Class 7G, Z
Lesson topic: " Mutual arrangement two circles"
Purpose: to know possible cases of mutual arrangement of two circles; apply knowledge to solve problems.
Objectives: Educational: to help students create and consolidate a visual representation of the possible cases of the location of two circles, students will be able to:
Establish a connection between the mutual arrangement of circles, their radii and the distance between their centers;
Analyze the geometric design and mentally modify it,
Develop planimetric imagination.
Students will be able to apply theoretical knowledge to problem solving.
Type of lesson: a lesson of introducing and consolidating new knowledge of the material.
Equipment: presentation for the lesson; compasses, ruler, pencil and textbook for each student.
Tutorial: . "Geometry Grade 7", Almaty "Atamura" 2012
During the classes.
Organizing time. Checking homework.
3. Actualization of basic knowledge.
Repeat the definitions of a circle, circle, radius, diameter, chord, distance from a point to a line.
1) 1) What cases of the location of a straight line and a circle do you know?
2) What line is called a tangent?
3) What line is called a secant?
4) The theorem about the diameter perpendicular to the chord?
5) How does the tangent pass with respect to the radius of the circle?
6) Fill in the table (on cards).
- Students under the guidance of a teacher solve and analyze problems.
1) The line a is a tangent to a circle with center O. Point A is given on a line a. The angle between the tangent and segment OA is 300. Find the length of segment OA if the radius is 2.5 m.
2) Determine the relative position of the line and the circle if:
- 1. R=16cm, d=12cm 2. R=5cm, d=4.2cm 3. R=7.2cm, d=3.7cm 4. R=8cm, d=1.2cm 5. R=5cm, d=50mm
a) a line and a circle do not have common points;
b) the line is tangent to the circle;
c) a line intersects a circle.
- d is the distance from the center of the circle to the straight line, R is the radius of the circle.
3) What can be said about the relative position of the line and the circle, if the diameter of the circle is 10.3 cm, and the distance from the center of the circle to the line is 4.15 cm; 2 dm; 103 mm; 5.15 cm, 1 dm 3 cm.
4) Given a circle with center O and point A. Where is point A if the radius of the circle is 7 cm, and the length of the segment OA is: a) 4 cm; b) 10 cm; c) 70 mm.
4. Together with the students, find out the topic of the lesson, formulate the objectives of the lesson.
5. Introduction of new material.
Practical work in groups.
Construct 3 circles. For each circle, build one more circle, so that 1) 2 circles do not intersect, 2) 2 circles touch, 3) two circles intersect. Find the radius of each circle and the distance between the centers of the circles, compare the results. What can be the conclusion?
2) Summarize and write in a notebook, cases of mutual arrangement of two circles.
Mutual arrangement of two circles on a plane.
The circles do not have common points (they do not intersect). (R1 and R2 are circle radii)
If R1 + R2< d,
d - The distance between the centers of the circles.
c) The circles have two common points. (intersect).
If R1 + R2 > d,
Question. Can two circles have three points in common?
6. Consolidation of the studied material.
Find an error in the data or in the statement and correct it by giving reasons for your opinion:
a) Two circles are touching. Their radii are R = 8 cm and r = 2 cm, the distance between the centers is d = 6.
B) Two circles have at least two points in common.
C) R = 4, r = 3, d = 5. The circles have no common points.
D) R \u003d 8, r \u003d 6, d \u003d 4. The smaller circle is located inside the larger one.
E) Two circles cannot be located so that one is inside the other.
7. The results of the lesson. What did you learn in the lesson? What rule has been established?
How can two circles be located? When do the circles have one common point? What is the common point of two circles called? What touches do you know? When do the circles intersect? What circles are called concentric?
Lesson topic: " Mutual arrangement of two circles on a plane.
Target :
educational - mastering new knowledge about the relative position of two circles, preparing for control work
Educational - development of computational skills, development of logical and structural thinking; formation of skills for finding rational solutions and achieving final results; development cognitive activity and creative thinking .
Educational – the formation of students' responsibility, consistency; development of cognitive and aesthetic qualities; formation information culture students.
Correctional - develop spatial thinking, memory, hand motor skills.
Lesson type: learning new educational material, fastening.
Type of lesson: mixed lesson.
Teaching method: verbal, visual, practical.
Form of study: collective.
Means of education: board
DURING THE CLASSES:
1. Organizational stage
- greetings;
- checking the readiness for the lesson;
2.
Updating of basic knowledge.
What topics did we cover in the previous lessons?
General form circle equations?
Perform orally:
Blitz Poll
3. Introduction of new material.
What do you think and what figure we will consider today .... What if there are two?
How can they be located???
Children show with their hands (neighbors) how circles can be located (physical education)
Well, what do you think we should consider today?? Today we should consider the relative position of the two circles. And find out what is the distance between the centers depending on the location.
