Induction of Mathematics Knowledge Points in Grade Two of Junior High School
Chapter 11 Triangle
I. Knowledge framework:
Second, the concept of knowledge:
1, triangle: A figure composed of three line segments that are not on the same line end to end is called a triangle.
2. Trilateral relationship: the sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is less than the third side.
3. Height: Draw a vertical line from the vertex of the triangle to the line where the opposite side is located, and the line segment between the vertex and the vertical foot is called the height of the triangle.
4. midline: in a triangle, the line segment connecting the vertex and its relative midpoint is called the midline of the triangle.
5. Angular bisector: The bisector of the inner angle of a triangle intersects the opposite side of this angle, and the line segment between the vertex and the intersection of this angle is called the angular bisector of the triangle.
6. Stability of triangle: The shape of triangle is fixed, and this property of triangle is called stability of triangle.
7. Polygon: On the plane, a figure composed of some line segments connected end to end is called polygon.
8. Interior Angle of Polygon: The angle formed by two adjacent sides of a polygon is called its interior angle.
9. Exterior angle of polygon: The angle formed by the extension line of one side of polygon and its adjacent side is called the exterior angle of polygon.
10, diagonal of polygon: the line segment connecting two non-adjacent vertices of polygon is called diagonal of polygon.
1 1, regular polygon: a polygon with equal angles and sides in a plane is called a regular polygon.
12, plane mosaic: covering a part of the plane with some non-overlapping polygons is called covering the plane with polygons.
13, formula and properties:
⑴ Sum of triangle internal angles: The sum of triangle internal angles is 180.
(2) the nature of the triangle exterior angle:
Property 1: One outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it.
Property 2: The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
⑶ Formula for the sum of polygon internal angles: the sum of polygon internal angles is equal to 180.
(4) Sum of polygon external angles: the sum of polygon external angles is 360.
5] Diagonal number of polygons: ① Diagonal lines can be drawn from a vertex of a polygon.
Lines divide polygons into triangles. ② The polygon * * * has a diagonal line.
Chapter 12 congruent triangles
I. Knowledge framework:
Second, the concept of knowledge:
1, basic definition:
(1) Conformity: two figures can completely overlap and are called congruences.
⑵ congruent triangles: Two triangles that can completely coincide are called congruent triangles.
⑶ Corresponding vertices: The mutually coincident vertices in congruent triangles are called corresponding vertices.
⑷ Corresponding edges: The overlapping edges in congruent triangles are called corresponding edges.
5] Correspondence angle: The mutually coincident angles in congruent triangles are called correspondence angles.
2, the basic nature:
⑴ Stability of the triangle: When the lengths of the three sides of the triangle are determined, the shape and size of the triangle are determined. This property is called the stability of triangles.
⑵ The nature of congruent triangles: the corresponding edges of congruent triangles are equal, and the corresponding angles are equal.
3, congruent triangles's judgment theorem:
⑴ Edge edge (): three sides correspond to the congruence of two triangles.
⑵ Corner (): Two triangles with equal included angles are congruent.
(3) corner (): the intersection of two corners and their edges corresponding to two triangles.
(4) Corner edge (): the opposite side of two angles and one of them corresponds to the congruence of two triangles.
5] hypotenuse and right-angled edge (): hypotenuse and a right-angled edge correspond to the congruence of two right-angled triangles.
4. Angle bisector:
(1) Painting:
⑵ Property theorem: A point on the bisector of an angle is equal to the distance on both sides of the angle.
(3) The inverse theorem of the property theorem: the point with equal distance from the inside of the angle to both sides of the angle is on the bisector of the angle.
5, the basic method of proof:
(1) Clarify the known and verified in the proposition. (including implicit conditions, such as male * * * edge, male * * * angle, opposite top.
Angle, angle bisector, midline, height, isosceles triangle, etc.
⑵ Draw a picture according to the meaning of the question, and use digital symbols to indicate the known and verified.
(3) After analysis, find out the method of proof from the known and write the proof process.
Chapter 13 Axisymmetric
I. Knowledge framework:
Second, the concept of knowledge:
1, basic concept:
(1) Axisymmetric graph: If a graph is folded along a straight line, the parts on both sides of the straight line can overlap each other, and this graph is called an axisymmetric graph.
