Two straight lines and a circle. Mutual arrangement of a straight line and a circle

A) a chord B) diameter

C) secant; D) tangent.

2. Through a point lying on a circle, you can draw ...... .. tangents

A) one; B) two

3. If the distance from the center of the circle to the straight line is less than the length of the radius of the circle, then the straight line ...

D) there is no correct answer.

4. If the distance from the center of the circle to the straight line is greater than the radius of the circle, then the straight line ...

A) touches the circle at one point; C) intersects the circle at two points;

C) does not intersect with a circle;

D) there is no correct answer.

5. Circles do not intersect and do not touch if ...

BUT)R 1 + R 2 = d ; AT)R 1 + R 2 < d ;

WITH)R 1 + R 2 > d ; D)d=0 .

6. Tangent and radius drawn at to the point of contact...

A) are parallel B) are perpendicular

C) match D) there is no correct answer.

7. 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?

8. What is the relative position of two circles if the distance between the centers is 4 and the radii are 11 and 7:

9. What can be said about the relative position of the line and the circle, if the diameter of the circle is 7.2 cm, and the distance from the center of the circle to the line is 0.4 dm:

10. Given a circle with center O and point A. Where is point A located if the radius of the circle is 7 cm, and the length of the segment OA is 70 mm?

A) inside a circle B) on a circle.

C) outside the circle; D) there is no correct answer.

Option 2

1. A straight line that has only one common point with a circle and is perpendicular to the radius is called ...

A) a chord B) diameter

C) secant; D) tangent.

2. From a point not lying on the circle, you can draw to the circle …….. tangents

A) one; B) two

C) none D) there is no correct answer.

3. If the distance from the center of the circle to the line is equal to the radius of the circle, then the line

A) touches the circle at one point; C) intersects the circle at two points;

C) does not intersect with a circle;

D) there is no correct answer.

4. Circles intersect at two points if ...

BUT)R 1 + R 2 = d ; AT)R 1 + R 2 < d ;

WITH)R 1 + R 2 > d ; D)d=0 .

5. Circles touch at one point if ...

BUT)R 1 + R 2 = d ; AT)R 1 + R 2 < d ;

WITH)R 1 + R 2 > d ; D)d=0 .

6. Circles are called concentric if ...

BUT)R 1 + R 2 = d ; AT)R 1 + R 2 < d ;

WITH)R 1 + R 2 > d ; D)d=0 .

7. 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?

A) 8 cm; C) 2 s m; C) 15 cm; D) 3 cm.

8. What is the relative position of two circles if the distance between the centers is 10 and the radii are 8 and 2:

A) external touch; B) internal touch;

C) intersect D) do not intersect.

9. What can be said about the relative position of the line and the circle, if the diameter of the circle is 7.2 cm, and the distance from the center of the circle to the line is 3.25 cm:

A) touch B) do not intersect.

C) intersect D) there is no correct answer.

10. 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 4 cm?

A) inside a circle

B) on a circle.

C) outside the circle;

D) there is no correct answer.

Rating: 10 b. - "5", 9 - 8 b. - "4", 7 - 6 b. - "3", 5 b. and below - "2"

Verification work No. 2

1) Fill in the table. Choose one of the options (6b):

a)mutual arrangement of two circles:

2. Find the length of the chord if the diameter is perpendicular to it, and one of the segments cut off by the diameter from it is 0.4 dm.

3) Solve one problem to choose from (2b):

1. Construct circles whose centers are less than the difference between their radii. Mark the distance between the centers of the circle. Make a conclusion.

2. Construct circles, the distance between the centers of which is equal to the difference between the radii of these circles. Mark the distance between the centers of the circle. Make a conclusion.

Rating: 10 - 9 b. - "5", 8 - 7 b. - "4", 6 - 5 b. - "3", 4 b. and below - "2"

RATING LIST

Recall an important definition - the definition of a circle]

Definition:

A circle centered at point O and radius R is the set of all points in the plane that are at a distance R from point O.

Let us pay attention to the fact that the set is called a circle. all points that satisfy the described condition. Consider an example:

Points A, B, C, D of the square are equidistant from point E, but they are not a circle (Fig. 1).

Rice. 1. Illustration for example

In this case, the figure is a circle, since it is all a set of points equidistant from the center.

If we connect any two points of the circle, we get a chord. The chord passing through the center is called the diameter.

MB - chord; AB - diameter; MnB - arc, it is contracted by the chord MB;

The corner is called central.

Point O is the center of the circle.

Rice. 2. Illustration for example

Thus, we remembered what a circle is and its main elements. Now let's move on to considering the relative position of the circle and the line.

Given a circle with center O and radius r. The line P, the distance from the center to the line, that is, the perpendicular OM, is equal to d.

We assume that the point O does not lie on the line P.

Given a circle and a straight line, we need to find the number of common points.

