A sentence that explains the meaning of a particular expression or name is called definition. We have already encountered definitions, for example, the definition of an angle, adjacent angles, isosceles triangle etc. Let's give a definition of another geometric figure - a circle.

Definition

This point is called center of the circle, and the segment connecting the center with any point on the circle is radius of the circle(Fig. 77). From the definition of a circle it follows that all radii have the same length.

Rice. 77

A segment connecting two points on a circle is called its chord. A chord passing through the center of a circle is called its diameter.

In Figure 78, segments AB and EF are chords of the circle, segment CD is the diameter of the circle. Obviously, the diameter of a circle is twice its radius. The center of a circle is the midpoint of any diameter.

Rice. 78

Any two points on a circle divide it into two parts. Each of these parts is called an arc of a circle. In Figure 79, ALB and AMB are arcs bounded by points A and B.

Rice. 79

To depict a circle in a drawing, use compass(Fig. 80).

Rice. 80

To draw a circle on the ground, you can use a rope (Fig. 81).

Rice. 81

The part of the plane bounded by a circle is called a circle (Fig. 82).

Rice. 82

Constructions with compasses and ruler

We have already dealt with geometric constructions: we drew straight lines, plotted segments equal to data, drew angles, triangles and other figures. At the same time, we used a scale ruler, a compass, a protractor, and a drawing square.

It turns out that many constructions can be performed using only a compass and a ruler without scale divisions. Therefore, in geometry, those construction tasks are specially distinguished that can be solved using only these two tools.

What can you do with them? It is clear that the ruler allows you to draw an arbitrary straight line, as well as construct a straight line passing through two given points. Using a compass, you can draw a circle of arbitrary radius, as well as a circle with a center at a given point and a radius equal to a given segment. By performing these simple operations, we can solve many interesting construction problems:

construct an angle equal to the given one;
through a given point draw a line perpendicular to the given line;
divide this segment in half and other tasks.

Let's start with a simple task.

Task

On a given ray, from its beginning, plot a segment equal to the given one.

Solution

Let us depict the figures given in the problem statement: ray OS and segment AB (Fig. 83, a). Then, using a compass, we construct a circle of radius AB with center O (Fig. 83, b). This circle will intersect the ray OS at some point D. The segment OD is the required one.

Rice. 83

Examples of construction problems

Constructing an angle equal to a given one

Task

Subtract an angle from a given ray equal to a given one.

Solution

This angle with vertex A and ray OM are shown in Figure 84. It is required to construct an angle equal to angle A, so that one of its sides coincides with ray OM.

Rice. 84

Let us draw a circle of arbitrary radius with its center at vertex A of the given angle. This circle intersects the sides of the angle at points B and C (Fig. 85, a). Then we draw a circle of the same radius with the center at the origin of this ray OM. It intersects the beam at point D (Fig. 85, b). After this, we will construct a circle with center D, the radius of which is equal to BC. Circles with centers O and D intersect at two points. Let us denote one of these points by the letter E. Let us prove that the angle MOE is the desired one.

Rice. 85

Consider triangles ABC and ODE. Segments AB and AC are the radii of a circle with center A, and segments OD and OE are radii of a circle with center O (see Fig. 85, b). Since by construction these circles have equal radii, then AB = OD, AC = OE. Also by construction BC = DE.

Therefore, Δ ABC = Δ ODE on three sides. Therefore, ∠DOE = ∠BAC, i.e., the constructed angle MOE is equal to the given angle A.

The same construction can be done on the ground if you use a rope instead of a compass.

Constructing an angle bisector

Task

Construct the bisector of the given angle.

Solution

This angle BAC is shown in Figure 86. Let’s draw a circle of arbitrary radius with the center at vertex A. It will intersect the sides of the angle at points B and C.

Rice. 86

Then we draw two circles of the same radius BC with centers at points B and C (only parts of these circles are shown in the figure). They will intersect at two points, at least one of which lies inside the corner. Let us denote it by the letter E. Let us prove that the ray AE is the bisector of the given angle BAC.

Consider triangles ACE and ABE. They are equal on three sides. Indeed, AE is the general side; AC and AB are equal as the radii of the same circle; CE = BE by construction.

