Circle Theorem: Chord of a Circle, Theorem 1, Theorem 2, Theorem 3, Theorem 4, Theorem 5 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Circle Theorem

  • A circle can be defined as the locus of all points in a plane being at equidistant from a fixed point.
  • The fixed point is the centre and whereas radius is the constant distance between any point on the circle and its centre.
  • Circumference is the perimeter of a circle.
  • Tangent of a circle is perpendicular to the radius at any point on it.
  • Circle theorem includes:
    • Concept of tangents
    • Sectors
    • Angles
    • The chord of a circle and proofs

Chord of a Circle

  • The chord of a circle is the line segment joining any two points along the circumference of the circle.
  • The largest chord passing through centre of the circle is its diameter.

Circle Theorems and Proofs

(1) In a circle two equal chords subtend equal angles at the centre of the circle.

The Circle Two Equal Chords Subtend Equal Angles


In the above figure it՚s given in and

From equation (i) and (ii) , we get by (SSS rule) of congruency

we get

(2) The perpendicular to a chord bisects the chord if drawn from the centre of the circle.

The Perpendicular to a Chord Bisects


In the above figure , therefore

Also, it is given that in and .

From Equation (i) , (ii) and (iii) we get,

by R. H. S Axiom of congruency

Hence, by

(3) Equal chords of a circle are equidistant (equal distance) from the centre of the circle.

Equal Chords of a Circle Are Equidistant


and are joined by construction

It՚s given that in and

(Perpendicular to a chord bisects it)

(Perpendicular to a chord bisects it)


From Equation (i) and (ii) ,

(Radii of the same circle)

since and

By R. H. S Axiom of Congruency

Hence, by