Eccentricity: it is the intersection of the perpendicular lines of the three sides of a triangle, that is, the center of the circumscribed circle.
Eccentricity theorem: the perpendicular lines of three sides of a triangle intersect at one point. This point is called the outer center of the triangle.
Properties of the outer center of triangle:
1, the intersection of the perpendicular lines of three sides of a triangle at a point, which is the center of the circumscribed circle of the triangle;
2. There is only one circumscribed circle of a triangle, that is, for a given triangle, its outer center exists, but there are countless inscribed triangles of a circle, and the outer centers of these triangles coincide;
3. The outer center of the acute triangle is in the triangle;
The outer center of an obtuse triangle is outside the triangle;
The outer center of a right triangle coincides with the midpoint of the hypotenuse.
At △ABC
4、OA=OB=OC=R
5、∠BOC=2∠BAC、∠AOB=2∠ACB、∠COA=2∠CBA
6、S△ABC=abc/4R
Concept of inequality of knowledge point 2 in junior middle school mathematics volume I
1. Inequality: A formula that uses an inequality symbol to represent inequality is called inequality.
2. Solution set of inequality: For an unknown inequality, any unknown value suitable for this inequality is called the solution of this inequality.
3. For an unknown inequality, the set of all its solutions is called the solution set of this inequality.
4. The process of finding the solution set of inequality is called solving inequality.
5. The method of expressing inequality with number axis.
Basic properties of inequality
1. Add and subtract the same number or the same algebraic expression on both sides of the inequality, and the direction of the inequality remains unchanged.
2. Both sides of the inequality are multiplied or divided by the same positive number, and the direction of the inequality remains unchanged.
3. When both sides of inequality are multiplied or divided by the same negative number, the direction of inequality will change.
4. Explanation: ① In the unary linear inequality, unlike the equality, the equal sign is unchanged, but changes with the operation of addition or multiplication. (2) If the inequality is multiplied by 0, the inequality becomes an equal sign. Therefore, in the problem, if the number of multiplication is required, it depends on whether there is a one-dimensional inequality in the problem. If there is, then the number multiplied by the inequality is not equal to 0, otherwise the inequality is not established.
linear inequality
1, the concept of one-dimensional linear inequality: Generally speaking, an inequality contains only one unknown, the degree of the unknown is 1, and both sides of the inequality are algebraic expressions. This inequality is called one-dimensional linear inequality.
2. The general steps to solve the linear inequality of one variable are: 1, denominator 2, bracket 3, shift term 4, merge similar terms 5, and convert the coefficient of x term into 1.
Unary linear inequality system
1, the concept of one-dimensional linear inequality group: combine several one-dimensional linear inequalities into one one-dimensional linear inequality group.
2. The common part of the solution set of several linear inequalities is called the solution set of linear inequalities.
3. The process of finding the solution set of inequality group is called solving inequality group.
4. When any number x can't make the inequality hold at the same time, we say that the inequality group has no solution or its solution is an empty set.
5. Solving one-dimensional linear inequalities.
1 Find the solution set of each inequality in the inequality group.
2 Use the number axis to find the common part of the solution set of these inequalities, that is, the solution set of this inequality group.
6. Inequality and unequal groups
Inequality: ① If the symbol > =, 0 and a≠ 1) is used, then X is called the logarithm of n with the base of A, and it is recorded as x=log(a)(N), where A should be written at the lower right of log. Where a is called the base of logarithm and n is called real number. Usually, we call the logarithm with the base of 10 as the ordinary logarithm, and the logarithm with the base of e as the natural logarithm.
Mathematics learning skills
1. Combination of seeking advice and self-study.
In the process of learning, we should strive for the guidance and help of teachers, but we should not rely too much on teachers. We should take the initiative to study, explore and acquire, and seek the help of teachers and classmates on the basis of our own serious study and research.
2. Combination of learning and thinking
In the process of learning, we should carefully study the content of teaching materials, ask questions and trace back to the source. For every concept, formula and theorem, we should understand its context, cause and effect, internal relations, and mathematical ideas and methods involved in the derivation process. When solving problems, we should try our best to adopt different ways and methods, and overcome the rigid learning methods of books and machinery.
