Every subject has its own learning method, but it is always inseparable. Mathematics, like Chinese and English, needs memorization, recitation and practice. The following are some study materials I have compiled for you, hoping to help you.

The induction of mathematics knowledge points in the second volume of the ninth grade

Knowledge point 1. concept

Figures with the same shape are called similar figures. (i.e. graphs with equal corresponding angles and equal corresponding edge ratios)

Interpretation: (1) Two graphs are similar, and one graph can be seen as being enlarged or reduced by the other graph.

(2) Conformity can be regarded as a special similarity, that is, not only the same shape, but also the same size.

(3) Judging whether two figures are similar depends on whether they are the same shape, which has nothing to do with other factors.

Knowledge point 2. Proportional line segment

For four line segments A, B, C and D, if the ratio of the lengths of two of them is equal to the ratio of the lengths of the other two, that is (or a:b=c:d), then these four line segments are called proportional line segments.

Knowledge point 3. Properties of similar polygons

Properties of similar polygons: the corresponding angles of similar polygons are equal, and the proportions of corresponding edges are equal.

Interpretation: (1) Understand the definition of similar polygons correctly and make clear the corresponding relationship.

(2) It is clear that the correspondence of similar polygons comes from writing, and the similarity ratio is sequential.

Knowledge point 4. Similar triangles's concept

A triangle with equal corresponding angles and equal ratio of corresponding sides is called similar triangles.

Interpretation: (1) similar triangles is one of the similar polygons;

(2) similar triangles should be understood by combining the properties of similar polygons;

(3) similar triangles should have the same shape, but different sizes;

(4) Similarity is indicated by "√" and pronounced as "similar to";

(5) The ratio of corresponding sides in similar triangles is called similarity ratio.

Knowledge point 5. Similar triangles's judgment method

(1) Definition: Two triangles with equal corresponding angles and proportional corresponding sides are similar;

(2) The triangle formed by cutting the other two sides (or extension lines of other two sides) with a straight line parallel to one side of the triangle is similar to the original triangle.

(3) If two angles of a triangle are equal to two angles of another triangle, then the two triangles are similar.

(4) If two sides of a triangle are proportional to two sides of another triangle, and the included angles are equal, then the two triangles are similar.

(5) If three sides of a triangle are proportional to three sides of another triangle, then the two triangles are similar.

(6) Two right triangles divided by the height on the hypotenuse are similar to the original triangle.

Knowledge point 6. The nature of similar triangles

(1) The corresponding angles are equal, and the ratio of the corresponding sides is equal;

(2) The ratio corresponding to the height, the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio;

(3) The ratio of similar triangles perimeter is equal to the similarity ratio; The area ratio is equal to the square of the similarity ratio.

(4) Projective theorem

Jiangsu education edition mathematics ninth grade knowledge points

1 quadratic root: the formal formula is quadratic root;

Attribute: It is non-negative;

2 multiplication and division of quadratic root:

Quadratic root addition and subtraction: when adding and subtracting quadratic roots, first merge the simplest quadratic roots of Huawei, and then merge the quadratic roots with the same number of roots.

4 Helen-Qin Jiushao formula: where s is the area and p is.

1: An equation with algebraic expressions on both sides of the equal sign and only one unknown number. The number of unknowns is 2.

2 matching method: match one side of the equation into a completely flat way, and then square both sides;

Factorization: the product of two factors on the left and zero on the right.

3 The application of quadratic equation in one variable in practical problems

Vieta Theorem: Let it be the two roots of the equation, then there is.

1: graphic transformation in which a graphic rotates by an angle around a certain point.

Property: the distance from the corresponding point to the center is equal;

The included angle between the corresponding point and the line segment connecting the rotation center is equal to the rotation angle.

Graphic consistency before and after rotation.

2. Center symmetry: if one graph rotates around a point by 180 degrees and coincides with another graph, then the two graphs are symmetrical about the center of this point;

Centrally symmetric figure: a figure rotates 180 degrees around a certain point and can coincide with the original figure, so it is called centrosymmetric;

Three coordinates of a point with symmetrical origin.

1 Definition of circle, center, radius, diameter, arc, chord and semicircle

2 Diameter perpendicular to the chord

A circle is a figure, and any straight line with a diameter is its axis of symmetry;

The diameter perpendicular to the chord divides the chord in two and squares the two arcs opposite to the chord;

The diameter of the chord is perpendicular to the chord, and the two arcs opposite the chord are equally divided.

