line segments that make up a triangle should be "not in a straight line"; If they are on the same line, the triangle does not exist;
(2) The three line segments are connected in turn, which means that there is a common * * * endpoint between the three line segments, and this common * * * endpoint is the vertex of the triangle.
Triangles can be divided into three categories according to the size of internal angles: acute triangle, right triangle and obtuse triangle.
2. The relationship between the three sides of a triangle
According to the axiom "the shortest line segment in a straight line connecting two points", we can get a property theorem of triangle trilateral relationship, that is, the sum of any two sides of a triangle is greater than the third side.
Another property of triangular trilateral relationship: the difference between any two sides of a triangle is smaller than the third side.
For these two properties, we should fully understand and master their essence to avoid making mistakes in application.
Let the lengths of three sides of a triangle be a, b and c respectively:
① Generally speaking, for a side of a triangle, there must be | b-c | < a < b+c; On the other hand, only when | b-c | < a < b+c holds, the three line segments A, B and C can form a triangle;
(2) Especially, if the line segment A is known to be the largest, as long as B+C > A is satisfied, then the three line segments A, B and C can form a triangle; If it is known that line segment A is the smallest, as long as | b-c | < a is satisfied, then these three line segments can form a triangle.
3. On the sum of the interior angles of a triangle
The sum of the three internal angles of a triangle is 180.
① The two acute angles of a right triangle are complementary;
② A triangle has at most one right angle or an obtuse angle;
③ At least two internal angles in a triangle are acute angles.
4. About the midline, height and midline of the triangle
① The bisector, midline and height of a triangle are all line segments, not straight lines or rays;
② Any triangle has three bisectors, three median lines and three heights;
③ The three bisectors and three median lines of any triangle are all inside the triangle. But the height of the triangle has different positions: the three heights of the acute triangle are all inside the triangle, as shown in figure1; A right triangle has a height inside the triangle, and the other two heights are just its two sides, as shown in Figure 2; One height of an obtuse triangle is inside the triangle and the other two heights are outside the triangle, as shown in Figure 3.
(4) In a triangle, three median lines intersect at a point, three angular bisectors intersect at a point, and three straight lines with heights intersect at a point.
Two. Graphic consistency
Figures that can completely overlap are called conformal. Congruent figures have the same shape and size. Only two figures with the same shape but different sizes, or two figures with the same area but different shapes, are not congruent figures.
Four. Congruent triangle
+0. About the concept of congruent triangles
Two triangles that can completely coincide are called congruent triangles. Overlapping vertices are called corresponding points, overlapping edges are called corresponding edges, and overlapping angles are called corresponding angles.
The so-called "perfect coincidence" means that all sides are equal and all angles are equal. So it can also be said that a triangle with two equal sides and angles is called congruent triangles.
2. The corresponding edges of congruent triangles are equal, and the corresponding angles are equal. ※.
3. The nature of congruent triangles is often used to prove that two line segments are equal and two angles are equal.
5. Explore the conditions of triangle congruence
1. Three sides correspond to the intersection of two equal triangles, abbreviated as "side" or "SSS". ※
2. Two triangles with equal angles between two sides are abbreviated as "corner edge" or "SAS". ※
3. The two corners of two triangles correspond to their clamping edges. These two triangles are congruent and abbreviated as "Angle" or "ASA". ※
4. The opposite side of two angles and one of them corresponds to the congruence of two triangles, which is called "corner edge" or "AAS" for short. ※
Make a triangle
1. Given two angles and their edges, draw a triangle with the congruence condition "Angle and Angle" ("ASA").
2. Knowing the two sides and their included angles, draw a triangle by using the congruence condition "edge and angle" ("SAS") of the triangle.
3. Given three sides, draw the search triangle by using the congruence condition "edge" ("SSS") of the triangle.
Eight. Explore the conditions of right-angled triangles.
1. The hypotenuse and right-angled side correspond to the coincidence of two right-angled triangles. . called "hypotenuse, right angle" or "HL". ※. This only applies to right-angled triangles.
2. Right triangle is a kind of triangle, which has the properties of a general triangle, so it can also be judged by "SAS", "ASA", "AAS" and "SSS". ※.
Other judgment methods of right triangle can be summarized as follows:
① Two right-angled sides correspond to congruences of two equal right-angled triangles;
② Two right-angled triangles with an acute angle and an equilateral are congruent.
(3) Three sides correspond to congruences of two right-angled triangles.
Chapter VII Axisymmetry in Life
1. If a graph is folded along a straight line and the parts on both sides of the straight line can overlap each other, then the graph is called an axisymmetric graph. This straight line be call the axis of symmetry. ※.
