In general, to imagine what a circle is, take a look at a ring or hoop. You can also take a round glass and cup, put upside down on a sheet of paper and circle with a pencil. With a multiple increase, the resulting line will become thick and not quite even, and its edges will be blurred. A circle as a geometric figure does not have such a characteristic as thickness.
Circumference: 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 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 a circle, then the resulting segment will be called a chord. The chord passing through the center of the circle is the diameter indicated by the letter D. The diameter divides the circle into two equal arcs and is twice as long as the radius. Thus, D = 2R, or R = D / 2.
Chord properties
- If you draw a chord through any two points of the circle, and then a radius or diameter perpendicular to the last, 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 those enclosed between them, will be equal.
- We draw two chords PR and QS intersecting within the 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 variables such as radius, diameter and constant Ο, which reflects the constancy of the ratio of the circumference of a circle to its diameter.
Thus, L = ΟD, or L = 2ΟR, where L is the circumference, D is the diameter, R is the radius.
The formula for the circumference can be considered as the initial one when finding the radius or diameter along the given circumference: D = L / Ο, R = L / 2Ο.
What is a circle: basic tenets
1. A straight line and a circle can be located on a plane as follows:
- do not have 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, 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 be in contact only at one point, which is located on a segment connecting the centers of these circles.
4. At any turns relative to the center, the circle goes into itself.
5. What is a circle in terms of symmetry?
- the same curvature of the line at any of the points;
- central symmetry with respect to point O;
- mirror symmetry with respect 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 the circumference, that is, cut off by a chord-diameter, is always 90 Β°.
7. If we compare closed curved lines of the same length, it turns out that the circle delimits a portion of the plane of the largest area.
A circle inscribed in a triangle and described around it
The idea of ββwhat a circle is will be incomplete without a description of 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 intersection point of the bisectors of the angles of the triangle.
- The center of the circle described near the triangle is located at the intersection of the median perpendiculars to each side of the triangle.
- If we describe a circle near 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 circled circles will be at the same point, if the base for the construction is an equilateral triangle.
Basic statements about circles and quadrangles
- A circle around a convex quadrangle can be described only when the sum of its opposite internal angles is 180 Β°.
- It is possible to construct a circle inscribed in a convex quadrangle if the sum of the lengths of its opposite sides is the same.
- A circle around a parallelogram can be described if its angles are straight.
- You can enter a circle in a parallelogram if all its sides are equal, that is, it is a rhombus.
- You can construct a circle through the corners of the trapezoid only if it is isosceles. In this case, the center of the circumscribed circle will be located at the intersection of the axis of symmetry of the quadrangle and the median perpendicular drawn to the side.