Case 1 - the distance from the center of the circle to the straight line is less than the radius of the circle:

In the first case, when the distance d is less than the radius of the circle r, the point M lies inside the circle. From this point we will set aside two segments - MA and MB, the length of which will be. We know the values ​​of r and d, d is less than r, which means that the expression exists and the points A and B exist. These two points lie on a straight line by construction. Let's check if they lie on a circle. Calculate the distance between OA and OB using the Pythagorean theorem:

Rice. 3. Case 1 illustration

The distance from the center to two points is equal to the radius of the circle, so we have proved that points A and B belong to the circle.

So, the points A and B belong to the line by construction, they belong to the circle by what has been proved - the circle and the line have two common points. Let us prove that there are no other points (Fig. 4).

Rice. 4. Illustration for the proof

To do this, take an arbitrary point C on a straight line and assume that it lies on a circle - the distance OS = r. In this case, the triangle is isosceles and its median ON, which does not coincide with the segment OM, is the height. We have obtained a contradiction: two perpendiculars are dropped from the point O to the line.

Thus, on the line P there are no other common points with the circle. We have proved that in the case when the distance d is less than the radius r of the circle, the line and the circle have only two common points.

Case two - the distance from the center of the circle to the straight line is equal to the radius of the circle (Fig. 5):

Rice. 5. Case 2 illustration

Recall that the distance from a point to a line is the length of the perpendicular, in this case OH is the perpendicular. Since, by condition, the length OH is equal to the radius of the circle, then the point H belongs to the circle, so the point H is common to the line and the circle.

Let us prove that there are no other common points. On the contrary: suppose that the point C on the line belongs to the circle. In this case, the distance OC is r, and then OC is OH. But in a right triangle, the hypotenuse OS is greater than the leg OH. We got a contradiction. Thus, the assumption is wrong and there is no point other than H that is common to the line and the circle. We have proved that in this case the common point is unique.

Case 3 - the distance from the center of the circle to the straight line is greater than the radius of the circle:

The distance from a point to a line is the length of the perpendicular. We draw a perpendicular from the point O to the straight line P, we get the point H, which does not lie on the circle, since OH is, by condition, greater than the radius of the circle. Let us prove that any other point of the line does not lie on the circle. This is clearly seen from right triangle, whose hypotenuse OM is greater than the leg OH, and therefore greater than the radius of the circle, so the point M does not belong to the circle, like any other point on the line. We have proved that in this case the circle and the line do not have common points (Fig. 6).

Rice. 6. Case 3 illustration

Consider theorem . Suppose that the line AB has two points in common with the circle (Fig. 7).

Rice. 7. Illustration for the theorem

We have a chord AB. Point H, according to the condition, is the middle of the chord AB and lies on the diameter CD.

It is required to prove that in this case the dimeter is perpendicular to the chord.

Proof:

Consider an isosceles triangle OAB, it is isosceles, since .

Point H, by condition, is the middle of the chord, which means the middle of the median AB of an isosceles triangle. We know that the median of an isosceles triangle is perpendicular to its base, which means it is the height: hence, thus, it is proved that the diameter passing through the middle of the chord is perpendicular to it.

fair and converse theorem : if the diameter is perpendicular to the chord, then it passes through its midpoint.

Given a circle with center O, its diameter CD and chord AB. It is known that the diameter is perpendicular to the chord, it is necessary to prove that it passes through its middle (Fig. 8).

Rice. 8. Illustration for the theorem

Proof:

Consider an isosceles triangle OAB, it is isosceles, since . OH, by condition, is the height of the triangle, since the diameter is perpendicular to the chord. The height in an isosceles triangle is also a median, so AH = HB, which means that the point H is the midpoint of the chord AB, which means that it is proved that the diameter perpendicular to the chord passes through its midpoint.

The direct and inverse theorem can be generalized as follows.

Theorem:

A diameter is perpendicular to a chord if and only if it passes through its midpoint.

So, we have considered all cases of mutual arrangement of a straight line and a circle. In the next lesson, we will consider the tangent to a circle.

Bibliography

1. Aleksandrov A.D. etc. Geometry Grade 8. - M.: Education, 2006.
2. Butuzov V.F., Kadomtsev S.B., Prasolov V.V. Geometry 8. - M.: Enlightenment, 2011.
3. Merzlyak A.G., Polonsky V.B., Yakir S.M. Geometry grade 8. - M.: VENTANA-GRAF, 2009.

1. edu.glavsprav.ru ().
2. Webmath.exponenta.ru().
3. Fmclass.ru ().

Homework

Task 1. Find the lengths of two segments of the chord, into which the diameter of the circle divides it, if the length of the chord is 16 cm, and the diameter is perpendicular to it.

Task 2. Indicate the number of common points of a straight line and a circle if:

a) the distance from the straight line to the center of the circle is 6 cm, and the radius of the circle is 6.05 cm;

b) the distance from the straight line to the center of the circle is 6.05 cm, and the radius of the circle is 6 cm;

c) the distance from the straight line to the center of the circle is 8 cm, and the radius of the circle is 16 cm.

Task 3. Find the length of the chord if the diameter is perpendicular to it, and one of the segments cut off by the diameter from it is 2 cm.

Didactic goal: formation of new knowledge.

Lesson goals.