⑵ Two figures form an axis symmetry: fold one figure along a certain straight line if it can be connected with another figure.
If two figures overlap, then they are said to be symmetrical about this line.
(3) The midline of the line segment: the straight line passing through the midpoint of the line segment and perpendicular to the line segment is called the midline of the line segment.
⑷ isosceles triangle: A triangle with two equal sides is called an isosceles triangle. Two equal sides are called waist, the other side is called bottom, the included angle between the two sides is called top angle, and the included angle between bottom and waist is called bottom angle.
5. equilateral triangle: A triangle with three equilateral sides is called an equilateral triangle.
2, the basic nature:
The essence of (1) symmetry;
(1) No matter whether an axisymmetric figure or two figures are symmetrical about a straight line, the axis of symmetry is the perpendicular to the line segment connected by any pair of corresponding points.
② Symmetric figures are congruent.
(2) The nature of the vertical line in the line segment:
(1) The point on the vertical line of the line segment is equal to the distance between the two endpoints of the line segment.
(2) The point with equal distance from the two endpoints of a line segment is on the middle vertical line of this line segment.
(3) Coordinate properties of axisymmetrical points.
(4) the nature of isosceles triangle:
(1) isosceles triangles have equal waists.
② The two base angles of an isosceles triangle are equal (equilateral and equiangular).
③ The bisector of the top corner of the isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide.
④ The isosceles triangle is an axisymmetric figure, and the symmetry axis is the combination of three lines (1).
5] the properties of equilateral triangle:
① All three sides of an equilateral triangle are equal.
② All three internal angles of an equilateral triangle are equal, equal to 60.
③ There are three lines on each side of an equilateral triangle.
④ The equilateral triangle is an axisymmetric figure, and the symmetry axis is the combination of three lines (three lines).
3. Basic judgment:
Determination of (1) isosceles triangle;
A triangle with equal sides is an isosceles triangle.
(2) If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal (equiangular pair).
Equilateral).
(2) Determination of equilateral triangle:
A triangle with three equilateral sides is an equilateral triangle.
(2) A triangle with three equal angles is an equilateral triangle.
③ An isosceles triangle with an angle of 60 is an equilateral triangle.
4. Basic methods:
(1) perpendicular to a known straight line:
(2) The midline of the known line segment:
(3) Symmetry axis: connect two corresponding points and make the middle perpendicular of the connecting line segment.
(4) Make a symmetrical figure of a known figure about a straight line:
5] Make a point on a straight line to make the sum of the distances from this point to two known points on the same side of the straight line the shortest.
Induction of Mathematics Knowledge Points in Grade Two of Junior High School
1. Symmetry axis: 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 is called the axis of symmetry.
2. Nature:
(1) The symmetry axis of an axisymmetric graph is the median vertical line of any pair of line segments connected by corresponding points.
(2) The distance between the point on the bisector of the angle and both sides of the angle is equal.
(3) The distance between any point on the vertical line in the line segment and the two end points of the line segment is equal.
(4) The point with equal distance from the two endpoints of a line segment is on the middle vertical line of this line segment.
(5) The corresponding line segment and the corresponding angle on the axisymmetric figure are equal.
3. The nature of isosceles triangle: the two base angles of isosceles triangle are equal (equilateral and equilateral).
4. The bisector of the top angle of the 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.
5. Determination of isosceles triangle: equilateral and equilateral.
6. Characteristics of equilateral triangle angles: three internal angles are equal, equal to 60.
7. Determination of equilateral triangle: A triangle with three equal angles is an isosceles triangle.
An isosceles triangle with an angle of 60 is an equilateral triangle.
A triangle with two angles of 60 is an equilateral triangle.
8. In a right triangle, the right side facing an angle of 30 is equal to half of the hypotenuse.
9. The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse.
Induction of Mathematics Knowledge Points in Grade Two (3)
Collection, collation and description of data
I. Knowledge framework
Two. The concept of knowledge
1. Comprehensive survey: The survey method for all the subjects is called comprehensive survey.
2. Sampling survey: The survey method of investigating some data and estimating the whole according to some data is called sampling survey.
