§ 1 Circle. Basic Concepts
In mathematics, there are sentences that explain the meaning of a particular name or expression. Such sentences are called definitions.
Let us define the concept of a circle. A circle is a geometric figure consisting of all points of a plane located at a given distance from a given point.
This point, let's call it point O, is called the center of the circle.
The segment connecting the center with any point on the circle is called the radius of the circle. There are many such segments that can be drawn, for example, OA, OB, OS. They will all be the same length.
A segment connecting two points on a circle is called a chord. MN is the chord of the circle.
The chord passing through the center of the circle is called the diameter. AB is the diameter of the circle. The diameter consists of two radii, which means that the length of the diameter is twice the radius. The center of a circle is the midpoint of any diameter.
Any two points on a circle divide it into two parts. These parts are called arcs of a circle.
ANB and AMB are arcs of a circle.
The part of the plane that is bounded by a circle is called a circle.
To depict a circle in a drawing, a compass is used. The circle can also be drawn on the ground. To do this, just use a rope. Secure one end of the rope to a peg driven into the ground, and draw a circle with the other end.
§ 2 Constructions with compasses and ruler
In geometry, many constructions can be performed using only a compass and a ruler without scale divisions.
Using only a ruler, you can draw an arbitrary straight line, as well as an arbitrary straight line passing through a given point, or a straight line passing through two given points.
A compass allows you to 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.
Separately, each of these tools makes it possible to make the simplest constructions, but with the help of these two tools you can already perform more complex operations, for example,
solve construction problems such as
Construct an angle equal to the given one,
Construct a triangle with the given sides,
Divide the segment in half
Through a given point draw a line perpendicular to the given line, etc.
Let's consider the problem.
Task: On a given ray, from its beginning, plot a segment equal to the given one.
Given a ray OS and a segment AB. It is necessary to construct a segment OD equal to the segment AB.
Using a compass, we construct a circle of radius, equal to length segment AB, with center at point O. This circle will intersect the given ray OS at some point D. Segment OD is the required segment.
List of used literature:
- Geometry. Grades 7-9: textbook. for general education organizations / L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev et al. - M.: Education, 2013. - 383 p.: ill.
- Gavrilova N.F. Lesson developments in geometry grade 7. - M.: “VAKO”, 2004. - 288 p. - (To help the school teacher).
- Belitskaya O.V. Geometry. 7th grade. Part 1. Tests. – Saratov: Lyceum, 2014. – 64 p.
A circle is a closed curved line, each point of which is located at the same distance from one point O, called the center.
Straight lines connecting any point on a circle to its center are called radii R.
The straight line AB connecting two points of a circle and passing through its center O is called diameter D.
The parts of circles are called arcs.
The straight line CD connecting two points on a circle is called chord.
Direct MN, which has only one common point with a circle is called tangent.
The part of the circle bounded by the chord CD and the arc is called segment.
The part of a circle bounded by two radii and an arc is called sector.
Two mutually perpendicular horizontal and vertical lines intersecting at the center of a circle are called axes of the circle.
The angle formed by two radii KOA is called central angle.
Two mutually perpendicular radius make an angle of 90 0 and limit 1/4 of the circle.
We draw a circle with horizontal and vertical axes, which divide it into 4 equal parts. Drawing with a compass or square at 45 0, two mutually perpendicular lines divide the circle into 8 equal parts.
Dividing a circle into 3 and 6 equal parts (multiples of 3 to three)
To divide a circle into 3, 6 and a multiple of them, draw a circle of a given radius and the corresponding axes. Division can begin from the point of intersection of the horizontal or vertical axis with the circle. The specified radius of the circle is plotted 6 times successively. Then the resulting points on the circle are sequentially connected by straight lines and form a regular inscribed hexagon. Connecting points through one gives an equilateral triangle, and dividing the circle into three equal parts.
The construction of a regular pentagon is carried out as follows. We draw two mutually perpendicular circle axis equal to the diameter of the circle. We divide right half horizontal diameter in half using arc R1. From the resulting point “a” in the middle of this segment with radius R2, draw a circular arc until it intersects with the horizontal diameter at point “b”. With radius R3, from point “1”, draw a circular arc until it intersects with a given circle (point 5) and obtain the side of a regular pentagon. The distance "b-O" gives the side of a regular decagon.
Dividing a circle into N number of identical parts (constructing a regular polygon with N sides)
This is done as follows. We draw horizontal and vertical mutually perpendicular axis of the circle. From the top point “1” of the circle, draw a straight line at an arbitrary angle to the vertical axis. We lay out equal segments of arbitrary length on it, the number of which is equal to the number of parts into which we divide the given circle, for example 9. We connect the end of the last segment to the lower point of the vertical diameter. We draw lines parallel to the resulting one from the ends of the set aside segments until they intersect with the vertical diameter, thus dividing the vertical diameter of a given circle into a given number of parts. With a radius equal to the diameter of the circle, from the bottom point of the vertical axis we draw an arc MN until it intersects with the continuation of the horizontal axis of the circle. From points M and N we draw rays through even (or odd) division points of the vertical diameter until they intersect with the circle. The resulting segments of the circle will be the required ones, because points 1, 2, …. 9 divide the circle into 9 (N) equal parts.
In construction problems, a compass and a ruler are considered ideal tools, in particular, a ruler has no divisions and has only one side of infinite length, and a compass can have an arbitrarily large or arbitrarily small opening.