3. Combine learning with application and be diligent in practice.
In the process of learning, we should accurately grasp the essential meaning of abstract concepts and understand the evolution process of abstraction from actual model to theory. For theoretical knowledge, we should look for concrete examples in a wider scope, make them concrete, and try our best to apply theoretical knowledge and thinking methods to practice.
4. broaden your horizons, accept the appointment, and return to the appointment from Bo.
Textbooks are the main source of knowledge, but not the only source. In the process of learning, in addition to studying textbooks carefully, we should also read relevant extracurricular materials to expand our knowledge. At the same time, on the basis of extensive reading, do research seriously and master its knowledge structure.
5. There are both imitation and innovation.
Imitation is an indispensable learning method in mathematics learning, but it must not be copied mechanically. On the basis of digestion and understanding, use your brains and put forward your own opinions and opinions, instead of sticking to the existing framework and existing model.
6. Review in time to enhance memory.
The content of study in class must be digested on the same day, reviewed first, then practiced, and the review work must be carried out frequently. Every time you finish a unit, you should summarize and sort out the knowledge you have learned so as to make it systematic and profound.
7. Summarize the learning experience and evaluate the learning effect.
Summarizing and evaluating in learning is conducive to the establishment of knowledge system, the mastery of problem-solving laws, the adjustment of learning methods and attitudes, and the improvement of judgment ability. In the process of learning, we should pay attention to summing up the gains and experiences of listening to lectures, reading books and solving problems.
Knowledge point 4 (triangle midline theorem) in the first volume of mathematics in grade three
The center line of the triangle is parallel to the third side of the triangle and equal to half of the third side.
(Nature of parallelogram)
① The opposite sides of the parallelogram are equal;
② The diagonals of parallelograms are equal;
(3) The diagonal of the parallelogram is equally divided.
(The nature of rectangle)
① A rectangle has all the properties of a parallelogram;
② All four corners of a rectangle are right angles;
③ The diagonals of the rectangles are equal.
Judgement and properties of squares
1, determination method:
1 rectangle with equal adjacent sides;
2 rhombic, with adjacent sides vertical;
3 a rectangle with a vertical diagonal;
4 diamonds with equal diagonal lines;
2. Nature:
1 side: four sides are equal and the opposite sides are parallel;
2 angles: all four angles are equal angles and right angles, and adjacent angles are complementary;
3 Diagonal lines are equally divided and perpendicular to each other, and each long diagonal line is equally divided into a set of internal angles.
Judgement theorem of isosceles triangle
(Determination method of isosceles triangle)
1. A triangle with two equal sides is an isosceles triangle.
2. Decision Theorem: If a triangle has two equal angles, then it is an isosceles triangle.
Angular bisector: The ray bisecting an angle is called the angular bisector of the angle.
There are several points to note in the definition. The learning method is that the bisector of an angle is a ray, not a line segment or a straight line. Many times there will be a straight line in the topic, which is the symmetry axis of the bisector, which also involves the trajectory problem. The bisector of an angle is a point with equal distance to both sides of the angle.
Property theorem: the distance between the point on the bisector of an angle and both sides of the angle is equal.
Decision theorem: the points with equal distance to both sides of the angle are on the bisector of the angle.
Standard deviation and variance
What is the limit range? The difference between the data in a set of data and the minimum value is called range, that is, range = value-minimum value.
Calculator-general steps to find standard deviation and variance;
1. Open the calculator, press the "On" key, and press "Mode" and "2" to enter the statistical SD state.
2. Before starting data input, be sure to press Shift, CLR, 1 and = to clear the statistical memory.
3. Input data: press the number keys to input numerical values, and then press the "M+" key to complete data input once. If you want to enter the same data for this, you can also press "Shiet" after step 3. , and then enter the frequency of data, and then press the "M+" key.
4. When all data are input, press "Shift" and "2" and select "Standard Deviation" to get the standard deviation of the required data;
5. The square of standard deviation is the variance.
The difference of knowledge points in the first volume of mathematics in grade three 5 1, inevitable events, impossible events and random events.