Lesson topic: « Mutual arrangement of two circles. Problem solving. »
1. Concentric circles
2. Non-intersecting circles
3.External touch
4. Intersecting circles
5. Internal touch
So let's conclude
4. Formation of skills and abilities
Find an error in the data or in the statement and correct it by giving reasons for your opinion:
a) Two circles are touching. Their radii are R = 8 cm and r = 2 cm, the distance between the centers is d = 6.
B) Two circles have at least two points in common.
C) R = 4, r = 3, d = 5. The circles have no common points.
D) R \u003d 8, r \u003d 6, d \u003d 4. The smaller circle is located inside the larger one.
E) Two circles cannot be located so that one is inside the other.
5. Consolidation of skills and abilities.
The circles touch externally. The radius of the smaller circle is 3 cm, the radius of the larger one is 5 cm. What is the distance between the centers?
Solution: 3+5=8(cm)
The circles touch internally. The radius of the smaller circle is 3 cm. The radius of the larger circle is 5 cm. What is the distance between the centers of the circles?
Solution: 5-3=2(cm)
The circles touch internally. The distance between the centers of the circles is 2.5 cm. What are the radii of the circles?
answer: (5.5 cm and 3 cm), (6.5 cm and 4 cm), etc.
CHECKING UNDERSTANDING
1) How can two circles be located?
2) When do the circles have one common point?
3) What is the common point of two circles called?
4) What touches do you know?
5) When do the circles intersect?
6) What circles are called concentric?
Additional tasks on the topic: Vectors. Coordinate Method '(if there is time)
1)E(4;12),F(-4;-10), G(-2;6), H(4;-2) Find:
a) vector coordinatesEF, GH
b) vector lengthFG
c) the coordinates of the point O - the middleEF
point coordinatesW– middleGH
d) the equation of a circle with a diameterFG
e) equation of a straight lineFH
6. Homework
& 96 #1000. Which of these equations are circle equations. Find Center and Radius
7. Summing up the lesson (3 min.)
(give a qualitative assessment of the work of the class and individual students).
8. Stage of reflection (2 minutes.)
(initiate students' reflection on their own emotional state, their activities, interaction with the teacher and classmates using drawings)Let a circle and a point not coinciding with its center C be given (Fig. 205). Three cases are possible: the point lies inside the circle (Fig. 205, a), on the circle (Fig. 205, b), outside the circle (Fig. 205, c). Let's draw a straight line, it will intersect the circle at points K and L (in case b), the point will coincide with one of which will be the closest to the point in comparison with all other points of the circle), and the other - the most remote.
So, for example, in Fig. 205, and the point K of the circle is the closest to . Indeed, for any other point on the circle, the broken line is longer than the segment CAG: but also, therefore, on the contrary, for the point L we find (again, the broken line is longer than the straight line segment). We leave the analysis of the remaining two cases to the reader. Note that the largest distance is equal to the smallest if or if .
Let's move on to the analysis possible cases arrangement of two circles (Fig. 206).
a) The centers of the circles coincide (Fig. 206, a). Such circles are called concentric. If the radii of these circles are not equal, then one of them lies inside the other. If the radii are equal, they coincide.
b) Now let the centers of the circles be different. We connect them with a straight line, it is called the line of centers of a given pair of circles. The mutual arrangement of the circles will depend only on the ratio between the value of the segment d connecting their centers and the values of the radii of the circles R, r. All possible essentially different cases are shown in fig. 206 (we consider).
1. The distance between the centers is less than the difference in radii:
(Fig. 206, b), a small circle lies inside a large one. This also includes the case a) coincidence of centers (d = 0).
2. The distance between the centers is equal to the difference of the radii:
(Fig. 206, s). The small circle lies inside the large one, but has one common point with it on the line of centers (they say that there is internal contact).
3. The distance between the centers is greater than the difference of the radii, but less than their sum:
(Fig. 206, d). Each of the circles lies partly inside, partly outside the other.
Circles have two intersection points K and L, located symmetrically about the line of centers. A segment is a common chord of two intersecting circles. It is perpendicular to the line of centers.
4. The distance between the centers is equal to the sum of the radii:
(Fig. 206, e). Each of the circles lies outside the other, but they have a common point on the line of centers (external tangency).
5. The distance between the centers is greater than the sum of the radii: (Fig. 206, e). Each of the circles lies entirely outside the other. Circles do not have common points.
The above classification follows completely from the above. above the question about the largest and smallest distance from a point to a circle. It is only necessary to consider two points on one of the circles: the closest and the farthest from the center of the second circle. For example, consider the case By condition . But the point of the small circle farthest from O is at a distance from the center O. Therefore, the entire small circle lies inside the large one. Other cases are considered in the same way.
In particular, if the radii of the circles are equal, then only the last three cases are possible: intersection, external touch, external location.