3. Population: All the investigated objects are called population.
4. Individuals: Each survey object that constitutes the population is called an individual.
5. Sample: All the extracted individuals constitute a sample.
6. Sample size: The number of individuals in a sample is called sample size.
7. Frequency: Generally speaking, the number of times the data falls into different groups is called the frequency of that group.
8. Frequency: The ratio of frequency to total data is frequency.
9. Number of groups and distance between groups: When counting data, the data is divided into several groups according to a certain range, and the number of groups is called the number of groups, and the difference between the two ends of each group is called the distance between groups.
Summarize the knowledge points of junior two mathematics.
Root of number
Definition of 1. square root: If x2=a, X is called the square root of A (that is, the square root of A is X); Note: (1)a is the square number of X, (2) It is known that X is a power, A is a root, and the power sum root is a reciprocal operation.
2. The nature of the square root:
The square root of (1) positive number is a pair of opposites;
(2) The square root of 0 is still 0;
(3) Negative numbers have no square root.
3. Representation of square root: The square root of A is expressed as sum. Note: it can be regarded as a number, and it can also be regarded as an operation of opening two times.
4. arithmetic square root: the positive square root of a positive number is called the arithmetic square root of a, which is expressed as. Note: The arithmetic square root of 0 is still 0.
5. Three important non-negative numbers: a2≥0, |a|≥0 and ≥0. Note: The sum of non-negative numbers is 0, which means all of them are 0.
6. Two important formulas:
( 1) ; (a≥0)
(2) .
7. Definition of cube root: If x3=a, then X is called the cube root of A (that is, the cube root of A is X). Note: (1)a is called the cubic number of X; (2) The cubic root of A is expressed as: that is, the cubic power of A is opened.
8. The nature of the cube root:
(1) The cube root of a positive number is a positive number;
(2) The cube root of 0 is still 0;
(3) The cube root of a negative number is a negative number.
9. Characteristics of cube root:
10. Irrational number: Infinitely circulating decimals are called irrational numbers. An inexhaustible number with a prescription is an irrational number.
1 1. Real number: rational numbers and irrational numbers are collectively called real numbers.
Classification of real numbers: (1) (2).
13. The properties of the number axis: the points on the number axis correspond to real numbers one by one.
14. Approximation of irrational numbers: if the real number calculation result contains irrational numbers and the topic has no approximation requirements, the result should be expressed by irrational numbers; If the topic requires approximation, the result should be expressed as an approximation of irrational numbers. Note: In the approximate calculation of (1), one more bit should be reserved for the intermediate process; (2) Need to remember:
triangle
A-level concept of geometry: (requires deep understanding and skillful use, mainly used for geometric proof)
1. Definition of triangle angle bisector:
The bisector of a triangle intersects the opposite side of the corner, and the line segment between the intersection of the vertex and the corner is called the bisector of the triangle. Example of geometric expression (as shown in the figure):
(1)∵ Advertising sharing ∠BAC
∴∠BAD=∠CAD
(2)∫∠BAD =∠CAD
∴AD is the angular bisector.
2. The definition of triangle midline:
In a triangle, the line segment connecting the vertex and the midpoint of its opposite side is called the center line of the triangle (as shown in the figure).
Example of geometric expression:
(1) ∵AD is the center line of the triangle.
∴ BD = CD
(2)∫BD = CD
∴AD is the center line of a triangle.
3. The definition of triangle height line:
Draw a vertical line from the vertex of a triangle to its opposite side. The line segment between the vertex and the vertical foot is called the high line of the triangle.
(pictured)
Example of geometric expression:
(1) ∵AD is the height of ABC.
∴∠ADB=90
(2)∫∠ADB = 90
∴AD is the height of Δ δABC.
4. Trilateral relation theorem of triangle. ※:
The sum of the two sides of the triangle is greater than the third side, and the difference between the two sides of the triangle is smaller than the third side (as shown in the figure).
