The center of the circle with a compass and straightedge. Constructions with a compass and a ruler. Construction of the middle of the segment

This lesson is devoted to the study of the circle and the circle. Also, the teacher will teach you to distinguish between closed and open lines. You will get acquainted with the basic properties of a circle: center, radius and diameter. Learn their definitions. Learn to determine the radius if the diameter is known, and vice versa.

If you fill in the space inside the circle, for example, draw a circle with a compass on paper or cardboard and cut it out, then we get a circle (Fig. 10).

Rice. 10. Circle

A circle is the part of a plane bounded by a circle.

Condition: Vitya Verkhoglyadkin drew 11 diameters in his circle (Fig. 11). And when he counted the radii, he got 21. Did he count correctly?

Rice. 11. Illustration for the problem

Solution: radii should be twice as many as diameters, so:

Vitya counted incorrectly.

Bibliography

1. Mathematics. Grade 3 Proc. for general education institutions with adj. to an electron. carrier. At 2 h. Part 1 / [M.I. Moro, M.A. Bantova, G.V. Beltyukova and others] - 2nd ed. - M.: Education, 2012. - 112 p.: ill. - (School of Russia).
2. Rudnitskaya V.N., Yudacheva T.V. Mathematics, 3rd grade. - M.: VENTANA-GRAF.
3. Peterson L.G. Mathematics, 3rd grade. - M.: Juventa.

1. Mypresentation.ru ().
2. Sernam.ru ().
3. School-assistant.ru ().

Homework

1. Mathematics. Grade 3 Proc. for general education institutions with adj. to an electron. carrier. At 2 h. Part 1 / [M.I. Moro, M.A. Bantova, G.V. Beltyukova and others] - 2nd ed. - M.: Enlightenment, 2012., Art. 94 No. 1, art. 95 no. 3.

2. Solve the riddle.

We live together with my brother,

We have so much fun together

We will put a mug on the sheet (Fig. 12),

Let's circle it with a pencil.

Get what you need -

It's called...

3. It is necessary to determine the diameter of the circle if it is known that the radius is 5 m.

4. * Using a compass, draw two circles with radii: a) 2 cm and 5 cm; b) 10 mm and 15 mm.

In the manufacture or processing of wood parts, in some cases it is required to determine where their geometric center is located. If the part has a square or rectangular shape, then this is not difficult to do. It is enough to connect the opposite corners with diagonals, which at the same time intersect exactly in the center of our figure.
For products that have the shape of a circle, this solution will not work, because they do not have corners, and therefore diagonals. In this case, some other approach based on other principles is needed.

And they exist, and in many variations. Some of them are quite complex and require several tools, others are easy to implement and do not require a whole set of devices to implement them.
Now we will look at one of the easiest ways to find the center of a circle with just a regular ruler and pencil.

The sequence of finding the center of the circle:

1. First, we need to remember that a chord is a straight line connecting two points of a circle, and not passing through the center of the circle. It is not difficult to reproduce it at all: you just need to put a ruler on a circle anywhere so that it intersects the circle in two places, and draw a straight line with a pencil. A segment inside a circle will be a chord.
In principle, one chord can be dispensed with, but in order to increase the accuracy of establishing the center of the circle, we will draw at least a pair, and even better - 3, 4 or 5 chords of different lengths. This will allow us to level the errors of our constructions and more accurately cope with the task.

2. Next, using the same ruler, we find the midpoints of the chords we reproduced. For example, if the total length of one chord is 28 cm, then its center will be at a point that is 14 cm in a straight line from the intersection of the chord with the circle.
Having determined the centers of all chords in this way, we draw perpendicular lines through them, using, for example, a right triangle.

3. If we now continue these lines perpendicular to the chords in the direction towards the center of the circle, then they will intersect at approximately one point, which will be the desired center of the circle.

4. Having established the location of the center of our particular circle, we can use this fact for various purposes. So, if you place the leg of a carpenter's compass at this point, then you can draw an ideal circle, and then cut out a circle using the appropriate cutting tool and the point of the center of the circle that we have determined.

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

Definition

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

Rice. 77

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

In figure 78, the segments AB and EF are the chords of the circle, the 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 a compass and a ruler

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

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

What can be done with them? It is clear that the ruler allows one to draw an arbitrary line, as well as to construct a 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 building problems:

construct an angle equal to a 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, set aside a segment equal to the given one.

Solution

Let's depict the figures given in the condition of the problem: the ray OS and the segment AB (Fig. 83, a). Then, with 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 building tasks

Constructing an angle equal to a given one

Task

Set aside from the given ray an angle equal to the given one.

Solution

This angle with vertex A and the 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 the ray OM.

Rice. 84

Let's draw a circle of arbitrary radius with the center at the vertex A of the given angle. This circle intersects the sides of the corner at points B and C (Fig. 85, a). Then we draw a circle of the same radius with the center at the beginning of the given ray OM. It intersects the beam at point D (Fig. 85, b). After that, we 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 required one.