2. Possibility
Generally, in a large number of repeated experiments, if the frequency of event A occurs,
Will be stable near a constant p, then this constant p is called the probability of event A, and it is recorded as p (a) = p 。
Note: (1) probability is a quantitative reflection of the probability of random events.
(2) Probability is the value that the frequency of events tends to be stable gradually in a large number of repeated experiments, that is, the probability of events can be estimated by the frequency of events in a large number of repeated experiments, but it cannot be simply equated.
3, the method of probability
(1) enumeration method to find probability (list method, tree drawing method)
(2) Estimation of probability by frequency: Generally speaking, the frequency of events in a large number of repeated experiments can be used to estimate the probability of events. On the other hand, the frequency of events in a large number of repeated experiments is stable around a certain constant (the probability of events), which shows that the probability is a fixed value and the frequency varies with the number of experiments. This is an approximation of probability, and the two cannot be simply equated.
Judgement method of right triangle, the knowledge point of the first volume of mathematics in the third grade;
Judgment 1: Definition, a triangle with an angle of 90 is a right triangle.
Decision 2: Decision Theorem: A triangle with sides A, B and C is a right triangle with hypotenuse C. If three sides A, B and C of the triangle satisfy a2+b2=c2, the triangle is a right triangle. (Inverse Pythagorean Theorem).
Decision 3: If the opposite side of the 30 internal angle of a triangle is half of a certain side, the triangle is a right triangle with this long side as the hypotenuse.
Decision 4: A triangle whose two acute angles are complementary angles (the sum of the two angles is equal to 90) is a right triangle.
Decision 5: If two straight lines intersect and the product of their slopes is negative reciprocal, then the two straight lines are perpendicular to each other. therefore
Decision 6: If the median line of one side of a triangle is equal to half of its side, then the triangle is a right triangle.
Decision 7: A triangle with an angle of 30 degrees is equal to half of the hypotenuse of the triangle, then this triangle is a right triangle. (Unlike Decision 3, this theorem is applied to triangles with known hypotenuse. )
Knowledge points of the first volume of mathematics in the third grade 7 1. Numerical classification and concept number list;
Note: Classification principle: 1) Proportionality (no weight, no leakage) 2) Standard.
2. Non-negative number: the collective name of positive real number and zero. (Form: x0)
Property: If the sum of several non-negative numbers is 0, then each non-negative number is 0.
3. Reciprocity: ① Definition and characterization
② attribute: a.a1/a (a1); 1/a，aC.0
4. Reciprocal: ① Definition and representation
② Property: the position of aB.a and -a on the number axis when A.a0; The sum of c is 0 and the quotient is-1.
5. Number axis: ① Definition (three elements)
② Function: a. Visually compare real numbers; B. clearly reflect the absolute value; C. establish a one-to-one correspondence between points and real numbers.
6. Odd numbers, even numbers, prime numbers and composite numbers (positive integer natural numbers)
Definition and expression:
Odd number: 2n- 1
Even number: 2n(n is a natural number)
7. Absolute value: ① Definition (two kinds):
Algebraic definition:
Geometric definition: the geometric meaning of the absolute value top of the number A is the distance from the point corresponding to the real number A on the number axis to the origin.
②│a│0, and the symbol │ │ is a sign of non-negative numbers; ③ There is only one absolute value of number A; (4) To deal with any kind of topic, as long as there is a │ │ in it, the key step is to remove this │ │ symbol.
The difference between knowledge points 8 1, inevitable events, impossible events and random events in the first volume of mathematics in grade three.
2. Possibility
Generally speaking, in a large number of repeated experiments, if the frequency of event A will be stable near a certain constant P, then this constant P is called the probability of event A, and it is recorded P(A)= p (a) = p (a) = p. 。
Note: (1) probability is a quantitative reflection of the probability of random events.
(2) Probability is the value that the frequency of events tends to be stable gradually in a large number of repeated experiments, that is, the probability of events can be estimated by the frequency of events in a large number of repeated experiments, but it cannot be simply equated.