From the equality of triangles ACE and ABE it follows that ∠CAE = ∠BAE, i.e. ray AE is the bisector of the given angle BAC.

Comment

Is it possible to divide a given angle into two equal angles using a compass and ruler? It is clear that it is possible - to do this you need to draw the bisector of this angle.

This angle can also be divided into four equal angles. To do this, you need to divide it in half, and then divide each half in half again.

Is it possible to divide a given angle into three equal angles using a compass and ruler? This task, called angle trisection problems, has attracted the attention of mathematicians for many centuries. Only in the 19th century was it proven that such a construction is impossible for an arbitrary angle.

Construction of perpendicular lines

Task

Given a straight line and a point on it. Construct a line passing through a given point and perpendicular to a given line.

Solution

This straight line a and given point M belonging to this line are shown in Figure 87.

Rice. 87

On the rays of straight line a, emanating from point M, we plot equal segments MA and MB. Then we construct two circles with centers A and B of radius AB. They intersect at two points: P and Q.

Let's draw a straight line through point M and one of these points, for example, straight line MR (see Fig. 87), and prove that this straight line is the desired one, i.e., that it is perpendicular to the given straight line a.

In fact, since the median PM of the isosceles triangle RAB is also the height, then PM ⊥ a.

Constructing the midpoint of a segment

Task

Construct the midpoint of this segment.

Solution

Let AB be the given segment. Let's construct two circles with centers A and B of radius AB. They intersect at points P and Q. Let's draw a straight line PQ. The point O of the intersection of this line with the segment AB is the desired midpoint of the segment AB.

In fact, triangles APQ and BPQ are equal on three sides, therefore ∠1 =∠2 (Fig. 89).

Rice. 89

Consequently, the segment PO is the bisector of the isosceles triangle ARB, and therefore the median, i.e. point O is the middle of the segment AB.

Tasks

143. Which of the segments shown in Figure 90 are: a) chords of the circle; b) diameters of a circle; c) radii of the circle?

Rice. 90

144. Segments AB and CD are the diameters of a circle. Prove that: a) the chords BD and AC are equal; b) chords AD and BC are equal; c) ∠BAD = ∠BCD.

145. Segment MK is the diameter of a circle with center O, and MR and RK are equal chords of this circle. Find ∠POM.

146. Segments AB and CD are the diameters of a circle with center O. Find the perimeter of triangle AOD if it is known that CB = 13 cm, AB = 16 cm.

147. On a circle with center O, points A and B are marked so that angle AOB is a right angle. Segment BC is the diameter of a circle. Prove that the chords AB and AC are equal.

148. Two points A and B are given on a straight line. On the continuation of ray BA A, lay off a segment BC so that BC = 2AB.

149. Given a line a, a point B not lying on it, and a segment PQ. Construct point M on line a so that BM = PQ. Does a problem always have a solution?

150. Given a circle, a point A not lying on it, and a segment PQ. Construct a point M on the circle so that AM = PQ. Does a problem always have a solution?

151. Danes acute angle YOU and the XY beam. Construct the angle YXZ so that ∠YXZ = 2∠BAC.

152. Obtuse angle AOB is given. Construct the ray OX so that the angles HOA and HOB are equal obtuse angles.

153. Given a line a and a point M not lying on it. Construct a line passing through point M and perpendicular to line a.

Solution

Let's construct a circle with a center at a given point M, intersecting a given line a at two points, which we denote by the letters A and B (Fig. 91). Then we will construct two circles with centers A and B passing through point M. These circles intersect at point M and at another point, which we will denote by the letter N. Let us draw a line MN and prove that this line is the desired one, i.e. it is perpendicular to straight line a.

Rice. 91

In fact, triangles AMN and BMN are equal on three sides, so ∠1 = ∠2. It follows that the segment MC (C is the point of intersection of lines a and MN) is the bisector of the isosceles triangle AMB, and therefore its height. Thus, MN ⊥ AB, i.e. MN ⊥ a.

