What does a circle consist of? II

Demo material: compass, material for experiment: round objects and ropes (for each student) and rulers; circle model, colored crayons.

Target: Studying the concept of “circle” and its elements, establishing connections between them; introduction of new terms; developing the ability to make observations and draw conclusions using experimental data; nurturing cognitive interest in mathematics.

Lesson progress

I. Organizational moment

Greetings. Setting a goal.

II. Oral counting

III. New material

Among all kinds of flat figures, two main ones stand out: the triangle and the circle. These figures have been known to you since early childhood. How to define a triangle? Through segments! How can we determine what a circle is? After all, this line bends at every point! The famous mathematician Grathendieck, recalling his school years, noted that he became interested in mathematics after learning the definition of a circle.

Let's draw a circle using a geometric device - compass. Constructing a circle with a demonstration compass on the board:

mark a point on the plane;
We align the leg of the compass with the tip with the marked point, and rotate the leg with the stylus around this point.

The result is a geometric figure - circle.

(Slide No. 1)

So what is a circle?

Definition. Circumference - is a closed curved line, all points of which are at equal distances from a given point on the plane, called center circles.

(Slide No. 2)

How many parts does a plane divide a circle into?

Point O- center circles.

OR - radius circle (this is a segment connecting the center of the circle with any point on it). In Latin radius- wheel spoke.

AB – chord circle (this is a segment connecting any two points on a circle).

DC – diameter circle (this is a chord passing through the center of the circle). Diameter comes from the Greek “diameter”.

DR– arc circle (this is a part of a circle bounded by two points).

How many radii and diameters can be drawn in a circle?

The part of the plane inside the circle and the circle itself form a circle.

Definition. Circle - This is the part of the plane bounded by a circle. The distance from any point on the circle to the center of the circle does not exceed the distance from the center of the circle to any point on the circle.

How do a circle and a circle differ from each other, and what do they have in common?

How are the lengths of the radius (r) and diameter (d) of one circle related to each other?

d = 2 * r (d– diameter length; r – radius length)

How are the lengths of a diameter and any chord related?

Diameter is the largest of the chords of a circle!

The circle is an amazingly harmonious figure; the ancient Greeks considered it the most perfect, since the circle is the only curve that can “slide on its own”, rotating around the center. The main property of a circle answers the questions why compasses are used to draw it and why wheels are made round, and not square or triangular. By the way, about the wheel. This is one of the greatest inventions of mankind. It turns out that coming up with the wheel was not as easy as it might seem. After all, even the Aztecs, who lived in Mexico, did not know the wheel until almost the 16th century.

The circle can be drawn on checkered paper without a compass, that is, by hand. True, the circle turns out to be a certain size. (Teacher shows on the checkered board)

The rule for depicting such a circle is written as 3-1, 1-1, 1-3.

Draw a quarter of such a circle by hand.

How many cells is the radius of this circle equal to? They say that the great German artist Albrecht Dürer could draw a circle so accurately with one movement of his hand (without rules) that a subsequent check with a compass (the center was indicated by the artist) did not show any deviations.

Laboratory work

You already know how to measure the length of a segment, find the perimeters of polygons (triangle, square, rectangle). How to measure the length of a circle if the circle itself is a curved line, and the unit of measurement of length is a segment?

There are several ways to measure circumference.

The trace from the circle (one revolution) on a straight line.

The teacher draws a straight line on the board, marks a point on it and on the boundary of the circle model. Combines them, and then smoothly rolls the circle in a straight line until the marked point A on a circle will not be on a straight line at a point IN. Segment AB will then be equal to the circumference.

Leonardo da Vinci: "The movement of carts has always shown us how to straighten the circumference of a circle."

Assignment to students:

a) draw a circle by circling the bottom of a round object;

b) wrap the bottom of the object with thread (once) so that the end of the thread coincides with the beginning at the same point on the circle;

c) straighten this thread to a segment and measure its length using a ruler, this will be the circumference.

The teacher is interested in the measurement results of several students.

However, these methods of directly measuring the circumference are inconvenient and give rough results. Therefore, since ancient times, they began to look for more advanced ways to measure circumference. During the measurement process, we noticed that there is a certain relationship between the length of a circle and the length of its diameter.

d) Measure the diameter of the bottom of the object (the largest of the chords of the circle);

e) find the ratio C:d (accurate to tenths).

Ask several students for the results of calculations.

Many scientists and mathematicians tried to prove that this ratio is a constant number, independent of the size of the circle. The ancient Greek mathematician Archimedes was the first to do this. He found a fairly accurate meaning for this ratio.

This relationship began to be denoted by a Greek letter (read “pi”) - the first letter of the Greek word “periphery” is a circle.

C – circumference;

d – diameter length.