Rice. 85

Consider triangles ABC and ODE. The segments AB and AC are the radii of a circle with center A, and the segments OD and OE are the 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 performed on the ground, if instead of a compass we use a rope.

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 a 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 intersect at two points, at least one of which lies inside the corner. We 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 common side; AC and AB are equal as radii of the same circle; CE = BE by construction.

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

Comment

Can a given angle be divided into two equal angles using a compass and straightedge? It is clear that it is possible - for this you need to draw a 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 straightedge? This task, called angle trisection problems, has attracted the attention of mathematicians for many centuries. It was only in the 19th century that it was proved that such a construction is impossible for an arbitrary angle.

Construction of perpendicular lines

Task

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

Solution

The given line a and the given point M belonging to this line are shown in Figure 87.

Rice. 87

On the rays of the straight line a, emanating from the point M, we set aside 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 us draw a line through the point M and one of these points, for example, the line MP (see Fig. 87), and prove that this line is the desired one, that is, that it is perpendicular to the given line a.

Indeed, since the median PM of an isosceles triangle PAB is also the altitude, then PM ⊥ a.

Construction of the middle of the segment

Task

Construct the midpoint of this segment.

Solution

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

Indeed, triangles APQ and BPQ are equal in three sides, so ∠1 = ∠2 (Fig. 89).

Rice. 89

Consequently, the segment RO is the bisector of the isosceles triangle ARV, and hence the median, that is, the point O is the midpoint of the segment AB.

Tasks

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

Rice. 90

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

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

148. Two points A and B are given on a straight line. On the continuation of the beam BA, set aside the segment BC so that BC \u003d 2AB.

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

151. Acute angle BAC and ray XY are given. Construct the angle YXZ so that ∠YXZ = 2∠BAC.

152. Obtuse angle AOB is given. Construct the ray OX so that the angles XOA and XOB 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 straight line a at two points, which we denote by the letters A and B (Fig. 91). Then we construct two circles with centers A and B passing through the point M. These circles intersect at the point M and at one more point, which we denote by the letter N. Let's draw the line MN and prove that this line is the desired one, i.e. it is perpendicular to straight line a.

Rice. 91

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

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

Answers to tasks

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

Goals:

to consolidate the concepts of “circle”, “circle” among students; to derive the concept of "radius of a circle"; learn to build circles of a given radius; develop the ability to reason, analyze.

Personal UUD:
to form a positive attitude towards mathematics lessons;
interest in subject-research activities;

Meta-subject tasks

Regulatory UUD:
accept and save the learning task;
find several solutions in cooperation with the teacher and the class;

Cognitive UUD:
setting and solving problems:
independently identify and formulate the problem;
general education:
find the necessary information in the textbook;
build a circle of a given radius using a compass;
brain teaser:
to form the concept of "radius";
classify, compare;
draw your own conclusions;

Communicative UUD:
actively participate in teamwork, using speech means;
argue your point of view;

Item Skills:
identify the essential features of the concepts "circle radius";
build circles with different radii;
recognize radii in a drawing.

During the classes

Motivation for learning activities

- Let's check if everyone is ready for the lesson?

"Emotional entry into the lesson":

Smile like the sun.

Frown like clouds

Cry like rain

Surprised as if you saw a rainbow

Now repeat after me

Game "Friendly echo"

2.Updating knowledge

Verbal counting

a) 60-40 36+12 10+20 58-12 90-50 31+13

Unravel the pattern. Continue the row.

Answer: 20, 48,30,46,40,44 50.42

b) Solve the problem:

1. On the first day, the store sold 42 kg of fruit, and on the second day, 2 kg more. How many kilograms were sold on the second day?

What needs to be changed so that the task is solved in 2 steps.

Balls - 16 pcs.

Jump ropes - 28 pcs.

Find a solution to this problem.

28-16 28+16

Change the question so that the problem can be solved by subtraction.

3. Statement of the learning task

1. Name the geometric shapes

Circle circumference oval ball

Which figure is missing?

What do the figures have in common? (Circle, circumference, ball have the same shape)

What is the difference?

2. In

What points are on the circle? What are the points outside the circle?

What does point O mean? (circle center)

What is the name of the segment OB?

How many radii can be drawn in a circle?

Which segment is not a radius? Why?

What can be the conclusion?

Conclusion: all radii have the same length .

3. How many circles are in the picture?

How are circles different? (size)

What determines the size of a circle?

What can be the conclusion?

Conclusion: the larger the circle, the larger its radius.

Determine the topic of the lesson.

Topic: Constructing a circle of given radius using a compass.

What tasks can we set ourselves for this lesson?

4. Work on the theme

a) Construction of a circle.

What do you need to know to draw a circle of a given size?

Draw a circle with a radius of 3 cm.

b) Preparation for project activities

1) Consider the drawing

What shapes does a butterfly consist of? Circles with the same radius?

2) Work in pairs.

Restore the order of the stages above the project.

Project presentation or demonstration

Intention (to make a sketch)

Build figures to implement the plan

Consider what radius the shapes should have

c) Work on the project.

Work in groups according to the compiled algorithm