154. Given a triangle ABC. Construct: a) bisector AK; b) median VM; c) height CH of the triangle. 155. Using a compass and ruler, construct an angle equal to: a) 45°; b) 22°30".

Answers to problems

152. Instruction. First, construct the bisector of angle AOB.

Test No. 4 on the topic “Circle”

Testing theoretical knowledge.

At the board: prove the property of a tangent to a circle, the theorem on the inscribed angle, the theorem on segments of intersecting chords, the perpendicular bisector to a segment, the theorem on circles inscribed in a triangle and circumscribed about a triangle.

Class (frontal conversation).

The relative position of a straight line and a circle. Definition of a tangent to a circle and its properties. What angle is called central? What angle is called inscribed? What is its equal degree measure? Four wonderful points of the triangle. Which circle is called an inscribed circle? Described? Which polygon is called circumscribed? Inscribed? What properties do the sides of a quadrilateral circumscribed about a circle have? What properties do the angles of a quadrilateral inscribed in a circle have? State the theorem about segments of intersecting chords.

T-1. Fill in the blanks (ellipses) to make the correct statement.

OPTION 1.

1. A point equidistant from all points on a circle is called its....

2. A segment connecting two points on a circle is called its....

3. All radii of a circle....

4. In the figure 0(r) is a circle, AB is a tangent to it; point B is called...

6. The angle between the tangent to the circle and the radius drawn to the point of tangency is....

7. In the figure, AB is the diameter of the circle, C is the point lying on the circle. Triangle DIA... (type of triangle).

8. In the figure, AB = 2BC, AB is the diameter of the circle. Angle CAB is...

9. In the figure, chords AB and CD intersect at point M. Angle ACD is equal to angle....

10. In the figure O is the center of the circle. Arc AmB is 120°. Angle ABC is equal.

11. In the figure, AK = 24 cm, KB = 9 cm, CK = 12 cm. Then KD = ...

12*. In the figure, AB = BC = 13 cm, height BD = 12 cm. Then VC = ..., KS = ....

OPTION 2.

1. A geometric figure, all points of which are located at the same distance from a given point, is called....

2. A chord passing through the center of a circle is called....

3. All circle diameters....

4. In Figure 0(d) is a circle, B is the point of tangency between straight line AB and the circle. The straight line AB is called... to the circle.

6. Tangent to a circle and radius drawn to the point of tangency, ....

7. In the figure, AB is a tangent, OA is a secant passing through the center of the circle. Triangle OVA... (type of triangle).

8. In the figure OS = CA, AB is the tangent to the circle with center O. Angle BAC is equal to....

9. Chords AB and CD of a circle intersect at point K. Angle ADC is equal to angle....

10. In the figure O is the center of the circle, angle CBA is 40°. Arc CmB is equal to...

11. In the figure AM = 15 cm, MB = 4 cm, MC = 3 cm. Then DM = ... .

12*. In the figure, AB = BC, BD is the height of triangle ABC, BC = 8 cm, KS = 5 cm. Then BD = ..., DC = ....

T-2. Determine whether the following statements are true or false.

OPTION 1.

1. A straight line that has only one common point with a circle is called a tangent to this circle.

2. A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

3. The figure shows a circle. Then Ð DAC = Ð DBC.

4. Every line passing through the middle of a chord of a circle is perpendicular to it.

5. A ray touches a circle if it has only one common point with it.

6. In the figure, AB is the diameter of the circle, Ð 1 = 30°. Then р 2 = 60°.

7. The figure shows a circle. Then Ð DAB = Ð DOB.

8. In the figure O is the center of the circle. If ÈVS = 60°, then Ð SVA = 60°.

9. In the figure, the diameter AB of the circle is 10 cm, the chord AC = 8 cm. Then the area of ​​triangle ABC is 24 cm2.

10. Two chords of a circle AB and CD intersect at point O so that AO = 16 cm, BO = 9 cm, OD = 24 cm. Then CO = 6 cm.

11*. The tangency point of a circle inscribed in an isosceles triangle divides the side into segments of 5 cm and 8 cm, counting from the base. Then the area of ​​the triangle is 60 cm2.

OPTION 2.