3, the method of probability
(1) enumeration method to find probability (list method, tree drawing method)
(2) Estimation of probability by frequency: On the one hand, the probability of events can be estimated by the frequency of events in a large number of repeated experiments. On the other hand, the frequency of events in a large number of repeated experiments is stable around a certain constant (the probability of events), which shows that the probability is a fixed value and the frequency varies with the number of experiments. This is an approximation of probability, and the two cannot be simply equated.
Monomial and Polynomial of 9 Knowledge Points in Junior Middle School Mathematics Volume I
A formula containing only a few numbers and letters (including multiplication) is called a monomial. A single number or letter is also a monomial.
The numerical factor in the monomial is called the numerical coefficient of this monomial or letter factor, or coefficient for short.
When the single coefficient is 1 or-1, "1" is usually omitted.
In a monomial, the sum of the exponents of all the letters is called the degree of the monomial.
If there are several monomials, whether their coefficients are the same or not, as long as they contain the same letters and the indexes of the same letters are the same respectively, then these monomials are called similar monomials, and all constants are simply called similar terms.
1, polynomial
A formula consisting of the algebraic sum of finite monomials is called a polynomial.
Each monomial in a polynomial is called a polynomial term, and the term without letters is called a constant term.
The monomial can be regarded as a special case of polynomial.
Increase or decrease the coefficient of similar monomials, while the index of letters in monomials remains unchanged.
In a polynomial, the number of different unknowns is called the number of elements of the polynomial, and the number of monomials contained in the polynomial is called the degree of monomials contained in terms of the polynomial, which is called the degree of the polynomial.
2, the value of the polynomial
Any polynomial is a formula that connects known numbers with unknown numbers through addition, subtraction, multiplication and division.
3. Identities of polynomials
For two unary polynomials fx and gx, if the unknown number X takes any value A, and if their values are all equal, that is, fa=ga, the two polynomials are said to be equal, which is denoted as fx==gx, or abbreviated as fx=gx.
Property 1 If fx==gx, then for any value a, there is fa=ga.
Property 2 If fx==gx, then the coefficients of two polynomials must be equal.
4. The root of unary polynomial
Generally speaking, the value of unknown x that can make the value of polynomial fx equal to 0 is called the root of polynomial fx.
Polynomial addition, subtraction and multiplication
1, addition and subtraction of polynomials
2. Polynomial multiplication
In the multiplication of monomial, the coefficient is the coefficient of the product, and for the same letter factor, the index is the factor of the product.
3. Polynomial multiplication
Multiply polynomials by multiplying each term of one polynomial by each term of another polynomial, and then add the products.
Commonly used multiplication formula
Formula one variance formula
a+ba—b=a^2—b^2
The product of the sum of two numbers and the difference between the two numbers is equal to the square difference between the two numbers.
The knowledge point of the first volume of mathematics in grade three is 10I. Define and define expressions.
Generally speaking, there is the following relationship between independent variable X and dependent variable Y: Y = AX 2+BX+C.
A, b and c are constants, and a≠0, a determines the opening direction of the function. When a0 is used, the opening direction is upward, and when a0 is used, the opening direction is downward. IaI can also determine the opening size. The greater the IaI, the smaller the opening and the larger the opening. Y is called the quadratic function of X.
The right side of a quadratic function expression is usually a quadratic trinomial.
Two. Three Expressions of Quadratic Function
General formula: y = ax 2+bx+c (a, b and c are constants, and a≠0).
Vertex: y = a(x-h)2+k[ vertex P(h, k) of parabola]
Intersection point: y = a(x-x)(x-x)[ only applicable to parabolas with intersection points A(x, 0) and B(x, 0) with the x axis]
Note: Among these three forms of mutual transformation, there are the following relations:
k=(4ac-b^2)/4a x，x=(-b √b^2-4ac)/2a
Three. Quadratic function image
Making the image of quadratic function y = x 2 in plane rectangular coordinate system, we can see that the image of quadratic function is parabola.
