The induction of mathematics knowledge points in the second volume of the ninth grade
circle
★ Key points ① Important properties of the circle; (2) the positional relationship between straight lines and circles, and between circles; ③ Angle theorem related to circle; ④ Theorem of proportional line segment related to circle.
☆ Summary ☆
First, the basic properties of the circle
Definition of 1. circle (two kinds)
2. Related concepts: chord and diameter; Arc, equal arc, upper arc, lower arc, semicircle; Distance from chord to center; Equal circle, same circle, concentric circle.
3. "Three-point circle" theorem
4. Vertical Diameter Theorem and Its Inference
5. "Equivalence" theorem and its inference
6. Angle related to circle: (1) Definition of central angle (equivalence theorem)
(2) the definition of the angle of circle (the theorem of the angle of circle, the relationship with the angle of center)
⑶ Definition of chord angle (chord angle theorem)
Second, the positional relationship between a straight line and a circle
1. Properties of Tangents (Key Points)
2. Tangent (key point) judgment theorem
3. Tangent length theorem
Third, the position relationship between circles.
1. Five positional relationships and their judgments and properties: (emphasis: tangency)
2. The property theorem of the tangent (intersection line) connecting two circles.
3. Common tangent of two circles: (1) Definition (2) Property
Four, proportional line segment related to the circle
1. Intersecting chord theorem
2. Cutting line theorem
Verb (abbreviation for verb) and regular polygon
The inscribed and circumscribed polygons of 1. circle (triangle, quadrilateral)
2. The circumscribed circle, inscribed circle and properties of triangle.
3. The properties of circumscribed quadrangles and inscribed quadrangles of a circle
4. Regular polygon and its calculation
Central corner: review outline of junior high school mathematics
Half of the Inner Corner: Review Outline of Junior High School Mathematics (right)
(Solve Rt△OAM to find out the relevant elements, junior high school mathematics review outline, junior high school mathematics review outline, etc. )
Six, a set of calculation formulas
1. circumference formula
2. Formula of circular area
3. Sector area formula
4. Arc length formula
5. Calculation method of arch area
6. The side development diagram of cylinder and cone and related calculation.
Knowledge points of mathematics unit in the first volume of the ninth grade
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.
Mathematics learning methods in grade three
conceptual class
We should attach importance to the teaching process, actively experience the process of knowledge generation and development, find out the ins and outs of knowledge, understand the process of knowledge generation, understand the derivation process of formulas, theorems and laws, change the method of rote learning, and experience the fun of learning knowledge from the process of knowledge formation and development; In the process of solving the problem, I felt the joy of success.
Exercise class
It is necessary to master the trick of "I would rather watch it once, not do it once, not tell it once, not argue it once". In addition to listening to the teacher and watching the teacher do it, you should also do more exercises yourself, and you should actively and boldly tell everyone about your experience. When encountering problems, you should argue with your classmates and teachers, stick to the truth and correct your mistakes. Pay attention to the problem-solving thinking process displayed by the teacher in class, think more, explore more, try more, find creative proofs and solutions, and learn the problem-solving methods of "making a mountain out of a molehill", that is, take objective questions such as multiple-choice questions and fill-in-the-blank questions seriously, and never be careless, just like treating big questions, so as to write wonderfully; For a topic as big as a comprehensive question, we might as well decompose the "big" into "small" and take "retreat" as "advance", that is, decompose or retreat a relatively complex question into the simplest and most primitive one, think through these small questions and simple questions, find out the law, and then make a leap and further sublimation, thus forming a big question, that is, settle for second best. If we have this ability to decompose and synthesize, coupled with solid basic skills, what problems can't beat us?
recite
In the process of mathematics learning, we should have a clear review consciousness and gradually develop good review habits, so as to gradually learn to learn to learn. Mathematics review should be a reflective learning process. We should reflect on whether the knowledge and skills we have learned have reached the level required by the curriculum; It is necessary to reflect on what mathematical thinking methods are involved in learning, how these mathematical thinking methods are used, and what are the characteristics in the process of application; It is necessary to reflect on basic issues (including basic graphics, images, etc.). ), whether the typical problems have been really understood, and which problems can be attributed to these basic problems; We should reflect on our mistakes, find out the reasons and formulate corrective measures. In the new semester, we will prepare a "case card" for math learning, write down the mistakes we usually make, find out the "reasons" and prescribe a "prescription". We will often take it out and think about where the mistakes are, why they are wrong and how to correct them. Through your efforts, there will be no "cases" in your mathematics by the time of the senior high school entrance examination. And math review should be carried out in the process of applying math knowledge, so as to deepen understanding and develop ability. Therefore, in the new year, we should do a certain number of math exercises under the guidance of teachers, so as to draw inferences from others and use them skillfully to avoid the tactics of "practicing" rather than "repeating".
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