1. A straight line, the distance to which from the center of a circle is equal to the radius of this circle, is tangent to it.

2. The radius drawn to the point of tangency of the line and the circle is perpendicular to this line.

3. The figure shows a circle. Then Ð DAC = Ð DBC.

5. A segment touches a circle if it has only one common point with it.

6. In the figure, AB is the diameter of the circle. Then if Ð 2 = 50°, then Ð1 = 40°.

7. The figure shows a circle. Then Ð ABC = ÐAOC.

8. In the figure O is the center of the circle. Then if ÐCAB - 60°, then È AC = 60°.

9. In the figure, the diameter BD of the circle is 13 cm. Then if the chord BC = 5 cm, then the area of ​​the triangle CBD is 30 cm2.

10. Two chords of a circle AB and CD intersect at point M so that MB = 3 cm, MA = 28 cm, CM = 21 cm. Then MD = 4 cm.

11*. The tangent point of a circle inscribed in an isosceles triangle divides the side into segments of 4 cm and 6 cm, counting from the vertex. Then the area of ​​this triangle is 48 cm2.

T-3.In each task, determine the correct answer from among those proposed.

OPTION 1.

1. In the figure, arc AC is 84°. What is the angle ABC subtended by this arc?

A) 84°; B) 42°; B) I don’t know.

2. In the figure, the angle MRC is 88°. What is the size of the arc MK, on ​​which the angle MKK rests?

A) 88°; B) 176°; B) I don’t know.

3. From point A, located at a distance of two radii from the center of the circle, a tangent AB is drawn. What is angle OAB?

A) 60°; B) 30°; B) I don’t know.

4. Two chords MA and MB are drawn from point M of the circle. The chord MA subtends an arc equal to 80°, and angle AMB is equal to 70°. Determine the arc subtended by the chord MB.

A) 210°; B) 140°; B) I don’t know.

5. In the figure, the diameter AB of the circle is 10 cm, the chord BC = 6 cm. Find the area of ​​triangle ACB.

A) 30 cm2; B) 24 cm2; B) I don’t know.

6. From point K of a circle with center O, two mutually perpendicular chords KM and KD are drawn. The distance from point O to the chord KM is 15 cm, and to the chord KD is 20 cm. What are the lengths of the chords KM and KD7

A) 30 cm and 40 cm; B) 15 cm and 20 cm; B) I don't know.

7. Two chords AB and CD are divided by a point O at their intersection so that AO = 9 cm, OB = 6 cm, CO = 3 cm. What is the length of segment OD7

A) 12 cm; B) 18 cm; B) I don’t know.

8. From point A to the circle, a tangent AB and a secant AC passing through the center of the circle are drawn. The distance from A to the circle is 4 cm, and the diameter of the circle is 12 cm. What is the length of the tangent?

A) 8 cm; B) 6 cm; B) I don’t know.

9*. Line AB touches a circle with center O and radius 5 cm at point A. Find the distance from point B to the circle if the length of the tangent is 12 cm.

A) 7 cm; B) 8 cm; B) I don’t know.

OPTION 2.

1. In the figure, arc AB is equal to 164°. What is the angle ACB subtended by this arc?

A) 168°; B) 82°; B) I don’t know.

2. In the figure, angle ABC is 44°. What is the arc AC on which the angle ABC rests?

A) 88°; B) 44°; B) I don’t know.

3. From point M, located at a distance of two radii from the center of the circle, a tangent MK is drawn. What is the COM angle?

A) 60°; B) 30°; B) I don’t know.

4. Two chords AM and AB are drawn from point A of the circle. The chord AM subtends an arc equal to 120°, and the angle MAB is equal to 80°. Determine the magnitude of the arc subtended by the chord AB.

A) 80°; B) 120°; B) I don’t know.

5. In the figure, the diameter AC of the circle is 13 cm, the chord AB = 12 cm. Find the area of ​​triangle ACB.

A) 78 cm2; B) 30 cm2; B) I don’t know.

6. From point A of a circle with center O, two mutually perpendicular chords AB and AC are drawn. The distance from point O to the chord AB is 40 cm, and to the chord AC is 25 cm. What are the lengths of the chords AB and AC?

