Beijing normal university printing plate ninth grade mathematics unit knowledge point volume I
Chapter I Evidence
I. Isosceles triangle
1. Definition: A triangle with two equal sides is an isosceles triangle.
2. Property: 1. The two base angles of an isosceles triangle are equal (abbreviated as "equilateral angles").
2. The bisector of the top angle of the isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide ("three lines are one")
3. The bisectors of the two base angles of an isosceles triangle are equal. (The midline of the two waists is equal and the height of the two waists is equal)
4. The point on the vertical line of the bottom of the isosceles triangle is equal to the distance between the two waists.
5. The included angle between waist height and waist bottom of isosceles triangle is equal to half of the top angle.
6. The sum of the distances from any point on the bottom of an isosceles triangle to two waists is equal to the height of one waist (which can be proved by equal area method).
7. An isosceles triangle is an axisymmetric figure with only one axis of symmetry, and the line where the bisector of the top angle is located is its axis of symmetry.
3. Judgment: In the same triangle, two triangles with equal angles are isosceles triangles (equilateral triangles for short).
Special isosceles triangle
equilateral triangle
1, definition: A triangle with three equilateral sides is called an equilateral triangle, also called a regular triangle.
(Note: If all three sides of a triangle are equal, it is called an equilateral triangle, but generally it is not called an isosceles triangle).
2. Properties: (1) The internal angles of equilateral triangles are all equal, which are all 60 degrees.
(2) The midline, the high line of each side of the equilateral triangle and the bisector of each corner coincide with each other.
(3) An equilateral triangle is an axisymmetric figure with three axes of symmetry, and the axis of symmetry is the straight line where the median line, height line or bisector of each side is located.
3. Judgment: (1) A triangle with three equal sides is an equilateral triangle.
(2) Three triangles with equal internal angles are equilateral triangles.
(3) An isosceles triangle with an angle of 60 degrees is an equilateral triangle.
(4) A triangle with two angles equal to 60 degrees is an equilateral triangle.
Second, congruence of right triangle.
There are five ways to judge whether 1 is congruent with a right triangle:
(1), two corners and their clamping edges are congruent with each other; (ASA)
(2) Both sides and their included angles are congruent; (SAS)
(3) Two triangles corresponding to three sides are congruent; (SSS)
(4) The opposite sides of two angles and one angle correspond to the congruences of two equal triangles; (AAS)
(5) The hypotenuse and the right angle correspond to the congruence of two equal triangles; (HL)
2. In a right triangle, if an inner angle is equal to 30 degrees, then the right side it faces is equal to half of the hypotenuse.
In a right triangle, the center line of the hypotenuse is equal to half of the hypotenuse.
Perpendicular bisector: A straight line perpendicular to a line segment and bisecting the line segment.
Property: The distance between the point on the vertical line of a line segment and the two endpoints of this line segment is equal.
Judgment: The point where the two ends of a line segment are at the same distance is on the middle vertical line of this line segment.
5. The perpendicular lines of the three sides of a triangle intersect at a point, and the distance from the point to the three vertices is equal, and the intersection point is the outer center of the triangle.
6. The distance from the point on the bisector of the angle is equal to both sides of the angle.
7. In an angle, if the distance from a point to both sides of the angle is equal, then it is on the bisector of the angle.
8. The bisector of an angle is the set of all points with equal distance to both sides of the angle.
9. The bisectors of the three angles of a triangle intersect at one point, and the distances from the intersection point to the three sides are equal. The intersection point is the center of the triangle.
10, the three midlines of the triangle intersect at one point, and the intersection point is the center of gravity of the triangle.
1 1 The three high lines intersect the triangle at a point, and the intersection point is the vertical center of the triangle.
Summary of mathematics knowledge points in the second volume of the ninth grade
The positional relationship between straight line and circle
(1) A straight line and a circle have nothing in common, which is called separation. AB is separated from circle O, d>r.
② A straight line and a circle have two common points, which are called intersections. This straight line is called the secant of a circle. AB intersects with ⊙O and d.
③ A straight line and a circle have only one common point, which is called tangency. This straight line is called the tangent of the circle, and this common point is called the tangent point. AB is tangent to ⊙O, and d = r. (d is the distance from the center of the circle to the straight line)
In the plane, the general method to judge the positional relationship between the straight line Ax+By+C=0 and the circle X 2+Y 2+DX+EY+F = 0 is:
1. You can get y=(-C-Ax)/B from Ax+By+C=0 (where b is not equal to 0), and substitute it into x 2+y 2+dx+ey+f = 0, and the equation about x becomes.
If b 2-4ac > 0, the circle and the straight line have two intersections, that is, the circle and the straight line intersect.
If b 2-4ac = 0, the circle and the straight line have 1 intersections, that is, the circle is tangent to the straight line.
If b 2-4ac
2. If B=0 indicates that the straight line is Ax+C=0, that is, x=-C/A, parallel to the Y axis (or perpendicular to the X axis), change X 2+Y 2+DX+EY+F = 0 to (X-A) 2+(Y-B) 2 = R, and let Y =
When x=-C/Ax2, the straight line deviates from the circle;
Mathematics review plan for the first volume of the third grade
1. review objectives: the following objectives should be achieved through overall review:
(1 Make the learned knowledge systematic and structured, and make students connect their three-year mathematical knowledge into an organic whole, which is more conducive to students' understanding;
(2) Practice more, consolidate basic knowledge and master basic skills;
(3) Do a good job in method teaching, guide students to summarize problem-solving methods and adapt to the changes of various types of questions;
(4) Do a good job in comprehensive problem training and improve students' ability to comprehensively apply knowledge to analyze problems.
