Speech Please on CIRCLE?

Question

Explanation on chord, arc, circumcircle and all related matters. Please its urgent

Answer 1

THE CIRCLE

The circle :
The circle is the set of points of a plane that are at
a constant distance from a fixed point in the same plane
The fixed point “M” is called the center of the circle
Points A , B, C are on circle M they belong to its circumferrence

Radius of a circle :
It is the line segment with an end point at the center
And the other end point on the circle .
All radii of a circle have the same length

Chord of a circle :
A chord is the line segment whose endpoints are on
The circle

Diameter of a circle :
It is a chord passing through the center of
The circle
•the diameter is the longest chord
•length of diameter = twice radius D = 2r

Symmetry in the circle ;:
Any straight line passing through the center
Of a circle is an axis of symmetry for this
Circle ( the straight line carrying the diameter )
IMPORTANT REMARK :
There is an infinite number of axis of symmetry for a circle
the circumcircle of a polygon is the circle which passes through all vertices of that polygon the polygon is said to be inscribed in that circle

Answer 2

circle-locus of points on a plain such that its distances is equal to a single point
chord-a line segment containing two points that is included in the circle
arc-a part of the circumference of the circle(curved line segment)
radii-distance from a point on the circle to the center of the circle
diameter-also known as the longest chord, it is th distance from a point on th circle passing through the center to another point on the same circle
circumcircle- circle created by connecting the corners of polygons
Ax^2 + By^2 + Cxy + Dx + Ey + F = 0 - equation of circle where A and B are equal, C=0

Answer 3

CHORD (Geometry)- line through arc: a straight line connecting two points on an arc or circle

ARC (geometry)- section of circle: a section of a circle, ellipse, or other curved figure

CIRCUMFERENCE (geometry)- distance around circle: the distance around the edge of a circle

RADIUS (mathematics geometry)- line from centre: a straight line extending from the centre of a circle to its edge or from the centre of a sphere to its surface. Symbol r

-(mathematics geometry) length of radius: the length of a radius. Symbol r

SECTOR (geometry) part of circle: a part of a circle bounded by two radii and the part of the circumference that lies between them

DIAMETER- line through centre of circle: a straight line running from one side of a circle or other rounded geometric figure through the centre to the other side, or the length of this line

Answer 4

Circle
From Wikipedia, the free encyclopedia
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Circle illustrationThis article is about the shape and mathematical concept of circle. For other uses, see Circle (disambiguation).
In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference(C). Usually, however, the circumference means the length of the circle, and the interior of the circle is called a disk. An arc is any continuous portion of a circle.

A circle is a special ellipse in which the two foci coincide (i.e., are the same point). Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

Contents [hide]
1 Analytic results
2 Properties
2.1 Chord properties
2.2 Tangent properties
2.3 Theorems
3 Inscribed angles
4 An alternative definition of a circle
5 Numbers and the circle
6 See also
7 External links



[edit] Analytic results

Area of the circle = π × area of the shaded square
Approximating the area of a circle by regular polygons
Area of a circle using infinitesimal area elementIn an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that

.
If the circle is centred at the origin (0, 0), then this formula can be simplified to

x2 + y2 = r2.
The circle centred at the origin with radius 1 is called the unit circle.

Expressed in parametric equations, (x, y) can be written using the trigonometric functions sine and cosine as

x = a + r cos(t)
y = b + r sin(t),
where t is a parametric variable, understood as the angle the ray to (x, y) makes with the x-axis.

In homogeneous coordinates each conic section with equation

ax2 + ay2 + 2b1xz + 2b2yz + cz2 = 0
is called a circle.

It can be proven that a conic section is a circle if and only if the point I(1,i,0) and J(1,-i,0) lie on the conic section. These points are called the circular points at infinity.

In polar coordinates the equation of a circle is:


The slope of a circle at a point (x, y) can be expressed with the following formula, assuming the centre is at the origin and (x, y) is on the circle:

.