A) 25 cm and 40 cm; B) 50 cm and 80 cm; B) I don't know.

7. Two chords MK and CD are divided by their intersection point P so that MP = 8 cm, PC = 4 cm. KR = 16 cm. What is the length of the segment PD.

A) 24 cm; B) 32 cm; B) I don’t know.

8. From point M to the circle, a tangent MA and a secant MC are drawn, passing through the center of the circle O. The distance from M to the center O is 20 cm, the radius of the circle is 12 cm. What is the length of the tangent?

A) 16 cm; B) 24 cm; B) I don’t know.

9*. Line AB touches a circle with center O and radius 5 cm at point B. Find the length of the tangent if the distance from point A to the circle is 8 cm.

A) 13 cm; B) 12 cm; B) I don’t know.

Cards for individual work.

Card 1.

1. How many common points can a straight line and a circle have? Formulate the property and sign of a tangent.

2. Segment BD - the height of an isosceles triangle ABC with base AC. What parts does a circle with center B and radius BD divide the side of the triangle into if AB = cm, BD = 5 cm?

3. The picture shows right triangle ABC, the sides of which touch a circle of radius 1 cm. What segments does the point of contact divide the hypotenuse of the triangle, equal to 5 cm, into?

Card 2.

1. What angle is called inscribed? State the inscribed angle theorem.

2. The vertices of a triangle with sides 2 cm, 5 cm and 6 cm lie on a circle. Prove that none of the sides of the triangle is the diameter of this circle.

3. The figure shows a circle with center O, AB is the tangent, and AC is the secant of this circle. Find the angles of triangle ABC if ÈBD=62°.

Card 3.

1. State the theorem about segments of intersecting chords.

2. Chords KL and MN of the circle intersect at point A. Find AK and AL if AM=2 dm, AN=6 dm, KL=7 dm.

3. The figure shows a circle with center O, AC is the diameter, and BC is the tangent to this circle. What parts is segment AB divided by point D if AC = 20 cm, BC = 15 cm?

Card 4.

1. State the theorem about a circle inscribed in a triangle.

2. Inscribe a circle in the given right triangle.

3. The base of an isosceles triangle is 16 cm, the side is 17 cm. Find the radius of the circle inscribed in this triangle.

Card 5.

1. Formulate a statement about the property of the circumscribed quadrilateral. Is the opposite statement true?

2. Find the area of ​​a rectangular trapezoid circumscribed about a circle if the sides of this trapezoid are 10 cm and 16 cm.

3. The area of ​​a quadrilateral ABCD circumscribed about a circle of radius 5 dm is equal to 90. Find the sides CD and AD of this quadrilateral if AB = 9 dm, BC = 10 dm.

Card 6.

1. State the theorem about the circumscribed circle of a triangle.

2. Construct a circle circumscribed about this obtuse triangle.

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Crossword.

Horizontal: 1. A straight line that has two common points with a circle. 2. Mapping the plane onto itself. 3. Double radius.

Vertical: 4. Angle unit or 1/60 minute. 5. Part of a circle limited by two radii and the arc of the circle’s circumference. 6. A segment connecting the center of a circle with any point on the circle. 7. Determination of a point on a circle.

Note: materials from the newspaper “Mathematics” were used in the development.

This video tutorial was created specifically for self-study topic "Circle". Students will be able to learn strictly geometric definition circles. The teacher will analyze in detail the solution of several typical problems for constructing a circle.

Circle- This geometric figure, consisting of a set of points that are equidistant from a given point.

Figure 1 shows a circle.

Rice. 1. Circumference

The abbreviated form of a given circle is Okr (O, r), which reads: “A circle with a center at point O and radius r.” A point from which other points are equidistant is called center circles. The segment connecting the center and a point lying on the circle is called radius. If you connect two points lying on a circle, you can draw a segment called chord. A chord passing through the center of a circle is called diameter.

Thus, the following notations exist:

O - center of the circle;

OM = r - radius of the circle;

OM = ON = r - radii of the circle;

MN - chord;

AM - diameter;

AM = 2r - relationship between radius and diameter.

