Title: Some Properties of Circle

Teaching purpose: to understand the definition of circle, master the positional relationship between point and circle, and cultivate students' ability to analyze and solve problems by combining numbers and shapes.

Emphasis and difficulty in teaching: understanding the definition of circle

Teaching emphasis: understanding two points: ① All points on the circle meet the requirement that the distance from the fixed point (center of the circle) is equal to the fixed length (radius);

② The point whose distance from the fixed point (center) is equal to the fixed length (radius) is on the circle with the fixed point as the center and the fixed length as the radius.

Teaching process:

First, review the old knowledge:

1, the definition of angular bisector and median vertical line (explained from the point of view of set)

2. Draw circles with radii of 1cm, 2cm and 3.5cm on a piece of transparent paper, and two students at the same table compare the sizes of the circles drawn (corresponding overlap). And answer: Why can these circles overlap respectively? And understand how this circle is formed.

Second, teach new lessons:

1, let the students take out the prepared wooden strips and demonstrate the formation of the circle according to the textbook, and then demonstrate the formation of the circle again with compasses.

Analysis and induction circle definition:

In a plane, a line segment rotates once around its fixed endpoint, and the figure formed by the rotation of the other endpoint is called a circle, where the fixed endpoint is called the center of the circle and the line segment is called the radius.

Note: "In-plane" can not be ignored. The circle centered on point O is marked as "⊙O" and pronounced as "circle O"

2, further observation, experience the formation of the circle, combined with the definition of garden, analysis:

1 The distance from each point on the circle to a fixed point (center) is equal to a fixed length (radius).

2. All points whose distance from a fixed point is equal to a fixed length are centered on the fixed point.

On a circle of fixed length and radius. This leads to the definition of a circle:

A circle is a set of points whose distance from a fixed point is equal to a fixed length.

For example, a point set with a distance of 1.5cm on the plane is a circle with o as the center and a radius of 1.5cm.

3. In the process of drawing a circle, I also realized that the distance from each point to the center of the circle is less than the radius, and all the points whose distance from the center of the circle is less than the radius are in the circle.

The interior of a circle is a collection of points whose distance from the center of the circle is less than the radius. Similarly, the outside of a circle is a collection of points whose distance from the center of the circle is greater than the radius.

4, preliminary grasp the relationship between circle and set:

(1) Find the set of points by knowing the graph.

For example, as shown in the figure, a circle with O as the center and a radius of 2cm,

It is a collection of points with O point as the center and 2cm length as the radius;

The inside of a circle with O as the center and a radius of 2 cm is from

The set of all points whose distance from the center O is less than 2 cm;

The outside of a circle with O as the center and a radius of 2 cm is from to.

A set of points whose center O is more than 2 cm away.

2 groups of known points, looking for patterns.

For example, a point set with a distance of 3cm from the known point O is a circle with a radius of 3cm centered on the point O.

5, the position relationship between point and circle:

Points are on the circle, points are inside the circle, and points are outside the circle.

The positional relationship between a point and a circle and the quantitative relationship between a point and the center of a circle are as follows:

Let the center of the circle be O, the radius be R, and the distance from point P to point O be D, then there is

Point p is in the circle op > R.

Point p is on the circle op = R.

Point p is outside the circle op < r

Example 1: Verification: The four vertices of a rectangle are on the same circle with the diagonal intersection as the center.

It is proved that multi-point * * * circle can be known from the definition of circle, that is, it is necessary to prove that points A, B, C and D are equidistant from point O.

Third, consolidate the exercises:

1. It is known that in △ABC, ∠ c = 90, AC = 2CM, BC = 4CM, where cm is the center line, c is the center of the circle, cm is the length of the circle, and four points A, B, C and M are outside the circle.

There is something in the circle, there is something in the circle.

2. Textbook P

3. What are the figures of the vertex circle we have learned?

33.5 degrees

Fourth, after-class summary:

Two definitions of 1 and circle

2. The internal and external definition of a circle.

3. The positional relationship between a point and a circle

4. The positional relationship between a point and a circle and the quantitative relationship between a point and the center of a circle.

5, proof of multipoint * * * circle

Verb (short for verb) Task:

Textbooks P 1, (1, 2), 2, 3, 4.

Instruction design description

This lesson mainly discusses the concept of circle, deeply understands the formation of circle, and enables students to get rid of the superficial understanding of circle in primary school and master a more complete definition of circle in junior high school knowledge.

The focus of teaching is to make students understand the two points of the circle. Simply put, the point whose distance from the center of the circle is equal to the radius is on the circle, and the distance from the point on the circle to the center of the circle is equal to the radius. When introducing the concept of circle, we first use set language to explain the circle, such as the set definition of angular bisector and vertical line we learned before, and then use graphic drawing to understand the definition of circle. The purpose of this design is to cultivate students' idea of combining numbers with shapes.

In teaching, let students demonstrate the formation of a circle by themselves and understand two necessary conditions for drawing a circle: fixed point and fixed length; Let the students understand the concept of circle by themselves, and at the same time, they will also understand the meaning of circle inside and outside. They can explain the inside and outside equally with the definition of set, and can lead to the positional relationship between points and circle. Then, students will know the definition of circle more clearly and understand the circle more completely in a series of processes. This example aims to enable students to master the definition of multipoint * * * circles and explore all other graphs of vertex * * * circles.

In a word, this course is mainly guided and taught by teachers. Through students' self-demonstration, we can understand the formation of the circle, cultivate students' summing-up ability, improve their ability to explore and solve problems, design a general framework, explore and study first, and then understand the application.