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
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;
Rotational transformation
1. Concept: In a plane, turning a figure around a fixed point by an angle in a certain direction is called rotation.
Description: (1) The rotation of the graph is determined by the rotation center and rotation angle; (2) In the process of rotation, the center of rotation remains unchanged. (3) In the process of rotation, the rotation direction is consistent. (4) When the rotation is stationary, the rotation angle of a point on the diagram is the same. (5) Rotation will not change the size and shape of the figure.
2. Property: (1) The distance from the corresponding point to the rotation center is equal;
(2) The included angle of the connecting line between the corresponding point and the rotation center is equal to the rotation angle;
(3) Graphic congruence before and after rotation.
3. Steps and methods of rotation drawing: (1) Determine the rotation center, rotation direction and rotation angle; (2) Find out the key points of the graph; (3) connecting the key points of the graph with the rotation center, and then rotating by a rotation angle according to the rotation direction to obtain the corresponding points of these key points; (4) Connect these corresponding points in turn according to the original image, and the resulting graph is the rotated graph.
Note: When drawing by rotation, the included angle between a pair of corresponding points and the rotation center is the rotation angle.
Mathematics learning methods in grade three
1, the idea of "equation"
Mathematics studies the spatial form and quantitative relationship of things. The most important quantitative relationship in junior high school is equality, followed by inequality. The most common equivalence relation is "equation". For example, uniform motion, distance, speed and time are equivalent, and a related equation can be established: speed and time = distance. In this equation, there are generally known quantities and unknown quantities. An equation containing unknown quantities like this is an "equation", and the process of finding the unknown quantities through the known quantities in the equation is to solve the equation. We were exposed to simple equations in primary school, but in the first year of junior high school, we systematically studied the solution of one-dimensional linear equations and summarized five steps of solving one-dimensional linear equations. If you learn and master these five steps, any one-dimensional linear equation can be solved smoothly. In the second and third day of junior high school, you will also learn to solve one-dimensional quadratic equations, binary quadratic equations and simple triangular equations. In high school, we will also learn exponential equation, logarithmic equation, linear equation, parametric equation, polar coordinate equation and so on. The solution ideas of these equations are almost the same, and they are all transformed into the form of linear equations or quadratic equations in one variable by certain methods, and then solved by the familiar five steps to solve linear equations in one variable or the root formula to solve quadratic equations in one variable. Energy conservation in physics, chemical equilibrium formula in chemistry, and a large number of practical applications in reality all need to establish equations and get results by solving them. Therefore, students must learn how to solve one-dimensional linear equations and two-dimensional linear equations, and then learn other forms of equations.
The so-called "equation" idea means that for mathematical problems, especially the complex relationship between unknown quantities and known quantities encountered in reality, we are good at constructing relevant equations from the viewpoint of "equation" and then solving them.
2. The idea of "combination of numbers and shapes"
In the world, "number" and "shape" are everywhere. Everything, except its qualitative aspect, has only two attributes: shape and size, which are left for mathematics to study. There are two branches of junior high school mathematics-algebra and geometry. Algebra studies "number" and geometry studies "shape". It is a trend to learn algebra by means of "shape" and geometry by means of "number". The more you learn, the more inseparable you are from "number" and "shape". In senior high school, a course called "Analytic Geometry" appeared, which used algebra to study geometric problems. In the third grade, after the establishment of the plane rectangular coordinate system, the learning of functions can not be separated from images. Often with the help of images, the problem can be clearly explained, and it is easier to find the key to the problem, thus solving the problem. In the future mathematics study, we should pay attention to the thinking training of "combination of numbers and shapes" As long as any problem is a little close to the "shape", it is necessary to draw a sketch to analyze it according to the meaning of the problem. This is not only intuitive, but also comprehensive, easy to find the breakthrough point, which is of great benefit to solving problems. Those who taste the sweetness will gradually develop the good habit of "combining numbers with shapes".
3. The concept of "correspondence"
The concept of "correspondence" has a long history. For example, we correspond a pencil, a book and a house to an abstract number "1", and two eyes, a pair of earrings and a pair of twins to an abstract number "2". With the deepening of learning, we also extend "correspondence" to a form, a relationship, and so on. For example, when calculating or simplifying, we will correspond the left side of the formula, A, Y and B, and then directly get the result of the original formula with the right side of the formula. This is to use the idea and method of "correspondence" to solve problems. The second and third grades will also see the one-to-one correspondence between points on the number axis and real numbers, the one-to-one correspondence between points on the rectangular coordinate plane and a pair of ordered real numbers, and the correspondence between functions and their images. The thought of "correspondence" will play an increasingly important role in future research.
Related articles on knowledge points in the second volume of junior high school mathematics;
★ Nine-grade mathematics knowledge points Volume II
★ Summarize the mathematics knowledge points in the second volume of the ninth grade.
★ The latest summary of mathematics knowledge points in Grade Three.
★ The arrangement of knowledge points in the second volume of ninth grade mathematics
★ People's Education Edition Grade Three Mathematics Knowledge Points
★ Summary of Mathematics Knowledge Points in Grade Three
★ Math review materials at the end of the ninth grade next semester
★ Summarize the mathematics knowledge points in the second volume of the third grade.
★ Review materials of mathematics knowledge in senior high school entrance examination of PEP.
★ Guidance and summary of mathematics learning methods in grade three.
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