To get a general idea of what a circle is, look at a ring or hoop. You can also take a round glass and a cup, put it upside down on a piece of paper and circle it with a pencil. With multiple magnification, the resulting line will become thick and not quite even, and its edges will be blurry. The circle as a geometric figure does not have such a characteristic as thickness.
Circumference: definition and main means of description
A circle is a closed curve consisting of a set of 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 indicated by the letter O.
The distance from any of the points of the circle to the center is called the radius and is denoted by the letter R.
If you connect any two points of the 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. So D=2R, or R=D/2.
Properties of chords
- If you draw a chord through any two points of the circle, and then draw a radius or diameter 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 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 enclosed between them, will be equal.
- Let's draw two chords PR and QS intersecting within a circle at point T. The product of the segments of one chord will always be equal to the product of the segments of the other 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 values such as radius, diameter, and the constant "π", reflecting the constancy of the ratio of the circumference of a circle 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 the circumference of a circle can be considered as the initial formula for finding the radius or diameter for a given circumference: D=L/π, R=L/2π.
What is a circle: basic postulates
1. A straight line and a circle can be located on a plane as follows:
- do not have common points;
- have one common point, while the line is called a tangent: if you draw a radius through the center and the pointtouch, it will be perpendicular to the tangent;
- have two common points, while the line is called a secant.
2. Through three arbitrary points lying in the same plane, at most one circle can be drawn.
3. Two circles can only touch at one point, which is located on the segment connecting the centers of these circles.
4. With any rotation about the center, the circle turns into itself.
5. What is a circle in terms of symmetry?
- same line curvature at any point;
- central symmetry about point O;
- mirror symmetry about the diameter.
6. If you construct two arbitrary inscribed angles based on the same circular arc, they will be equal. The angle based on an arc equal to half the circumference of the circle, that is, cut off by a chord-diameter, is always 90 °.
7. If we compare closed curved lines of the same length, then it turns out that the circle delimits the section of the plane of the largest area.
Circle inscribed in a triangle and described around it
An idea of what a circle is would be incomplete without describing the features of the relationship of this geometric figure with triangles.
- When constructing a circle inscribed in a triangle, its center will always coincide with the point of intersection of the bisectors of the angles of the triangle.
- The center of the circumscribed triangle is located at the intersectionmid-perpendiculars to each side of the triangle.
- If you describe a circle around a right triangle, 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 base for construction is an equilateral triangle.
Basic statements about the circle and quadrilaterals
- A circle can be circumscribed around a convex quadrilateral only if the sum of its opposite interior angles is 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.
- It is possible to describe a circle around a parallelogram if its angles are right.
- You can inscribe a circle into a parallelogram if all its sides are equal, that is, it is a rhombus.
- It is possible to construct a circle through the angles 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 symmetry axis of the quadrilateral and the median perpendicular drawn to the side.