2. The distance from the point on the bisector of the angle is equal to both sides of the angle. ※.
3. The distance between any point on the vertical line of the line segment and the two endpoints of the line segment is equal. ※.
4. Angle, line segment and isosceles triangle are all axisymmetric figures. ※.
5. The bisector of the top angle of an isosceles triangle, the height on the bottom edge and the midline on the bottom edge coincide with each other, which is called "three lines in one" for short. ※.
6. The line segments connected with the corresponding points on the axisymmetric figure are vertically bisected by the axis of symmetry. ※.
7. The corresponding line segments on the axisymmetric figure are equal, and the corresponding angles are also equal. ※.
Nine. Division in algebraic expressions
1. Single division
Single division, which is divided by the coefficient and the same base, is the factor of quotient. For letters only included in the division formula, it is the factor of quotient together with its index;
2. Polynomial divided by monomial
When a polynomial is divided by a single term, each term of the polynomial is divided by the single term, and then the obtained quotients are added. It is characterized by dividing a polynomial by a monomial and converting it into a monomial divided by a monomial, and the number of terms obtained is the same as that of the original polynomial. In addition, pay special attention to symbols.
Chapter II Parallel Lines and Intersecting Lines
A corner on the pool table
1. Concepts and properties of complementary angle and complementary angle. ※
If the sum of two angles is 90 degrees (or right angle), then the two angles are complementary;
If the sum of two angles is 180 (or flat angle), then the two angles are complementary;
Note: these two concepts are aimed at two angles, emphasizing the quantitative relationship between the two angles, regardless of the mutual position of the two angles.
Their main properties are: the same angle or the complementary angle of the same angle is equal;
The same angle or the complementary angle of the same angle is equal.
2. Explore the conditions of parallel lines
There are three conditions for two straight lines to be parallel to each other, that is, the judgment theorem of two straight lines being parallel to each other. ※:
(1) Same angle, two straight lines are parallel;
② The internal dislocation angles are equal and the two straight lines are parallel;
③ The internal angles on the same side are complementary, and the two straight lines are parallel.
Three. Characteristics of parallel lines
The characteristic of parallel lines is the property theorem of parallel lines, which has three * * *. ※:
(1) Two straight lines are parallel and the same angle is equal;
② Two straight lines are parallel and the internal dislocation angles are equal;
③ The two straight lines are parallel and complementary.
4. Use rulers as line segments and angles.
1. About ruler painting. ※
Drawing with a ruler means drawing with only compasses and a ruler without scales.
2. About the role of the ruler. ※
The function of ruler is to connect the line segment between two points; A line segment extend in two directions.
The function of compasses is to make a circle with any point as the center and any length as the radius; Draw an arc with any point as the center and any length as the radius.
Chapter III Data in Life
1. scientific notation: any positive number can be written in the form of a× 10n, where 1 ≤ A < 10, and n is an integer. This notation is called scientific notation. ※.
2. When taking the divisor of a number by rounding, it is said that the divisor is accurate to which place; For a divisor, all the numbers from the first non-zero number on the left to the most accurate number are called the significant digits of this number.
3. Statistical work includes:
(1) Set goals; ② Collecting data; (3) sorting out data; ④ Expression and description of data; ⑤ Analysis results.
Chapter IV Probability
+0. The probability of occurrence and non-occurrence of random events is not always 50%.
2. There are a lot of uncertain events in real life, and probability is a subject to study uncertain events. ※.
3. Understand the probability of inevitable events and impossible events. ※.
The probability of inevitable events is 1, that is, p (inevitable events) =1; The probability of an impossible event is 0, that is, p (impossible event) = 0; If a is an uncertain event, then 0
4. Understand the calculation method of geometric probability. ※
Event occurrence probability =
The fifth chapter triangle
I. Understanding triangles
The Concept of 1. Triangle and Its Angle Classification
A figure composed of three line segments that are not on the same line end to end is called a triangle.
There are two points to note here:
① The three line segments that make up a triangle should be "not in a straight line"; If they are on the same line, the triangle does not exist;
(2) The three line segments are connected in turn, which means that there is a common * * * endpoint between the three line segments, and this common * * * endpoint is the vertex of the triangle.
Triangles can be divided into three categories according to the size of internal angles: acute triangle, right triangle and obtuse triangle.
2. The relationship between the three sides of a triangle
According to the axiom "the shortest line segment in a straight line connecting two points", we can get a property theorem of triangle trilateral relationship, that is, the sum of any two sides of a triangle is greater than the third side.
Another property of triangular trilateral relationship: the difference between any two sides of a triangle is smaller than the third side.