Any two points cut a circle into two arcs, for example: arcs NLM and NAM for given points N and M.

Example 1: Figure 2 shows a circle. Determine the center, radius, chords, diameter and possible arcs.

Solution:

Rice. 2. Drawing for example 1

Let's define the main elements of this circle:

O - center of the circle;

OE = OD = OA = OC - radii of the circle;

EF, BA - chords;

DC - diameter.

IN at the moment Let's remember the definition of a circle. A circle is a part of a plane bounded by a circle. It is absolutely clear that the difference between a circle and a circle is as follows: a circle is a line, and a circle is a part of the plane that is limited by this line. For example, Figure 3 shows a circle.

Rice. 3. Circle

Example 2: The figure shows a circle with diameters AB and CD. Prove that the chords AC and BD are equal. Prove that chords BC and AD are equal. Prove that angles BAD and BCD are equal.

Rice. 4. Drawing for example 2

Solution:

First, let's find out that CO = OD = OB = OA, since the indicated segments are radii of the same circle. We will prove these statements using chains of triangles. For example, according to the first sign, since OB = OA as radii, CO = OD similarly, like vertical. From the equality of triangles it follows that AC = BD.

Next, we will prove that it is similar in terms of the first sign. OD = OA, CO = OB as radii, and like vertical. From the equality of triangles it follows that AD = BC.

Next we will prove that according to the third sign. BD is the common side of the triangles, AD = CB according to the proven statement in paragraph 2, AB = CD as the diameters of the circle. From the equality of triangles it follows that .

Q.E.D.

Example 3: segment MK is the diameter of a circle, and PM and RK are equal chords. Find the angle POM.

Rice. 5. Drawing for example 3

Solution:

By definition, it is isosceles, since RK = RM. Since OK - OM are the radii of circles, then PO is the median drawn to the base. According to the property of an isosceles triangle, the median drawn to the base is the height, respectively.

1. Help portal calc.ru ().

1. No. 99. Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolova, ed. Sadovnichego V.A. - M.: Education, 2010.
2. Two chords equal to the radius are drawn from a point on a given circle. Find the angle between them.
3. Prove that any ray emanating from the center of a circle intersects the circle at one point.
4. Prove that the diameter of a circle passing through the midpoint of a chord is perpendicular to it.

“Computer drawing” - Computer graphics. Hatch. this is the artist's weapon. Tasks: Result of the lesson, crossword puzzle “Mill”. Engraving. Main means of expression drawing - line. He studied at the Moscow School of Painting, then at the Stroganov School. Pencil. Illustration for the book. Integrated lesson: fine arts+ computer science.

“Saving drawings” - Which command should I choose? It is proposed to store all your files in a special folder “My Documents”. Move with mouse, copy (CTRL), delete (DELETE). Practical work“Saving a drawing to your hard drive.” To store information on a computer, long-term memory is used - a hard drive.

“Editing pictures” - 1. Select the required area, select an arbitrary area 2. Copy. Drawing a circle, square, straight line. Clear picture Select area to be deleted Delete. Circle Square Straight line. Copying. Setting drawing parameters. Creating and editing a drawing. Creating a drawing.

“3D drawings on asphalt” - Philip Kozlov - the first Russian Madonnari. In his youth, Kurt Wenner worked as an illustrator for NASA, where he created the initial images of future spaceships. 3d drawings on asphalt. Kurt Wenner is one of the most famous street artists who draws 3D drawings on asphalt using regular crayons.

“Beam straight segment” - Point O - the beginning of the beam. Points C and D are the ends of the segment CD. S. Dot. Straight, Segment, Beam. Point, Line. Straight. Numbers - coordinates of points: Ray PM. Coordinate. Name the segments, straight lines and rays shown in the figure. Segment OE is a unit segment, OE=1. Beam FR.

“Circumference” - Diameter. Find the circumference of this disk. Find the area of ​​the dial. Circumference. What is the diameter of the Moon? The number "pi" is called the Archimedean number. Find the diameter of the wheel. Find the diameter and area of ​​the arena. Find the diameter of the locomotive wheel. Moscow. The great ancient Greek mathematician Archimedes.