to outline to imagine that such a circle, look at the ring or hoop.You can also take a round glass bowl and put upside down on a piece of paper and a pencil to circle.Repeated increase resulting line will be thick and not very smooth, and its edges will be blurred.The circle as a geometric figure has such characteristics as thickness.
Circumference: definition and basic tools for describing
Circle - a closed curve consisting of a plurality of pixels arranged in the same plane and equidistant from the center of the circle.The center is on the same plane.As a rule, it is denoted by the letter O.
distance from any point of the circumference to the center is called the radius and denoted by the letter R.
If you connect any two points of the circle, then the resulting segment is called a chord.Chord passing through the center of the circle - is the diameter, denoted by D. The diameter divides the circle into two equal arc length and twice the size of the radius.Thus, D = 2R, or R = D / 2.
Properties chords
- If any two points of the circle to hold a chord, and then perpendicular to the latter - the radius or diameter, this segment will break and the chord and arc severed it into two equal parts.Converse is also true: if the radius (diameter) of the chord divides in half, it is perpendicular to it.
- If within the same circle to hold two parallel chords, the arc cut off them, as well as agreements between them are equal.
- Draw two chords PR and QS, intersecting within the circle at point T. The product segments of one chord will always be equal to the product segments of the other chord, ie PT's 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 these values as radius, diameter, and the constant "π", which reflects the constancy of the ratio of the circumference to its diameter.
Thus, L = πD, or L = 2πR, where L - is the circumference, D - diameter, R - radius.
Formula circumferential length can be considered as a starting point for finding the radius or the diameter for a given circumference: D = L / π, R = L / 2π.
What is the circle: basic postulates
1. lines and circles can be located on the plane as follows:
- not have points in common;
- have one point in common with the line is called the tangent: if we draw through the center and radius of the point of contact, it will be perpendicular to the tangent;
- have two points in common, and the line is called cutting.
2. After three arbitrary points lying in one plane can be made not more than one circle.
3. Two circles may touch only one point, which is located on the segment connecting the centers of the circles.
4. In all corners to the center circle into itself.
5. What is the circle with the point of view of symmetry?
- same curvature of the line at any point;
- central symmetry with respect to the point O;
- mirror symmetry with respect to the diameter.
6. If you build any two inscribed angles, based on the same arc of a circle, they will be equal.Angle subtended by an arc equal to half of the circumference, that is cut off by a chord, the diameter is always equal to 90 °.
7. If you compare the closed curved lines of the same length, it turns out that the circle separates the greatest area of land the plane.
circle inscribed in the triangle, and described by him
notion that this circle would be complete without a description of features of the relationship of the geometric shape with triangles.
- When building a circle inscribed in the triangle, its center will always coincide with the point of intersection of the bisectors of the angles of a triangle.
- center of the circle described about the triangle, is located at the intersection of the median perpendicular to each side of the triangle.
- If you describe a circle about a right triangle, then its center will be located in the middle of the hypotenuse, that is, the latter will be in diameter.
- centers inscribed and circumscribed circles will be at the same point, if the basis for the construction of a equilateral triangle.
main allegations of the circle and quadrangles
- convex quadrilateral around a circle can be described only when the sum of the opposite interior angles equals 180 °.
- Build inscribed in the convex quadrilateral circle is possible if the same sum of the lengths of the opposite sides.
- describe a circle around the parallelogram is possible, if the corners are straight.
- Fit to parallelogram circle can be in if all of its sides are equal, that is, it is a diamond.
- Construct a circle through the corners of the trapezoid is possible only if it is isosceles.The center of the circumscribed circle will be located at the intersection of the axis of symmetry of the quadrilateral and the median perpendicular drawn to the side.