3 Arc, chord and central angle

In the same circle or in the same circle, equal central angles have equal arcs and equal chords.

4 circle angle

In the same circle or equal circle, the circumferential angle of the same arc or equal arc is equal, which is equal to half the central angle of the arc;

The semicircle (or diameter) faces the right angle, and the 90-degree angle faces the diameter.

The positional relationship between five points and a circle

D>r, the point outside the circle

Points on the circle d=r

Point dR+r in the circle

Circumscribed =R+r

Intersecting R-r

Mathematics review materials for grade three

Inverse proportional function, similarity, acute trigonometric function and projection and view.

(1) Inverse proportional function: The image and properties of inverse proportional function are important contents of the mathematics proposition of senior high school entrance examination. The test questions are novel and flexible, accounting for about 3-8 points, which is quite difficult.

Survey content

(1) can draw the image of inverse proportional function and master its basic properties.

② The expression of the inverse proportional function can be determined according to the conditions.

③ Inverse proportional function can be used to solve practical problems.

(2) Similarity: The shape of a figure is an extremely important content in plane geometry, and it is also the key content in mathematics of the senior high school entrance examination. The general score is about 6- 12, and it is difficult to choose multiple-choice questions, fill-in-the-blank questions and answer questions.

Survey content

The key is the nature and discrimination method of (1) similar triangles.

② Understanding of similar polygons and application of golden section.

(3) It is difficult to synthesize similar shapes with triangles and parallelograms.

(3) acute angle trigonometric function

(4) Projection and view: The score is generally 3-6 points, and the test questions appear in the form of filling in the blanks, selecting and answering.

Survey content

① Three views of common geometry

(2) The unfolding and folding of common geometry is a hot topic in the examination, which deserves attention.

③ Use similarity to combine parallel projection and central projection to solve practical problems.

(Different regions have different scores for reference)

Multiple choice questions: 3 points, *** 14, with a total score of 42 points.

Fill in the blanks: 3 points, ***5, total score 15.

Answer: * * * 7 questions, with a total score of 63 points.

(A) the calculation and proof of line segments and angles

The short answer to the senior high school entrance examination is generally divided into two or three parts. The first part is basically simple questions and intermediate questions, aiming at examining the foundation. The second part The second part is often the difficult problem of starting to pull points.

(B) column equation (group) to solve application problems

In the senior high school entrance examination, equation is the most important part of junior high school mathematics, so it is also the compulsory content of the senior high school entrance examination. Judging from the senior high school entrance examination in recent years, there are many exams combined with current events, so candidates need to have some real life experiences.

(C) reading comprehension problems

Reading comprehension is a highlight of senior high school entrance examination. Reading comprehension is often to give a material or introduce a super-class knowledge or give an answer to a question, and then give the conditions of the question.

Cross-synthesis of multiple functions

The functions of junior high school contact mainly include linear function, quadratic function and inverse proportional function. This kind of topic itself is not too difficult, and it rarely appears as a finale topic. Generally, it appears as an intermediate question to test students' mastery of functions.

(5) Dynamic geometry

Judging from the senior high school entrance examinations over the years, dynamic geometry often appears as a final topic, and the scoring rate is also the lowest. Dynamic geometry is generally divided into two categories, one is algebraic synthesis, in the coordinate system, the moving straight line is generally solved by multiple functions. The other is the geometric synthesis problem, which sets moving points in trapezoid, rectangle and triangle to test students' comprehensive analysis ability.

(6) graphic position relationship

In middle school mathematics, the graphic position relationship mainly includes points, lines, triangles, rectangles and squares and their relationships. The senior high school entrance examination will be included in the questions of function, coordinate system and geometry, the most important of which is the various problems of triangle.

Related articles in the second volume of the third grade mathematics knowledge points:

★ What are the knowledge points summarized in the second volume of junior high school mathematics?

★ Beijing Normal University Junior High School Mathematics Knowledge Point Volume II

★ The latest summary of mathematics knowledge points in Grade Three.

★ The arrangement of knowledge points in the second volume of ninth grade mathematics

★ Outline of knowledge points of mathematics circle in the second volume of the ninth grade

★ The arrangement of mathematics knowledge points in the second volume of the eighth grade

★ People's Education Edition ninth grade mathematics knowledge points

★ sort out the knowledge points of mathematics in grade three.

★ Summary of ninth grade mathematics knowledge points

★ Knowledge points of junior high school mathematics.