Knowledge points in the first volume of mathematics in grade three 1 1: the concept of quadratic root
A(a0) formula is called quadratic radical.
Note: In the quadratic root, the open number can be a number, or it can be an algebraic expression such as monomial, polynomial and fraction. However, it must be noted that a0 is the precondition that A is a quadratic root, because negative numbers have no square root, such as 5, (x2+ 1).
(x- 1) (x 1) is a quadratic radical, while (-2) and (-x2-7) are not quadratic radicals.
Knowledge point 2: Value range
1. Conditions for quadratic roots to be meaningful: According to the meaning of quadratic roots, A is meaningful at a0, so to make quadratic roots meaningful, we only need to make the number of roots greater than or equal to zero.
2. The condition that the square root is meaningless: Because negative numbers have no arithmetic square root, when A￠0, A is meaningless.
Knowledge point 3: Nonnegativity of quadratic root a(a0)
A(a0) represents the arithmetic square root of A, that is, a(a0) is non-negative, that is, 0(a0).
Note: Because the square root A represents the arithmetic square root of A, the arithmetic square root of positive number is positive, and the arithmetic square root of 0 is 0, so the arithmetic square root of non-negative number (a0) is non-negative, that is, 0(a0), which is similar to absolute value or even power. This property is often used to solve problems. If a+b=0, then a = 0 and b = 0; If a+|b|=0, then a = 0 and b = 0；; If a+b2=0, then a = 0 and b = 0.
Knowledge Point 4: Properties of Quadratic Formula (1)
(a)2=a(a0)
The written language stipulates that the square of the arithmetic square root of a non-negative number is equal to this non-negative number.
Note: The property formula (a)2=a(a0) of quadratic square root is the conclusion obtained by using the definition of square root in reverse. The above formula can also be applied in reverse: if a0, then
A=(a)2, such as: 2 = (2) 2, 1/2 = (1/2) 2.
Knowledge point 5: the nature of quadratic radical
a2=|a|
Written language stipulates that the arithmetic square root of the square of a number is equal to the absolute value of this number.
note:
1. When simplifying a2, be sure to find out whether the base number A of the radical sign is positive or negative. If it is positive or 0, it is equal to A itself, that is, A2 = | A | = A (if A is negative, it is equal to the inverse of A -a, that is, A2 = | A | =-A (A | 0);
2. The value range of A in a2 can be any real number, that is, A2 must be meaningful no matter what value A takes;
3. When simplifying a2, it should be converted into |a| first, and then simplified according to the meaning of absolute value.
Knowledge point 6: (a) Similarities and differences between 2 and a2
1, the difference: (a)2 and a2 have different meanings, (a)2 stands for the square of the arithmetic square root of non-negative number A, and a2 stands for the arithmetic square root of the square of real number A; In (a)2, A in a2 can be a positive real number, 0 or a negative real number. But (a)2 and a2 are both negative, that is, (a)20, a20. So the result of its operation is different, (a)2=a(a0) and a2=|a|.
2. Similarity: When the number of roots is non-negative, that is a0, (a)2 = a￠0, (a)2 is meaningless, while A2 = | a | =-a. 。
Knowledge points in the first volume of mathematics in grade three 12 1, the general form of quadratic function: Y = AX2+BX+C. (a0)
2. Some concepts about quadratic function: the image of quadratic function is parabola, so it is also called parabola y = AX2+BX+C; The parabola is symmetrical about the symmetry axis and is defined by the symmetry axis. Half the image goes uphill and the other half goes downhill. Where c is the intercept of the quadratic function on the Y axis, that is, the image of the quadratic function must pass through the (0, c) point.
3. Characteristics of y=ax2 (a0): When b=0 and c=0 in y=ax2+bx+c (a0), the quadratic function is Y = AX2 (A is a special quadratic function with the following characteristics: (1) The image is symmetrical about Y; (2) Vertex (0,0);
4. Find the analytic formula of quadratic function: Given the coordinates of three points on the image of quadratic function, we can set the analytic formula y=ax2+bx+c, substitute the coordinates of these three points, solve the ternary linear equations about A, B and C, and then find the values of A, B and C, thus finding the analytic formula-undetermined coefficient method.
