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

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

Proof:

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

Proof:

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

Proof:

and are joined by construction

It՚s given that in and

(Perpendicular to a chord bisects it)

(Perpendicular to a chord bisects it)

(Given)

From Equation (i) and (ii) ,

(Radii of the same circle)

since and

By R. H. S Axiom of Congruency

Hence, by