Example of geometric expression:
( 1)∵a b+ BC & gt； Alternating current
∴……………
(2)AB-BC
∴……………
5. Definition of isosceles triangle:
A triangle with two equal sides is called an isosceles triangle (pictured)
Example of geometric expression:
(1) √ΔABC Δ ABC is an isosceles triangle.
∴ AB = AC
(2)∫AB = AC
∴δabc is an isosceles triangle
6. The definition of equilateral triangle:
A triangle with three equilateral sides is called an equilateral triangle (pictured).
Example of geometric expression:
(1) ∵ δ ABC is an equilateral triangle.
∴AB=BC=AC
(2)∫AB = BC = AC
∴δabc is an equilateral triangle
7. The theorem of triangle interior angle sum and its inference;
(1) The sum of the interior angles of the triangle is180; (pictured)
(2) Two acute angles of a right triangle are complementary; (pictured)
(3) An outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it; (pictured)
(4) The outer angle of a triangle is larger than any inner angle that is not adjacent to it. ※ 。
(1) (2) (3)(4) Examples of geometric expressions:
( 1)≈A+∠b+∠C = 180
∴…………………
(2)∫∠C = 90°
∴∠A+∠B=90
(3)≈ACD =∠A+∠B
∴…………………
(4)∫∠ACD & gt； ∠A
∴…………………
The fifth summary of mathematics knowledge points in the second day of junior high school
linear function
(1) proportional function: the general shape is y=kx(k is a constant, k? 0) is called the proportional function, where k is called the proportional coefficient;
(2) Image features of proportional function: some straight lines passing through the origin;
(3) Image attributes:
(1) when k >; 0, the image of function y=kx rises from left to right through the first and third quadrants, that is, y also increases with the increase of x; ② when k
(4) To find the analytic expression of the proportional function: only one non-origin point is known;
(5) Drawing the image of the proportional function: passing through the origin and point (1, k); (or another non-origin)
(6) linear function: the general shape is y=kx+b(k and b are constants, k? 0), called a linear function;
(7) Proportional function is a special linear function; (because when b=0, y=kx+b is y=kx).
(8) Linear function image features: partial straight lines;
(9) Nature:
(1) y=kx and y=kx+b have the same inclination angle, and y=kx+b can be regarded as y=kx translation |b| unit length; (When b>0, translate upward; When b<0, translate downward)
2 when k >; 0, the straight line y=kx+b rises from left to right, that is, y increases with the increase of x;
③ When k < 0, the straight line y=kx+b decreases from left to right, that is, Y decreases with the increase of X;
4 when b>0, the intersection of the straight line y=kx+b and the positive semi-axis of the y axis is (0, b);
⑤ When b
(10) Find the analytical formula of linear function: that is, find the values of k and b;
(1 1) Draw the image of the function once: two points are known;
On Equations (Groups) and Inequalities from the Perspective of Functions
(1) Solving a linear equation with one variable can be transformed into: when the value of a linear function is 0, find the value of the corresponding independent variable; From the image, this is equivalent to knowing the straight line y=kx+b and determining the value of the abscissa of its intersection with the x axis;
(2) Solving the linear inequality of one variable can be regarded as: when the linear function value is greater than (less than) 0, find the value range corresponding to the independent variable;
(3) Each bivariate linear equation corresponds to a univariate linear function, so it also corresponds to a straight line;
(4) Generally, each binary linear equation group corresponds to two linear functions, so it also corresponds to two straight lines. From the "number" point of view, solving equations is equivalent to considering the values of two functions when the independent variables are equal. What is the value of this function? From the perspective of "shape", solving equations is equivalent to determining the coordinates of the intersection of two straight lines;
Senior two mathematics knowledge points induction related articles;
1.
2. Summary of knowledge points in the first volume of Mathematics in the second day of junior high school
3. Summary of mathematics knowledge points in senior two.
4. Summary of mathematics knowledge points in the second day of junior high school.
5. Summarize the knowledge points of eighth grade mathematics.
6. Summarize all the knowledge points in the first volume of Grade Two Mathematics.
7. People's Education Edition, the second day of mathematics knowledge points induction.
8. Junior high school mathematics knowledge arrangement:
9. Sort out the knowledge points of the first volume of second-grade mathematics.