5. Vertex of quadratic function: y = a (x-h) 2+k (a) The vertex coordinates (h, k) of quadratic function, the equation of symmetry x=h and the maximum value of function y = k can be obtained directly from the vertex.
Knowledge points in the first volume of mathematics in grade three 13 First of all, we know that SIN (a+b) = SINA * COSB+COSA * SINB, SIN (a-b) = SINA * COSB-COSA * SINB.
We add these two expressions to get sin(a+b)+sin(a-b)=2sina*cosb.
So sin a * cosb = (sin (a+b)+sin (a-b))/2.
Similarly, if you subtract the two expressions, you get COSA * SINB = (SIN (A+B)-SIN (A-B))/2.
Similarly, we also know that COS (a+b) = COSA * COSB-SINA * SINB, COS (a-b) = COSA * COSB+SINA * SINB.
Therefore, by adding the two expressions, we can get cos(a+b)+cos(a-b)=2cosa*cosb.
So we get, COSA * COSB = (COS (A+B)+COS (A-B))/2.
Similarly, by subtracting two expressions, Sina * sinb =-(cos (a+b)-cos (a-b))/2 can be obtained.
In this way, we get the formulas of the sum and difference of four products:
Sina * cosb =(sin(a+b)+sin(a-b))/2
cosa * sinb =(sin(a+b)-sin(a-b))/2
cosa * cosb =(cos(a+b)+cos(a-b))/2
Sina * sinb =-(cos(a+b)-cos(a-b))/2
Well, with four formulas of sum and difference, we can get four formulas of sum and difference product with only one deformation.
Let a+b be X and A-B be Y in the above four formulas, then A = (X+Y)/2 and B = (X-Y)/2.
If a and b are represented by x and y respectively, we can get four sum-difference product formulas:
sinx+siny = 2 sin((x+y)/2)* cos((x-y)/2)
sinx-siny = 2cos((x+y)/2)* sin((x-y)/2)
cosx+cosy = 2cos((x+y)/2)* cos((x-y)/2)
cosx-cosy =-2 sin((x+y)/2)* sin((x-y)/2)
Knowledge points in the first volume of mathematics in grade three 14 1. Definition: Two groups of parallelograms with parallel opposite sides are called parallelograms.
2. The properties of parallelogram
(1) The opposite sides of the parallelogram are parallel and equal;
(2) The adjacent angles of the parallelogram are complementary and the diagonal angles are equal;
(3) diagonal bisection of parallelogram;
3. Determination of parallelogram
Parallelogram is an important content in geometry, and how to judge whether a parallelogram is a parallelogram according to its properties is a key point. Here are five ways to judge a parallelogram:
The first category: related to the opposite sides of a quadrilateral.
(1) Two groups of parallelograms with parallel opposite sides are parallelograms;
(2) Two groups of quadrangles with equal opposite sides are parallelograms;
(3) A group of quadrilaterals with parallel and equal opposite sides are parallelograms;
The second category: related to the diagonal of quadrilateral.
(4) Two groups of quadrangles with equal diagonal are parallelograms;
The third category: related to the diagonal of the quadrilateral.
(5) The quadrilateral whose diagonals bisect each other is a parallelogram.
Knowledge points in the first volume of mathematics in grade three: 15 1. One-dimensional quadratic equation: in an integral equation, there is only one unknown and the equation with the highest degree of the unknown is called one-dimensional quadratic equation. The general form of an unary quadratic equation is (). Where () is called quadratic term, () is called linear term, and () is called constant term; () is called the coefficient of quadratic term, and () is called the coefficient of linear term.
2. Discrimination of error-prone knowledge:
(1) To judge whether an equation is a quadratic equation, we should sort it out and make it into a general form before judging. Pay attention to the general form of quadratic equation.
(2) When solving equations by formula method and factorization method, they should be converted into general form first.
(3) When collocation method is used, the quadratic coefficient should be 1.
(4) When using direct cholesky decomposition, remember to take positive and negative values.