In the complex plane, a circle with a centre at c and radius r has the equation | z − c | 2 = r2. Since , the slightly generalized equation for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles.

All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. The constants of proportionality are 2π and π, respectively.

In other words:

Length of a circle's circumference is:

The area enclosed by a circle is:

Diameter of a circle is:


Early 'science,' particularly geometry and astronomy/astrology, was connected to the divine for most medieval scholars. The compass in this 13th Century manuscript is a symbol of God's act of Creation, as many believed that there was something intrinsically "divine" or "perfect" that could be found in circlesThe formula for the area of a circle [1] can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the centre of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the centre) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the circumference and radius and dividing by 2, we get the area, π r 2.

The formula for the area of circle can also be derived by using an infinitesimal area element dA and integrating it over the whole circle.

In Chinese mathematics the area of a circle is usually expressed as:

A=c/2•d/2
where A is the area, c is the circumference, and d is the diameter. c=2πr, and d=2r, so c/2•d/2=πr²=A, the familiar identity for area.


[edit] Properties
The circle is the shape with the highest area for a given length of perimeter.
The circle is a highly symmetric shape, every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.

[edit] Chord properties
Chords equidistant from the centre of a circle are equal (length).
Equal (length) chords are equidistant from the centre.
The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:
A perpendicular line from the centre of a circle bisects the chord.
The line segment (Circular segment) through the centre bisecting a chord is perpendicular to the chord.
If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.
If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental.
For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
An inscribed angle subtended by a semicircle is a right angle.

[edit] Tangent properties
The line drawn perpendicular to the end point of a radius is a tangent to the circle.
A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
Tangents drawn from a point outside the circle are equal in length.
Two tangents can always be drawn from a point outside of the circle.

[edit] Theorems
See also: Power of a point
The chord theorem states that if two chords, CD and EF, intersect at G, then . (Chord theorem)
If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then . (tangent-secant theorem)
If two secants, DG and DE, also cut the circle at H and F respectively, then . (Corollary of the tangent-secant theorem)
The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent chord property)
If the angle subtended by the chord at the centre is 90 degrees then l = √(2) × r, where l is the length of the chord and r is the radius of the circle.

Secant-secant theoremIf two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.

[edit] Inscribed angles

Inscribed angle theoremAn inscribed angle ψ is exactly half of the corresponding central angle θ (see Figure). Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles ψ in the Figure). Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle.


[edit] An alternative definition of a circle

Apollonius' definition of a circleApollonius of Perga showed that a circle may also be defined as the set of points having a constant ratio of distances to two foci, A and B.

The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar


Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to , the angle CPD is exactly , i.e., a right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.
As a point of clarification, note that C and D are determined by A, B, and the desired ratio; i.e. A and B are not arbitrary points lying on an extension of the diameter of an existing circle.


[edit] Numbers and the circle
The division of the circle into 360 degrees dates back to ancient India, as found in the Rig Veda:

Twelve spokes, one wheel, navels three.
Who can comprehend this?
On it are placed together
three hundred and sixty like pegs.
They shake not in the least.
(Dirghatama, Rig Veda 1.164.48)
This division is used in mathematics, but also in geography and in astronomy to measure the celestial sphere and equator (both in terms of latitude and longitude).

The enneagram expresses the circle as equal to 9 integers. The structure of the enneagram is based partly on the primary triangle of the circle at 0/360 degrees, 120 degrees and 240 degrees of the circle. In terms of integers, these points of the circle correspond to the numbers 0/9, 3 and 6. The rest of the Enneagram structure consists of connections between the other 6 integers of the 9-based number system - determined by the fraction 1/7 = 0.142857 (repeating).

Answer 5

circle main points
first find the radius
then the centre
then can easily find the circle
using (x-a)^2+(y-b)^2=r^2
where (a,b)isthe centre
r is the radius