Tutorials:

• to form mathematical concepts: a tangent to a circle, the relative position of a straight line and a circle, to achieve understanding and reproduction by students of these concepts through the implementation of practical research work.

Health saving:

• creating a favorable psychological climate in the classroom;

Developing:

• to develop students' cognitive interest, the ability to explain, generalize the results, compare, contrast, draw conclusions.

Educational:

• education by means of mathematics of personality culture.

Forms of study:

• content - conversation, practical work;
• on the organization of activities - individual, frontal.

Lesson Plan

Equipment:

• computer, screen, projector;
• Handout.

Educational Resources:

1. Mathematics. Textbook for grade 6 educational institutions; / G.V. Dorofeev, M., Enlightenment, 2009

2. Markova V.I. Features of teaching geometry in the context of the implementation of the state educational standard: guidelines, Kirov, 2010

3. Atanasyan L.S. Textbook "Geometry 7-9".

During the classes

1. Organizational moment. Knowledge update.

The teacher tells the topic of the lesson. Finds out what associations arise with the word “circle”.

What is the diameter of the circle if the radius is 2.4 cm?

What is the radius if the diameter is 6.8 cm?

2. Goal setting.

Students set their goals for the lesson, the teacher summarizes them and sets the goals of the lesson.

A program of activities in the lesson is drawn up.

3. Acquaintance with new material.

1) Working with models: “Show on models how a straight line and a circle can be located on a plane.”

How many points do they have in common?

2) Implementation of practical research work.

Target. Set the property of the relative position of the line and the circle.

Equipment: a circle drawn on a piece of paper and a stick as a straight line, a ruler.

1. In the figure (on a sheet of paper), set the relative position of the circle and the straight line.
2. Measure the radius of the circle R and the distance from the center of the circle to the straight line d.
3. Record the results of the study in a table.

4. Make a conclusion about the relative position of the straight line and the circle, depending on the ratio of R and d.

Conclusion: If the distance from the center of the circle to the line is equal to the radius, then the line touches the circle and has one common point with the circle. If the distance from the center of the circle to the line is greater than the radius, then the circle and the line have no common points. If the distance from the center of the circle to the line is less than the radius, the line intersects the circle and has two common points with it.

5. Primary comprehension, consolidation through problem solving.

1) Textbook assignments: No. 498, No. 499.

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=7.2dm, d=3.7dm
• 4. R=8 cm, d=1.2dm
• 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.

6. Reflection

What did you learn in the lesson?

What rule has been established?

Complete the following tasks on the cards:

Draw a line through every two points. How many common points does each of the lines have with the circle.

The line ______ and the circle have no common points.

The line ______ and the circle have only one ___________ point.

The lines ______, _______, ________, _______ and the circle have two common points.

7. Summing up. Setting homework:

1) analyze the cluster compiled at the beginning of the lesson, refine it taking into account the knowledge gained;

2) textbook: No. 500;

3) fill in the table (on cards).

Circle - geometric figure, consisting of all points of the plane located at a given distance from a given point.

This point (O) is called circle center.
Circle radius is a line segment that connects the center to a point on the circle. All radii have the same length (by definition).
Chord A line segment that connects two points on a circle. The chord passing through the center of the circle is called diameter. The center of a circle is the midpoint of any diameter.
Any two points on the circle divide it into two parts. Each of these parts is called circular arc. The arc is called semicircle if the segment connecting its ends is a diameter.
The length of a unit semicircle is denoted by π .
The sum of the degree measures of two circular arcs with common ends is 360º.
The part of the plane bounded by a circle is called around.
circular sector- a part of a circle bounded by an arc and two radii connecting the ends of the arc with the center of the circle. The arc that bounds the sector is called sector arc.
Two circles that have a common center are called concentric.
Two circles that intersect at right angles are called orthogonal.

Mutual arrangement of a straight line and a circle

1. If the distance from the center of the circle to the straight line is less than the radius of the circle ( d), then the line and the circle have two common points. In this case, the line is called secant in relation to the circle.
2. If the distance from the center of the circle to the line is equal to the radius of the circle, then the line and the circle have only one common point. Such a line is called tangent to circle, and their common point is called point of contact between a line and a circle.
3. If the distance from the center of the circle to the line is greater than the radius of the circle, then the line and the circle do not have common points
.

Central and inscribed angles

Central corner is the angle with the vertex at the center of the circle.
Inscribed angle An angle whose vertex lies on the circle and whose sides intersect the circle.

Inscribed angle theorem

An inscribed angle is measured by half the arc it intercepts.

• Consequence 1.
  Inscribed angles subtending the same arc are equal.


• Consequence 2.
  An inscribed angle that intersects a semicircle is a right angle.

Theorem on the product of segments of intersecting chords.

If two chords of a circle intersect, then the product of the segments of one chord is equal to the product of the segments of the other chord.

Basic Formulas

• Circumference:

• Arc length:

• Diameter:

• Arc length:

• Area of ​​a circle:

• Circular sector area:

Circle equation

• In a rectangular coordinate system, the equation for a circle of radius r centered on a point C(x o; y o) has the form:

• The equation for a circle of radius r centered at the origin is: