Summary of knowledge points on the mathematics circle for the high school entrance examination, exploration of the direction of examination, analysis of common mistakes, essential for training and improvement

High School Entrance Exam Mathematics | Summary of circle knowledge points, investigation direction exploration, easy Analysis of mistakes, essential for training and improvement

The examination of knowledge about circles in mathematics in the high school entrance examination is the key examination content in the geometry part of junior high school mathematics. He brought the abstract thinking in junior high school to the extreme. Of course, in the examination process of the high school entrance examination, the circle knowledge examination was not conducted separately, but was combined with other knowledge such as triangles and parallelograms, so the connection between the knowledge points in the examination was It is more compact and its complexity is more concentrated.

If you want to master the skills and methods of the circle knowledge test points in junior high school, you must have a clear understanding of each test point involved in the circle. In order to help students master the knowledge of circles as soon as possible , the test center's interpretation and analysis of the content of each circle test point can quickly help everyone break through abstract concepts and thinking.

So what are the important test points in studying and reviewing? What do you need to pay attention to in terms of understanding? Through the following analysis, we will help you defeat them one by one.

First, about the concept and properties of circles.

The concept and properties of a circle are mainly based on the basic interpretation of the original related content. This part of the content will help everyone to have a comprehensive understanding of the circle, as well as some basic concepts involved. It plays a basic and protective role in understanding the comprehensive question types of circles in the future. Possible deviations in the understanding of each basic content or areas that everyone has not noticed are pointed out one by one to help everyone have a deeper understanding of the concepts and properties.

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In the process of understanding these basic concepts and properties, you can use graphics to make theorem inferences and understand concepts more easily. Improve your understanding, and use graphics to more vividly simplify the difficulties in understanding abstract concepts, thereby improving your understanding.

As for the above basic concepts and properties of the examination method and the problem-solving skills used in the process of examining the question types, we also need to pay more attention to it, and it is also what everyone is most concerned about. content.

First, when using the vertical diameter theorem, how to master its calculation skills based on graphics. When using the law of purity, we need to know the application of the vertical diameter theorem and the Pythagorean theorem. Among them, the right triangle composed of radius, chord center distance and half of the chord satisfies the Pythagorean theorem. As long as you know any two of them, you can find the third quantity.

Second, when using the relationship between the central angles of Hu Xian circles to solve problems, in the same circle or equal circles, two circular angles and their two arcs, two As long as one set of quantities in a string is equal, the corresponding other two sets of quantities are also equal respectively. It is widely used when used to find equal angles or equal segments.

When proving the equality of each other in a circle, it is often necessary to prove that the central angles and chords of the circles are equal. When proving that the central angles and the chords are equal, half of the corresponding radius chords are often used The lines connecting the points construct a right triangle, and the Pythagorean theorem is used to prove that the triangles are congruent. In the same source, the circumferential angles subtended by the copper lake are equal, which is a hidden condition in the problem. In the process of proof, it is a very useful condition. When doing the problem, you must dig out this condition to see if there is a matching circumference angle.

Fourth, when using the properties of an inscribed quadrilateral in a circle to find angles, we need to know the complementary diagonals of a human inscribed quadrilateral. You can find some degrees related to the angles of the circle.

Fifth, when solving the problem of minimizing the sum of the lengths of two line segments in a circle, it is often solved by transforming and using the perpendicular diameter theorem and the shortest line segment between two points. This is mainly to test everyone's ability to flexibly apply the vertical diameter theorem.

In the process of learning this part of the content, you should also pay attention to the knowledge that is easily confused. For example, the diameter is the chord, but the chord is not necessarily the diameter. Everyone just needs to remember that only the chord passing through the center of the circle is the diameter, and the diameter is the longest chord. In the same circle, there are two situations for the circumferential angle subtended by a chord, but when solving problems, the solution is often missed due to poor consideration. When using the condition that the corollary of the perpendicular diameter theorem is to bisect a string, it must be pointed out that the bisected string is a non-diameter string, otherwise the proposition may not be true.