Historical information about the number π:

Archimedes, who lived in Syracuse (Sicily) from 287 to 212 BC, found the meaning without measurements, just by reasoning

In fact, the number π cannot be expressed as an exact fraction. The 16th century mathematician Ludolf had the patience to calculate it with 35 decimal places and bequeathed this value of π to be carved on his grave monument. In 1946 – 1947 two scientists independently calculated the 808 decimal places of pi. Now more than a billion digits of the number π have been found on computers.

The approximate value of π, accurate to five decimal places, can be remembered using the following line (based on the number of letters in the word):

π ≈ 3.14159 – “I know and remember this perfectly.”

Introduction to the Circumference Formula

Knowing that C:d = π, what will be the length of circle C?

(Slide No. 3) C = πd C = 2πr

How did the second formula come about?

Reads: circumference is equal to the product of the number π and its diameter (or twice the product of the number π and its radius).

Area of a circle is equal to the product of the number π and the square of the radius.

S= πr 2

IV. Problem solving

№1. Find the circumference of a circle whose radius is 24 cm. Round the number π to the nearest hundredth.

Solution:π ≈ 3.14.

If r = 24 cm, then C = 2 π r ≈ 2 3.14 24 = 150.72(cm).

Answer: circumference 150.72 cm.

No. 2 (orally): How to find the length of an arc equal to a semicircle?

Task: If you wrap a wire around the globe along the equator and then add 1 meter to its length, will a mouse be able to slip between the wire and the ground?

Solution: C = 2 πR, C+1 = 2π(R+x)

Not only a mouse, but also a large cat will slip into such a gap. And it would seem, what does 1 m mean compared to 40 million meters of the earth's equator?

V. Conclusion

What main points should you pay attention to when constructing a circle?
What parts of the lesson were most interesting to you?
What new did you learn in this lesson?

Solution to crossword puzzle with pictures(Slide No. 3)

It is accompanied by a repetition of the definitions of circle, chord, arc, radius, diameter, formulas for circumference. And as a result - the keyword: “CIRCLE” (horizontally).

Lesson Summary: grading, comments on homework. Homework: p. 24, No. 853, 854. Conduct an experiment to find the number π 2 more times.

To get a general idea of what a circle is, look at a ring or hoop. You can also take a round glass and cup, place it upside down on a piece of paper and trace it with a pencil. With repeated magnification, the resulting line will become thick and not entirely smooth, and its edges will be blurred. A circle as a geometric figure does not have such a characteristic as thickness.

Circle: definition and basic means of description

A circle is a closed curve consisting of many points located in the same plane and equidistant from the center of the circle. In this case, the center is in the same plane. As a rule, it is denoted by the letter O.

The distance from any point on the circle to the center is called the radius and is denoted by the letter R.

If you connect any two points on a circle, the resulting segment will be called a chord. The chord passing through the center of the circle is the diameter, denoted by the letter D. The diameter divides the circle into two equal arcs and is twice the length of the radius. Thus, D = 2R, or R = D/2.

Properties of chords

If a chord is drawn through any two points of the circle, and then a radius or diameter is drawn perpendicular to the latter, then this segment will split both the chord and the arc cut off by it into two equal parts. The converse statement is also true: if the radius (diameter) divides the chord in half, then it is perpendicular to it.
If two parallel chords are drawn within the same circle, then the arcs cut off by them, as well as those enclosed between them, will be equal.
Let's draw two chords PR and QS intersecting within the circle at point T. The product of segments of one chord will always be equal to the product of segments of another chord, that is, PT x TR = QT x TS.

Circumference: general concept and basic formulas

One of the basic characteristics of this geometric figure is the circumference. The formula is derived using quantities such as radius, diameter and the constant "π", reflecting the constancy of the ratio of the circumference to its diameter.

Thus, L = πD, or L = 2πR, where L is the circumference, D is the diameter, R is the radius.

The formula for circumference can be considered as the initial one when finding the radius or diameter for a given circumference: D = L/π, R = L/2π.

What is a circle: basic postulates

have no common points;
have one common point, and the straight line is called a tangent: if you draw a radius through the center and the point of tangency, then it will be perpendicular to the tangent;
have two common points, and the line is called a secant.

2. Through three arbitrary points lying in the same plane, no more than one circle can be drawn.

3. Two circles can touch only at one point, which is located on the segment connecting the centers of these circles.

4. For any rotations relative to the center, the circle turns into itself.

5. What is a circle in terms of symmetry?

the same curvature of the line at any point;
relative to point O;
mirror symmetry relative to diameter.

6. If you construct two arbitrary inscribed angles based on the same arc of a circle, they will be equal. An angle based on an arc equal to half, that is, cut off by a chord-diameter, is always equal to 90°.

7. If you compare closed curved lines of the same length, it turns out that the circle delimits the section of the plane with the largest area.

