Notes of Circles

Introduction to Circles

Circles

  • The set of all the points in a plane that is at a fixed distance from a fixed point makes a circle.
  • A Fixed point from which the set of points are at fixed distance is called centre of the circle.
  • A circle divides the plane into 3 parts: interior (inside the circle), the circle itself and exterior (outside the circle)

A circle is a collection (set) of all those points in a plane, each one of which is at a constant distance from a fixed point in the plane. 

The fixed point is called the centre and the constant distance is called the radius of the circle. 

All the points lying inside a circle are called its interior points and all those points which lie outside the circle are called its exterior points

The collection (set) of all interior points of a circle is called the interior of the circle while the collection of all exterior points of a circle is called the exterior of the circle

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A line can meet a circle at the most in two points. A line segment joining two points on a circle is called the chord of the circle. 

A chord passing through the center of the circle is called a diameter of the circle. A diameter of circle is its longest chord. 

Radius

The distance between the center of the circle and any point on its edge is called the radius.

Tangent and Secant

  • A line that touches the circle at exactly one point is called its tangent
  • A line that cuts a circle at two points is called as a secant.
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In the above figure: PQ is the tangent and AB is the secant.

Chord

The line segment within the circle joining any 2 points on the circle is called the chord.

Diameter

  • A Chord passing through the center of the circle is called the diameter.
  • The Diameter is 2 times the radius and it is the longest chord.

A line which meets a circle in two points is called a secant of the circle. 

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A polygon is a closed figure made up of three or more line segments. 

A polygon is called a regular polygon, if it has all its sides equal and has all its angles equal.

Arc

  1. The portion of a circle(curve) between 2 points is called an arc.
  2. Among the two pieces made by an arc, the longer one is called major arc and the shorter one is called minor arc.

A (continuous) part of a circle is called an arc of the circle. The arc of a circle is denoted by the symbol ‘’. 

An arc less than one-half of the whole arc of a circle is called a minor arc of the circle, and an arc greater than one-half of the whole arc of a circle is called a major arc of the circle. 

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The whole arc of a circle is called the circumference of the circle. 

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One-half of the whole arc of a circle is called a semi-circle of the circle. 

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Circumference

The perimeter of a circle is the distance covered by going around its boundary once. The perimeter of a circle has a special name: Circumference, which is π times the diameter which is given by the formula 2πr

Segment and Sector

  • A circular segment is a region of a circle which is “cut off” from the rest of the circle by a secant or a chord.
  • Smaller region cut off by a chord is called minor segment and the bigger region is called major segment.
  • A sector is the portion of a circle enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector.
  • For 2 equal arcs or for semicircles – both the segment and sector is called the semicircular region.

Any angle whose vertex is centre of the circle is called a central angle

The degree measure of a minor arc is the measure of the central angle subtended by the arc. 

The degree measure of a circle is 360°. The degree measure of a semi-circle is 180° 

The degree measure of a major arc is (360° – θ°), where θ° is the degree measure of the corresponding minor arc. 

Two circles are said to be congruent if and only if either of them can be superposed on the other so as to cover it exactly. 

Two arcs of a circle (or of congruent circles) are congruent if either of them can be superposed on the other so as to cover it exactly. 

The part of the plane region enclosed by an arc of a circle and its two bounding radii is called a sector of a circle. 

If the central angle of a sector is more than180° , then the sector is called a major sector and if the central angle is less than180° , then the sector is called a minor sector

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A chord of a circle divides it into two parts. Each part is called a segment

The part containing the minor arc is called the minor segment, and the part containing the major arc is called the major segment

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Circles and Their Chords

Theorem of equal chords subtending angles at the center.

Equal chords subtend equal angles at the center.

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Proof : AB and CD are the 2 equal chords. 

In ΔAOB and ΔCOD

OB=OC [Radii]

OA=OD [Radii]

AB=CD [Given] 

ΔAOB≅ΔCOD (SSS rule)

Hence, ∠AOB=∠COD [CPCT]

Theorem of equal angles subtended by different chords.

If the angles subtended by the chords of a circle at the center are equal, then the chords are equal.

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Proof : In ΔAOB and ΔCOD

OB=OC [Radii]

∠AOB=∠COD [Given] 

OA=OD [Radii]

ΔAOB≅ΔCOD(SAS rule)

Hence, AB=CD [CPCT]

Perpendicular from the center to a chord bisects the chord.

Perpendicular from the center of a circle to a chord bisects the chord.

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Proof: AB is a chord and OM is the perpendicular drawn from the center.

From Δ OMB and ΔOMA,

∠OMA=∠OMB=900

OA=OB (radii)

OM=OM (common)

Hence,ΔOMB≅ΔOMA (RHS rule)

Therefore AM=MB [CPCT]

A Line through the center that bisects the chord is perpendicular to the chord.

A line drawn through the center of a circle to bisect a chord, is perpendicular to the chord.

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Proof: OM drawn from center to bisect chord AB .