Acceptable constructions. The following operations are allowed in construction tasks:
1. Mark a point:
- arbitrary point of the plane;
- an arbitrary point on a given line;
- an arbitrary point on a given circle;
- the point of intersection of two given lines;
- points of intersection/tangency of a given line and a given circle;
- points of intersection/tangency of two given circles.
2. Using a ruler you can draw a straight line:
- an arbitrary straight line on a plane;
- an arbitrary straight line passing through a given point;
- a straight line passing through two given points.
3. Using a compass you can construct a circle:
- an arbitrary circle on a plane;
- an arbitrary circle with a center at a given point;
- an arbitrary circle with a radius equal to the distance between two given points;
- a circle with a center at a given point and a radius equal to the distance between two given points.
Solving construction problems. The solution to the construction problem contains three essential parts:
- Description of the method for constructing the required object.
- Proof that the object constructed in the described way is indeed the desired one.
- Analysis of the described construction method for its applicability to different versions of the initial conditions, as well as for the uniqueness or non-uniqueness of the solution obtained by the described method.
Constructing a segment equal to the given one. Let a ray with a beginning at point $O$ and a segment $AB$ be given. To construct a segment $OP = AB$ on a ray, you need to construct a circle with a center at point $O$ of radius $AB$. The point of intersection of the ray with the circle will be the required point $P$.
Constructing an angle equal to a given one. Let a ray with origin at point $O$ and angle $ABC$ be given. With the center at point $B$ we construct a circle with an arbitrary radius $r$. Let us denote the intersection points of the circle with the rays $BA$ and $BC$ as $A"$ and $C"$, respectively.
Let's construct a circle with a center at point $O$ of radius $r$. Let us denote the point of intersection of the circle with the ray as $P$. Let's construct a circle with a center at point $P$ of radius $A"B"$. We denote the intersection point of the circles as $Q$. Let's draw the ray $OQ$.
We get the angle $POQ$, equal to angle$ABC$, since triangles $POQ$ and $ABC$ are equal on three sides.
Constructing the perpendicular bisector to a segment. Let's construct two intersecting circles of arbitrary radius with centers at the ends of the segment. By connecting two points of their intersection, we obtain a perpendicular bisector.
Constructing the bisector of an angle. Let's draw a circle of arbitrary radius with the center at the vertex of the corner. Let's construct two intersecting circles of arbitrary radius with centers at the points of intersection of the first circle with the sides of the angle. By connecting the vertex of an angle with any of the intersection points of these two circles, we obtain the bisector of the angle.
Constructing the sum of two segments. To construct on a given ray a segment equal to the sum of two given segments, you need to apply the method of constructing a segment equal to a given one twice.
Constructing the sum of two angles. In order to plot an angle from a given ray equal to the sum of two given angles, you need to apply the method of constructing an angle equal to the given one twice.
Finding the midpoint of a segment. In order to mark the middle of a given segment, you need to construct a perpendicular bisector to the segment and mark the point of intersection of the perpendicular with the segment itself.
Constructing a perpendicular line through a given point. Let it be required to construct a line perpendicular to a given point and passing through a given point. We draw a circle of arbitrary radius with a center at a given point (regardless of whether it lies on a line or not), intersecting the line at two points. We construct a perpendicular bisector to a segment with ends at the points of intersection of the circle and the line. This will be the desired perpendicular line.
Constructing a parallel line through a given point. Let it be required to construct a line parallel to a given point and passing through a given point outside the line. We construct a line passing through a given point and perpendicular to a given line. Then we construct a straight line passing through this point, perpendicular to the constructed perpendicular. The resulting straight line will be the required one.
Goals:
consolidate the concepts of “circle” and “circle” among students; derive the concept of “radius of a circle”; learn to construct circles of a given radius; develop the ability to reason and analyze.
Personal UUD:
form positive attitude to mathematics lessons;
interest in subject research activities;
Meta-subject tasks
Regulatory UUD:
accept and save the learning task;
in collaboration with the teacher and class, find several solutions;
Cognitive UUD:
formulation and solution of problems:
independently identify and formulate the problem;
general education:
find the necessary information in the textbook;
construct a circle of a given radius using a compass;
brain teaser:
form the concept of “radius”;
carry out classification, comparison;
independently formulate conclusions;
Communication UUD:
actively participate in team work, using verbal means;
argue your point of view;
Subject Skills:
identify the essential features of the concepts “radius of a circle”;
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 the rain
Be 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 problem can be 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 is solved by subtraction.
3. Staging educational task
1. Name geometric figures
Circle circumference oval ball
Which figure is the odd one out?
What do the figures have in common? (Circle, circle, ball have the same shape)
What is the difference?
2. B
Which points belong to the circle? What points are 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 concluded?
Conclusion: all radii have the same length .
3. How many circles are there in the picture?
How are circles different? (size)
What determines the size of a circle?
What can be concluded?
Conclusion: the larger the circle, the larger its radius.
Determine the topic of the lesson.
Subject: Constructing a circle of a given radius using a compass.
What tasks can we set for ourselves for this lesson?
4. Work on the topic
a) Constructing 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) Look at the picture 
What shapes does a butterfly consist of? Circles with the same radius?
2) Work in pairs.
Restore the order of the stages of the project.
Project presentation or demonstration
Concept (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