Circle inscribed in and circumscribed by a triangle

The idea of what a circle is will be incomplete without a description of the features of its relationship with triangles.

When constructing a circle inscribed in a triangle, its center will always coincide with the intersection point of the triangle.
The center of a circle circumscribed about a triangle is located at the intersection of the median perpendiculars to each of the sides of the triangle.
If we describe a circle, then its center will be in the middle of the hypotenuse, that is, the latter will be the diameter.
The centers of the inscribed and circumscribed circles will be at the same point if the basis for the construction is

Basic statements about circles and quadrilaterals

A circle can be described around a convex quadrilateral only when the sum of its opposite internal angles equals 180°.
It is possible to construct a circle inscribed in a convex quadrilateral if the sum of the lengths of its opposite sides is the same.
You can describe a circle around a parallelogram if its angles are right.
A circle can be inscribed in a parallelogram if all its sides are equal, that is, it is a rhombus.
You can construct a circle through the corners of a trapezoid only if it is isosceles. In this case, the center of the circumscribed circle will be located at the intersection of the quadrilateral and the median perpendicular drawn to the side.

This is a closed flat line, each point of which is equidistant from the same point ( O), called center.

Straight ( O.A., O.B., OS. ..) connecting the center with the points of the circle are radii.

From this we get:

1. All radii of one circle are equal.

2. Two circles with the same radii will be equal.

3. Diameter equal to two radii.

4. Dot, lying inside the circle is closer to the center, and a point lying outside the circle is further from the center than points on the circle.

5. Diameter, perpendicular to the chord, divides this chord and both arcs contracted by it in half.

6. Arcs, enclosed between parallel chords, are equal.

When working with circles, the following theorems apply:

1. Theorem . A straight line and a circle cannot have more than two points in common.

From this theorem we obtain two logically following consequences:

No part circle cannot be combined with a line, because otherwise the circle with the line would have more than two points in common.

A line, no part of which can be combined with a straight line, is called crooked.

From the previous it follows that the circle is crooked line.

2. Theorem . Through any three points that do not lie on the same line, you can draw a circle, and only one.

How consequence from this theorem we obtain:

Three perpendicular to the sides triangle inscribed in a circle drawn through their midpoints intersect at one point, which is the center of the circle.

Let's solve the problem. It is required to find the center of the proposed circle.

Let's mark any three points A, B and C on the proposed one, draw two points through them chords, for example, AB and CB, and from the middle of these chords we indicate perpendiculars MN and PQ. The desired center, being equally distant from A, B and C, must lie on both MN and PQ, therefore, it is located at the intersection of these perpendiculars, i.e. at point O.

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

This point (O) is called center of the circle.
Circle radius- this is a segment connecting the center with any point on the circle. All radii have the same length (by definition).
Chord- a segment connecting two points on a circle. A chord passing through the center of a circle is called diameter. The center of a circle is the midpoint of any diameter.
Any two points on a circle divide it into two parts. Each of these parts is called arc of a circle. 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 arcs of a circle with common ends is equal to 360º.
The part of the plane bounded by a circle is called all around.
Circular sector- a part of a circle bounded by an arc and two radii connecting the ends of the arc to the center of the circle. The arc that limits the sector is called arc of the sector.
Two circles having a common center are called concentric.
Two circles intersecting at right angles are called orthogonal.

The relative position of a straight line and a circle

If the distance from the center of the circle to the straight line is less than the radius of the circle ( d), then the straight line and the circle have two common points. In this case the line is called secant in relation to the circle.
If the distance from the center of the circle to the straight line is equal to the radius of the circle, then the straight line and the circle have only one common point. This line is called tangent to the circle, and their common point is called point of tangency between a line and a circle.
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 and the circle have no common points

Central and inscribed angles

Central angle is an angle with its vertex at the center of the circle.
Inscribed angle- an angle whose vertex lies on a circle and whose sides intersect the circle.

Inscribed angle theorem

An inscribed angle is measured by the half of the arc on which it subtends.

Corollary 1.
Inscribed angles subtending the same arc are equal.

Corollary 2.
An inscribed angle subtended by 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:

C = 2∙π∙R

Circular arc length:

R = С/(2∙π) = D/2

Diameter:

D = C/π = 2∙R

Circular arc length:

l = (π∙R) / 180∙α,
Where α - degree measure of the length of a circular arc)

Circle area:

S = π∙R 2

Area of the circular sector:

S = ((π∙R 2) / 360)∙α

Equation of a circle

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

(x - x o) 2 + (y - y o) 2 = r 2

The equation of a circle of radius r with center at the origin has the form:

x 2 + y 2 = r 2

First, let's understand the difference between a circle and a circle. To see this difference, it is enough to consider what both figures are. These are an infinite number of points on the plane, located at an equal distance from a single central point. But, if the circle also consists of internal space, then it does not belong to the circle. It turns out that a circle is both a circle that limits it (circle(r)), and an innumerable number of points that are inside the circle.