From Δ OMA and ΔOMB,

OA=OB (Radii)

OM=OM (common)

AM=BM (Given)

Therefore, ΔOMA≅ΔOMB (SSS rule)

⇒∠OMA=∠OMB (C.P.C.T)

But, ∠OMA+∠OMB=1800

Hence, ∠OMA=∠OMB=900

⇒OM⊥AB

Circle through 3 points

  • There is one and only one circle passing through three given noncollinear points.
  • A unique circle passes through 3 vertices of a triangle ABC called as the circumcircle. The centre and radius are called the circumcenter and circumradius of this triangle, respectively.

Equal chords are at equal distances from the center.

Equal chords of a circle(or of congruent circles) are equidistant from the centre (or centres).

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Proof : Given, AB=CD, O is the centre. Join OA and OC.

Draw, OP⊥AB,OQ⊥CD

In ΔOAP andΔOCQ,

OA=OC (Radii)

AP=CQ ( AB = CD ⇒12AB=12CD, since OP and OQ bisects the chords AB and CD.)

ΔOAP≅ΔOCQ (RHS rule)

Hence, OP=OQ (C.P.C.T.C)

Chords equidistant from center are equal

Chords equidistant from the center of a circle are equal in length.

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Proof : Given OX = OY (The chords AB and CD are at equidistant)

OX⊥AB,OY⊥CD

In ΔAOX and ΔDOY

∠OXA=∠OYD  (Both 900)

OA=OD (Radii)

OX=OY (Given)

ΔAOX≅ΔDOY (RHS rule)

Therefore AX=DY (CPCT)

Similarly XB=YC 

So, AB=CD

Circles and Quadrilaterals

Angle subtended by an arc of a circle on the circle and at the center

The angle subtended by an arc at the centre is double the angle subtended by it on any part of the circle.

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Here PQ is the arc of a circle with centre O, that subtends ∠POQ at the centre. 

Join AO and extend it to B.

In ΔOAQ

OA=OQ….. [Radii]

Hence, ∠OAQ=∠OQA….[Property of isosceles triangle]

Implies ∠BOQ=2∠OAQ …..[Exterior angle of triangle = Sum of 2 interior angles]

Similarly, ∠BOP=2∠OAP

⇒∠BOQ+∠BOP=2∠OAQ+2∠OAP

⇒∠POQ=2∠PAQ  

Hence proved

Angles in same segment of a circle.

Angles in the same segment of a circle are equal.

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Consider a circle with centre O.

∠PAQ and ∠PCQ are the angles formed in the major segment PACQ with respect to the arc PQ.

Join OP and OQ

∠POQ=2∠PAQ=2∠PCQ …..[ Angle subtended by an arc at the centre is double the angle subtended by it in any part of the circle]

⇒∠PCQ=∠PAQ

Hence proved

Angle subtended by diameter on the circle

Angle subtended by diameter on a circle is a right angle.(Angle in a semicircle is a right angle)

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Consider a circle with center O, POQ is the diameter of the circle.

∠PAQ is the angle subtended by diameter PQ at the circuference.

∠POQ is the angle subtended by diameter PQ at the center.

∠PAQ=12∠POQ… [Angle subtended by arc at the centre is double the angle at any other part] 

∠PAQ=12×1800=900

Hence proved

Line segment that subtends equal angles at two other points

If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle.(i.e they are concyclic)

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Here ACB=ADB and all 4 points A,B,C,D are concyclic.

Cyclic Quadrilateral

A Quadrilateral is called a cyclic quadrilateral if all the four vertices lie on a circle.

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In a circle, if all four points A, B, C and D lie on the circle, then quadrilateral ABCD is a cyclic quadrilateral.

Sum of opposite angles of a cyclic quadrilateral

If sum of a pair of opposite angles of a quadrilateral is 180 degree, the quadrilateral is cyclic.

Sum of pair of opposite angles in quadrilateral

The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degree

Equal chords of a circle subtend equal angles at the centre. 

If the angles subtend by the chords of a circle at the centre are equal, then the chords are equal. 

In a circle, perpendicular from the center to a chord bisects the chord. 

The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. 

An infinite number of circles can be drawn through a given point, say P. 

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An infinite number of circles can be drawn through two given points, say A and B. 

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One and only one circle can be drawn through three non-collinear points. 

Perpendicular bisectors of two chords of a circle, intersect each other at the centre of the circle. 

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The length of the perpendicular from a point to a line is the distance of the line from the point. 

The chords corresponding to congruent arcs are equal. 

Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres). 

Chords equidistant from the centre of a circle are equal in length. 

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. 

If a line segment joining two points subtends equal angles at two other points lying on the same side of the line segment, the four points are concyclic, i.e., lie on the same circle. 

Angles in the same segment of a circle are equal. 

An angle in a semi-circle is a right angle. 

The arc of a circle subtending a right angle at any point of the circle in its alternate segment is a semicircle. 

A quadrilateral, all the four vertices of which lie on a circle is called a cyclic quadrilateral. The four vertices A, B, C and D are said to be concyclic points. 

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The opposite angles of a cyclic quadrilateral are supplementary. 

If the sum of any pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic. 4  Any exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.