For these two properties, we should fully understand and master their essence to avoid making mistakes in application.
Let the lengths of three sides of a triangle be a, b and c respectively:
① Generally speaking, for a side of a triangle, there must be | b-c | < a < b+c; On the other hand, only when | b-c | < a < b+c holds, the three line segments A, B and C can form a triangle;
(2) Especially, if the line segment A is known to be the largest, as long as B+C > A is satisfied, then the three line segments A, B and C can form a triangle; If it is known that line segment A is the smallest, as long as | b-c | < a is satisfied, then these three line segments can form a triangle.
3. On the sum of the interior angles of a triangle
The sum of the three internal angles of a triangle is 180.
① The two acute angles of a right triangle are complementary;
② A triangle has at most one right angle or an obtuse angle;
③ At least two internal angles in a triangle are acute angles.
4. About the midline, height and midline of the triangle
① The bisector, midline and height of a triangle are all line segments, not straight lines or rays;
② Any triangle has three bisectors, three median lines and three heights;
③ The three bisectors and three median lines of any triangle are all inside the triangle. But the height of the triangle has different positions: the three heights of the acute triangle are all inside the triangle, as shown in figure1; A right triangle has a height inside the triangle, and the other two heights are just its two sides, as shown in Figure 2; One height of an obtuse triangle is inside the triangle and the other two heights are outside the triangle, as shown in Figure 3.
(4) In a triangle, three median lines intersect at a point, three angular bisectors intersect at a point, and three straight lines with heights intersect at a point.
Two. Graphic consistency
Figures that can completely overlap are called conformal. Congruent figures have the same shape and size. Only two figures with the same shape but different sizes, or two figures with the same area but different shapes, are not congruent figures.
Four. Congruent triangle
+0. About the concept of congruent triangles
Two triangles that can completely coincide are called congruent triangles. Overlapping vertices are called corresponding points, overlapping edges are called corresponding edges, and overlapping angles are called corresponding angles.
The so-called "perfect coincidence" means that all sides are equal and all angles are equal. So it can also be said that a triangle with two equal sides and angles is called congruent triangles.
2. The corresponding edges of congruent triangles are equal, and the corresponding angles are equal. ※.
3. The nature of congruent triangles is often used to prove that two line segments are equal and two angles are equal.
5. Explore the conditions of triangle congruence
1. Three sides correspond to the intersection of two equal triangles, abbreviated as "side" or "SSS". ※
2. Two triangles with equal angles between two sides are abbreviated as "corner edge" or "SAS". ※
3. The two corners of two triangles correspond to their clamping edges. These two triangles are congruent and abbreviated as "Angle" or "ASA". ※
4. The opposite side of two angles and one of them corresponds to the congruence of two triangles, which is called "corner edge" or "AAS" for short. ※
Make a triangle
1. Given two angles and their edges, draw a triangle with the congruence condition "Angle and Angle" ("ASA").
2. Two sides and their included angles are known, and a triangle is obtained by using the congruence condition of triangle "edges and corners" ("SAS").
3. Given three sides, draw the search triangle by using the congruence condition "edge" ("SSS") of the triangle.
Eight. Explore the conditions of right-angled triangles.
1. The hypotenuse and right-angled side correspond to the coincidence of two right-angled triangles. . called "hypotenuse, right angle" or "HL". ※. This only applies to right-angled triangles.
2. Right triangle is a kind of triangle, which has the properties of a general triangle, so it can also be judged by "SAS", "ASA", "AAS" and "SSS". ※.
Other judgment methods of right triangle can be summarized as follows:
① Two right-angled sides correspond to congruences of two equal right-angled triangles;
② Two right-angled triangles with an acute angle and an equilateral are congruent.
(3) Three sides correspond to congruences of two right-angled triangles.
Chapter VII Axisymmetry in Life
1. If a graph is folded along a straight line and the parts on both sides of the straight line can overlap each other, then the graph is called an axisymmetric graph. This straight line be call the axis of symmetry. ※.
2. The distance from the point on the bisector of the angle is equal to both sides of the angle. ※.
3. The distance between any point on the vertical line of the line segment and the two endpoints of the line segment is equal. ※.
4. Angle, line segment and isosceles triangle are all axisymmetric figures. ※.
5. The bisector of the top angle of an isosceles triangle, the height on the bottom edge and the midline on the bottom edge coincide with each other, which is called "three lines in one" for short. ※.
6. The line segments connected with the corresponding points on the axisymmetric figure are vertically bisected by the axis of symmetry. ※.
7. The corresponding line segments on the axisymmetric figure are equal, and the corresponding angles are also equal. ※.