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Circles and their Properties

A circle is the path travelled by a point which moves in such a way that its distance from a fixed point remains constant. The fixed point is known as the centre and the fixed distance is called the radius.
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Before we move ahead, let us understand the basics defi­nitions of circle.

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Definition: The fixed point is called the centre. In the given diagram ‘O’ is the centre of the circle.

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Definition: The fixed distance is called a radius. In the given diagram OP is the radius of the circle. (point P lies on the circumference)

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Definition: The circumference of a circle is the distance around a circle, which is equal to 2πr. (r → radius of the circle)

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Definition: A line segment which intersects the circle in two distinct points, is called as secant. In the given diagram secant PQ intersects circle at two points at A and B.

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(R is the point of contact) Note: Radius is always perpendicular to tangent.

Definition: A line segment which has one common point with the circumference of a circle, i.e., it touches only at only one point is called as tangent of circle. The common point is called as point of contact. In the given diagram, PQ is a tangent which touches the circle at a point R.

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Definition: A line segment whose end points lie on the circle. In the given diagram AB is a chord.

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Definition: A chord which passes through the centre of the circle is called the diameter of the circle. The length of the diameter is twice the length of the radius. In the given diagram PQ is the diameter of the circle. (Ois the centre of the circle)

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Definition: Any two points on the circle divides the circle into two parts the smaller part is called as minor arc and the larger part is called as major arc. It is denoted as ‘’. In the given diagram PQ is arc.

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Defnation: A diameter of the circle divides the circle into two equal parts. Each part is called a semicircle.

Central Angle
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Definition: An angle formed at the centre of the circle, is called the central angle. In the given diagram AOB in the central angle.

Inscribed Angle
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Definition: When two chords have one common end point, then the angle included between these two chords at the common point is called the inscribed angle. ABC is the inscribed angle by the arc ADC

Measure of an Arc
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m(arc PRQ = m POQ
m(arc PSQ) = 360° − m (arc PRQ)

Definition: Basically, it is the central angle formed by an arc. e.g.,
  1. measure of a circle = 360°
  2. measure of a semicircle = 180°
  3. measure of a minor arc = POQ
  4. measure of a major arc = 360 − POQ

Intercepted Arc
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Definition: In the given diagram, AB and CD are the two intercepted arcs, intercepted by CPD. The end points of the arc must touch the arms of CPD, i.e., CP and DP.

Concentric Circles
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Definition: Circles having the same centre at a plane are called the concentric circles. 
In the given diagram, there are two circles with radii r1and r2 having the common (or same) centre. These are called as concentric circl Ces.


Congruent Circles
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Definition: Circles with equal radii are called as congruent circles.

Segment of a Circle
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Definition: A chord divides a circle into two regions. These two regions are called the segments of a circle:
  1. major segment
  2. minor segment.

Cyclic Quadrilateral
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Definition: A quadrilateral whose all the four vertices lie on the circle.

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Definition: A circle which passes through all the three vertices of a triangle. Thus the circumcentre is always equidistant from the vertices of the triangle.
OA = OB = OC (circumradius)

In Circle
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Definition: A circle which touches all the three sides of a triangle i.e., all the three sides of a triangle are tangents to the circle is called an incircle. Incircle is always equidistant from the sides of a triangle.

Now come to different formula and theorems attached to circle:
Circumference of a circle = 2πr
Area of a circle = πr2, where r is the radius.
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Area of a sector Description: 5891.png
Circumference of a sector Description: 5884.png
Perimeter of a sector Description: 5877.png
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Area of a segment = Area of a sector OADB – Area of triangle OAB
Area of a segment Description: 5862.png

Common Tangents and Secants of Circles

Depending upon the positioning of the circles, two or more than two circles can have a common tangent. Following is a list indicating the number of common tangents in case of two circles:
Sl. No. Position of two circles Number of common tangents
1 One circle lies entirely insidethe other circle Zero
2 Two circles touch internally One
3 Two circles intersect in two distinct points Two
4 Two circles touch externally Three
5 One circle lies entirely outside the other circle Four

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