Eighth grade mathematics | the next volume of Pythagorean theorem pre-study, knowledge point induction + five major problem type analysis, collection
The Pythagorean theorem for eighth-grade math is one of the few pieces of knowledge in junior high school mathematics that involves a combination of geometry and algebra. It uses the relationship between the three sides of the triangle and the problem of angle to combine, and the examination point of the Pythagorean theorem in the examination is mainly reflected in the practical application of the Pythagorean theorem.
Since the Pythagorean theorem involves the relationship between its three sides, triangles can be used as the basis for solving problems in the right triangles that involve the length of the edges.
Students who are preparing for the prephotre should not only understand the content involved in the Pythagorean theorem but also understand the difficulties in their learning and the practical application of the Pythagorean theorem, especially among the five major problem types involved in the Pythagorean theorem, the last two are the contents that everyone pays attention to, which is very typical in the practical application of the Pythagorean theorem.
The following teacher Tang will lead you to the basic knowledge of the Pythagorean theorem and the analysis of the five typical problem types for comprehensive explanation and understanding, through the study of these basic knowledge, how to answer and apply in the typical example problems, are the specific demonstration of the skills that everyone should master when learning the Pythagorean theorem.
First, the knowledge points of the Pythagorean theorem are summarized
The main knowledge point of the Pythagorean theorem is the specific content of the Pythagorean theorem, as well as the understanding of the inverse theorem of the Pythagorean theorem, we can prove it through geometric shapes, that is, the Pythagorean theorem, which involves several methods, and these methods must be sorted out one by one, and understand where the process of proof comes from.
Although in the examination, the proof of the Pythagorean theorem does not need to be mastered by everyone, and the examination will not be examined, but the proof method involved can be proved from the direction of algebra, and the left hand can also use the combination of numbers and shapes to prove, this idea is very good for broadening everyone's mathematical learning ideas, so for the process of proof, we must understand the idea of its proof from beginning to end.
As a special reminder, in the Pythagorean theorem, we all remember that the relationship between the three sides is that the c side equals the a side plus the b side. However, in the specific application process, the meaning represented by abc must be understood, can not be replaced by fixed letters, c represents the hypotenuse of the right triangle, a, b represents the two right angle sides of the right triangle, so when the letter changes, we must be vigilant to prevent low-level errors.
Second, the five major problem types of the Pythagorean theorem are analyzed
Type 1: Direct application of the Pythagorean theorem
The direct application of the Pythagorean theorem is actually a deep understanding of the concept of the Pythagorean theorem, learning to use the relationship between the three sides to find the length of any side, which is the basis for other operations and its application using the Pythagorean theorem.
Type 2: The structural application of the Pythagorean theorem
Based on the concept of the Pythagorean theorem and the simple calculation of the trilateral relationship, then in the actual operation process, when the Pythagorean law is not met, but the length of both sides is required, we can construct a right triangle with the help of auxiliary lines, and then use the relationship between the three sides of the right triangle to find the length of any side.
When constructing a right triangle, we generally follow two principles:
First, when there are right angles in the drawing, you should keep the right angles and choose to connect the other two endpoints to construct a right triangle.
Secondly, when there is no right angle stone in the triangle we need to do a vertical line on one side, then we try to retain the length of the line segment that exists in the problem condition, choose to do the vertical line of the other sides, so that the right triangle established can use the length of this side to solve the length of the other sides.
Type 3: Practical application of the Pythagorean theorem
The practical application of the Pythagorean theorem is actually to use the relationship between the three sides in the Pythagorean theorem to solve the distance problem in the actual problem, that is, to establish a model of a right triangle based on the distance problem, so when solving this practical problem, we must first establish a model of the right triangle in order to use its three-sided relationship to solve.
In the practical application of Pythagorean theorem, it is recognized that the most difficult learning is the problem of the shortest path, which is the problem, which requires students to have a rich spatial imagination and a full understanding of the problem type.
Type 4: Geometric conjecture of the Pythagorean theorem
The geometric conjecture of the Pythagorean theorem mainly uses the geometric meaning of the right triangle to express the relationship between the three sides of the triangle, so as to solve some of the more difficult knowledge points in the practical problem. This process is not difficult, but it is of great significance for the idea of establishing the combination of geometry and tree shape, and students must sort out this process clearly.
Type 5: Inverse proposition and Pythagorean theorem inverse theorem
The alternative classification of the classic question types of the Pythagorean theorem and the analysis of the examination method focus on the summary and application of the two major ideological methods.
In the actual test and test questions, what are the test points that it focuses on? We through the classic question type to give you a specific presentation, good methods or the actual scope of application must withstand the actual test in the question type, these problem solving skills and the key points of the problem solving how to grasp, we give you a detailed analysis from the problem type.
Behind these classic example questions, we have set up interview question types for everyone, that is, the question types can better test the changes in the actual application of the test center and the ways and methods of coping, only flexibly respond to each test center and master the rules of its questions and the test center test method. Then in the actual exam or test, the students' problem-solving and coping skills can be gradually improved.
In short, for the study of the Pythagorean theorem, we still need to pay attention to the foundation and the correct understanding of the Pythagorean theorem and the inverse theorem. In this part of the process of continuous muscle special proof through the proof of the Pythagorean theorem, different methods can reflect different mathematical thinking, of which the idea of tree-shape combination is the method that everyone should focus on when learning, which is of very far-reaching significance for the subsequent learning of algebra and geometry.