The examination of this part of the content in the high school entrance examination mainly focuses on the perpendicular diameter theorem, the circumferential angle theorem and the relationship between the central angle and arc of a circle. It is often used in the calculation of the radius and chord length of a circle. The cheese of the circumference angle is often examined together with other knowledge points. The question types involve multiple choice questions, fill-in-the-blank questions and simple short answer questions or proof questions. The difficulty level is Easy or medium difficulty.

Secondly, the positional relationship between points and circles, straight lines and circles.

Maybe when it comes to positional relationships, many people will think of the most commonly used tangents, the inscribed and circumscribed circles of triangles. Among them, the proof and application of tangents are the first type of comprehensive questions that are often tested, and the properties of the inscribed and circumscribed circles of triangles are a major focus and difficulty for students. Of course, in addition to these high-frequency test points, there are other test points as well. How should we understand the key content in order to have a thorough grasp? Teacher Tang leads everyone to gain a deep understanding through the following content interpretation.

Let’s focus on the knowledge points that are easier to confuse in this part of the content, so as to solve everyone’s most difficult problems. First of all, the circumcenter and incenter of a triangle are a headache for everyone. The outer shape refers to the focus of the perpendicular bisectors of the three sides of the triangle. Its distance from the three vertices is equal, while the incenter is the intersection of the three interior angle bisectors of the triangle. His distance to three sides is equal. These two contents correspond to the center of the circumcircle and the center of the inscribed circle of the triangle respectively.

Secondly, when proving that a straight line is a tangent to a circle, no matter whether the straight line passes through the original point or not, a point is connected to a point on the straight line from the center of the circle, which leads to an error. This ignores the tangent of the circle. basic concepts. Finally, it is easier for everyone to confuse line segments and circles. There is a common point and errors caused by the line segment and the circle having a common point.

This part of the content is a must-test in the high school entrance examination. It mainly examines the positional relationship between straight lines and circles, the determination and properties of tangents, the content of the inscribed circles of triangles and the tangent length theorem. The question types mainly involve multiple choice questions, fill-in-the-blank questions, and proof questions. The difficulty of the questions is medium to low, and the investigation directions mainly include problems related to tangents, reasoning proofs related to tangents, and solutions to innovative problems.

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Finally, calculations related to circles

This part of the content mainly involves calculations related to regular polygons, including arc length, central angle, area, sector area, cone side area, etc. So in addition to this part of the calculation content, what are the methods and techniques for solving actual calculation problems?

First, when calculating the regular polygon, you can calculate the radius of the regular n-gon and connect the center of the circle, Use the midpoint line segment of the side to divide the regular n-sided polygon into 2n right triangles, and then use the Pythagorean theorem to complete the calculation.

Second, when calculating using arc length, according to the formula of arc length, we only need to know any two quantities of arc length, central angle and radius to calculate the third quantity. This method is also suitable for calculations using the ascending area formula. Therefore, when studying, students should pay attention to the memory of these two formulas and the transfer of methods in application.

Third, find the shortest distance between two points on the side of the cone. This is a more difficult type of shortest path. The cone needs to be expanded into a graph using the two points. Solve for the shortest time. This is very similar to the method of using the Pythagorean Theorem to find the shortest path after unfolding the cube and cuboid that we learned before. Therefore, students studying this type of study can summarize and review them together.

The investigation of this part of the content mainly uses the circumference of the circle, the area of ​​the arc length circle, and the area calculation formula of the sector to solve related geometric problems and calculate simple geometric combination figures. It is also one of the compulsory contents of the high school entrance examination, mainly involving fill-in-the-blank, multiple choice, short-answer questions, with medium difficulty. Calculating the side area and surface area of ​​a cone, this part of the knowledge is related to the reality of life. It is also one of the hot test points in the high school entrance examination in recent years. It mainly tests students' mastery of basic knowledge and the use of this knowledge to solve practical problems. Ability.

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Written at the end, in addition to the full understanding and improvement of the basic concepts and properties of the knowledge test points of the junior high school circle, the importance of each section in the high school entrance examination test points, Students also need to have a deep understanding, which will help everyone grasp the key points in the review section during review. At the same time, Teacher Tang has given students a comprehensive summary of the methods and techniques for solving each part, focusing on the areas where each part of the methods and techniques are easily confused or wrong in actual application. We must be vigilant and form a summary of the overall thinking and skills of circle knowledge as soon as possible, then these problems will be easily solved.