Second, review methods and measures:
1. Excavate teaching materials, lay a solid foundation, attach importance to understanding basic knowledge and guide basic methods.
After more than two years of study, students have mastered certain basic knowledge, basic methods and basic skills, but their understanding of teaching materials is fragmentary and their exploration of the law of solving problems is superficial. Therefore, when organizing students to review in general, we should first guide students to systematically sort out the teaching materials, construct the knowledge structure, and make all kinds of concepts, axioms, theorems, formulas, common conclusions and problem-solving methods and skills reappear in students' minds. In teaching, based on the textbook, we should fully tap and give play to the potential functions of examples and exercises in the textbook, and guide students to summarize and sort out the basic knowledge and methods in the textbook to form a structure. Resolutely overcome the practice of emphasizing problems, skills, textbooks and basics.
2, * * * have participation, pay attention to the process.
Teachers should not do everything when reviewing for the senior high school entrance examination. In review, we should give full play to students' main role, highlight their main position, make students the protagonists of review activities, give students enough study time, let them talk and do, expose their thinking process and stimulate their thinking potential. Only in this way can the leading role of teachers be reflected and the guidance of teachers be truly implemented. Therefore, in the basic review, we give students as much time as possible to explore, so that students at all levels can satisfy their knowledge and improve the learning effect. Especially in the teaching process of comprehensive questions, get to the point, thoroughly understand and summarize in time. We must teach students ideas and methods, at the same time, teachers should evaluate them in place, start with the nuances, let students analyze, find out the reasons for the mistakes, know their own weak links, be familiar with the general analysis ideas, and discuss with students in depth, so we should pay attention to why we should solve them like this. Explain the train of thought and how to design the problem-solving format. How to find the breakthrough of the problem?
3. Strengthen training, pay attention to application and cultivate ability.
The ultimate goal of mathematics teaching is to cultivate students' innovative consciousness, applied consciousness and comprehensive ability. Teachers can cultivate consciously and purposefully. This can greatly accelerate the formation and development of mathematical ability, make all kinds of thinking methods reasonable and simple, and give full play to students' creative ability. This paper analyzes the ability questions of senior high school entrance examination in various provinces and cities in recent years: on the basis of students' existing knowledge, through reading comprehension and reasoning analysis, summarize the laws and conclusions; Connecting with practice, paying attention to application and cultivating the ability of exploration, discovery and innovation are the inevitable trends of the proposition of the senior high school entrance examination. Therefore, when organizing students to review, we should use innovative, practical, creative and open questions close to students' lives to activate students' thinking.
4. Implement the training of various mathematical thinking methods to improve students' mathematical quality.
Understanding and mastering all kinds of mathematical ideas and methods is the premise of forming mathematical skills and improving mathematical ability. Many mathematical ideas and methods have appeared and been applied to junior high school mathematics. For example, the idea of transformation, the idea of function, the idea of equation, the idea of combining numbers with shapes and so on. Mathematical methods include: substitution method, matching method, mirror image method, analytical method, undetermined coefficient method, analytical method and comprehensive method. These methods should be used flexibly according to needs. Therefore, in the review, we should train in layers according to the requirements.
(1 Take different training forms. On the one hand, we should always change the types of questions: fill-in-the-blank questions, true or false questions, multiple-choice questions, short-answer questions, proof questions and so on. Let students realize that although the questions have changed, the essential methods of solving problems have not changed, which will enhance students' interest in training. On the other hand, we should change the structure of the questions, such as changing the questions and changing the conditions.
(2) properly train problem groups. Special training for this method for a certain period of time can strengthen this method, leave a deep impression on students and master it quickly and firmly.
5. Do a good job in the classification and variant teaching of examples and exercises in textbooks.
In the teaching of mathematics review course, it is not only the need to improve the teaching quality in a large area, but also the means to deal with the exam. Therefore, according to the teaching purpose, teaching focus and students' reality, guide students to analyze and classify related examples, summarize the law of solving problems, and improve the review efficiency. For example, the exercises with variability can guide students to carry out variant training, so that students can perceive mathematical methods from many aspects and improve their ability to comprehensively analyze and solve problems. In the explanation, teachers should guide students to flexibly change representative issues, so as to draw inferences from others, cultivate students' adaptability, improve students' skills and skills, and tap the role of examples and exercises in teaching materials. We can start from the following aspects: (1) Find other solutions; (2) changing the form of the topic; (3) the conditions and conclusions of exchanging topics; (4) changing the conditions of the topic; 5] Further popularize and extend the conclusion; [6] A series of different problems; (7) analogy compilation, etc.
6, for all students, the implementation of hierarchical teaching.
According to the great differences of students' learning ability in mathematics, this paper specifically studies what knowledge and ability students at all levels lack most and what aspects of mathematics skills need to be improved most, finds out their differences and problems, and then selectively and emphatically implements breakthrough hierarchical teaching, putting forward different requirements for students at different levels. Top students can encourage them to study ahead of time, middle students can guide them and underachievers can help them. In particular, we should care about students with learning difficulties, through the cultivation of learning interest and the guidance of learning methods.
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