For any point L lying on the circle, the equality OL=R applies. (The length of the segment OL is equal to the radius of the circle).

A segment that connects two points on a circle is its chord.

A chord passing directly through the center of a circle is diameter this circle (D). The diameter can be calculated using the formula: D=2R

Circumference calculated by the formula: C=2\pi R

Area of a circle: S=\pi R^(2)

Arc of a circle is called that part of it that is located between its two points. These two points define two arcs of a circle. The chord CD subtends two arcs: CMD and CLD. Identical chords subtend equal arcs.

Central angle An angle that lies between two radii is called.

Arc length can be found using the formula:

Using degree measure: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
Using radian measure: CD = \alpha R

The diameter, which is perpendicular to the chord, divides the chord and the arcs contracted by it in half.

If chords AB and CD of a circle intersect at point N, then the products of segments of chords separated by point N are equal to each other.

AN\cdot NB = CN\cdot ND

Tangent to a circle

Tangent to a circle It is customary to call a straight line that has one common point with a circle.

If a line has two common points, it is called secant.

If you draw the radius to the tangent point, it will be perpendicular to the tangent to the circle.

Let's draw two tangents from this point to our circle. It turns out that the tangent segments will be equal to one another, and the center of the circle will be located on the bisector of the angle with the vertex at this point.

AC = CB

Now let’s draw a tangent and a secant to the circle from our point. We obtain that the square of the length of the tangent segment will be equal to the product of the entire secant segment and its outer part.

AC^(2) = CD \cdot BC

We can conclude: the product of an entire segment of the first secant and its external part is equal to the product of an entire segment of the second secant and its external part.

AC\cdot BC = EC\cdot DC

Angles in a circle

The degree measures of the central angle and the arc on which it rests are equal.

\angle COD = \cup CD = \alpha ^(\circ)

Inscribed angle is an angle whose vertex is on a circle and whose sides contain chords.

You can calculate it by knowing the size of the arc, since it is equal to half of this arc.

\angle AOB = 2 \angle ADB

Based on a diameter, inscribed angle, right angle.

\angle CBD = \angle CED = \angle CAD = 90^ (\circ)

Inscribed angles that subtend the same arc are identical.

Inscribed angles resting on one chord are identical or their sum is equal to 180^ (\circ) .

\angle ADB + \angle AKB = 180^ (\circ)

\angle ADB = \angle AEB = \angle AFB

On the same circle are the vertices of triangles with identical angles and a given base.

An angle with a vertex inside the circle and located between two chords is identical to half the sum of the angular values of the arcs of the circle that are contained within the given and vertical angles.

\angle DMC = \angle ADM + \angle DAM = \frac(1)(2) \left (\cup DmC + \cup AlB \right)

An angle with a vertex outside the circle and located between two secants is identical to half the difference in the angular values of the arcs of the circle that are contained inside the angle.

\angle M = \angle CBD - \angle ACB = \frac(1)(2) \left (\cup DmC - \cup AlB \right)

Inscribed circle

Inscribed circle is a circle tangent to the sides of a polygon.

At the point where the bisectors of the corners of a polygon intersect, its center is located.

A circle may not be inscribed in every polygon.

The area of a polygon with an inscribed circle is found by the formula:

S = pr,

p is the semi-perimeter of the polygon,

r is the radius of the inscribed circle.

It follows that the radius of the inscribed circle is equal to:

r = \frac(S)(p)

The sums of the lengths of opposite sides will be identical if the circle is inscribed in a convex quadrilateral. And vice versa: a circle fits into a convex quadrilateral if the sums of the lengths of opposite sides are identical.

AB + DC = AD + BC

It is possible to inscribe a circle in any of the triangles. Only one single one. At the point where the bisectors of the internal angles of the figure intersect, the center of this inscribed circle will lie.

The radius of the inscribed circle is calculated by the formula:

r = \frac(S)(p) ,

where p = \frac(a + b + c)(2)

Circumcircle

If a circle passes through each vertex of a polygon, then such a circle is usually called described about a polygon.

At the point of intersection of the perpendicular bisectors of the sides of this figure will be the center of the circumcircle.

The radius can be found by calculating it as the radius of the circle that is circumscribed about the triangle defined by any 3 vertices of the polygon.

There is the following condition: a circle can be described around a quadrilateral only if the sum of its opposite angles is equal to 180^( \circ) .

\angle A + \angle C = \angle B + \angle D = 180^ (\circ)

Around any triangle you can describe a circle, and only one. The center of such a circle will be located at the point where the perpendicular bisectors of the sides of the triangle intersect.