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.
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.
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.
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